From c004903acd3f03752239f5eac008445f2d37c5ae Mon Sep 17 00:00:00 2001 From: Jerome Lelong Date: Sat, 4 Nov 2023 15:55:07 +0100 Subject: [PATCH] Update docs and defs --- docs/manual-html/pnl-manual.css | 69 +- docs/manual-html/pnl-manual.html | 15785 +------------------------ docs/manual-html/pnl-manual18x.png | Bin 1452 -> 1445 bytes docs/manual-html/pnl-manual19x.png | Bin 477 -> 472 bytes docs/manual-html/pnl-manual21x.png | Bin 1653 -> 1661 bytes docs/manual-html/pnl-manual24x.png | Bin 3467 -> 3451 bytes docs/manual-html/pnl-manual25x.png | Bin 2096 -> 647 bytes docs/manual-html/pnl-manual26x.png | Bin 667 -> 2112 bytes docs/manual-html/pnl-manual27x.png | Bin 2129 -> 2170 bytes docs/manual-html/pnl-manual28x.png | Bin 1867 -> 1865 bytes docs/manual-html/pnl-manual29x.png | Bin 2643 -> 2623 bytes docs/manual-html/pnl-manual2x.png | Bin 1013 -> 1017 bytes docs/manual-html/pnl-manual30x.png | Bin 1720 -> 1726 bytes docs/manual-html/pnl-manual31x.png | Bin 1753 -> 1752 bytes docs/manual-html/pnl-manual32x.png | Bin 281 -> 279 bytes docs/manual-html/pnl-manual34x.png | Bin 3046 -> 3034 bytes docs/manual-html/pnl-manual35x.png | Bin 3791 -> 3791 bytes docs/manual-html/pnl-manual36x.png | Bin 2355 -> 2357 bytes docs/manual-html/pnl-manual38x.png | Bin 1780 -> 1776 bytes docs/manual-html/pnl-manual40x.png | Bin 4607 -> 4604 bytes docs/manual-html/pnl-manual43x.png | Bin 280 -> 280 bytes docs/manual-html/pnl-manual44x.png | Bin 356 -> 353 bytes docs/manual-html/pnl-manual45x.png | Bin 353 -> 356 bytes docs/manual-html/pnl-manual46x.png | Bin 1159 -> 1153 bytes docs/manual-html/pnl-manual47x.png | Bin 397 -> 395 bytes docs/manual-html/pnl-manual48x.png | Bin 519 -> 519 bytes docs/manual-html/pnl-manual50x.png | Bin 1756 -> 1756 bytes docs/manual-html/pnl-manual55x.png | Bin 759 -> 759 bytes docs/manual-html/pnl-manual56x.png | Bin 460 -> 460 bytes docs/manual-html/pnl-manual60x.png | Bin 2375 -> 2400 bytes docs/manual-html/pnl-manual61x.png | Bin 1669 -> 1667 bytes docs/manual-html/pnl-manual62x.png | Bin 1701 -> 1707 bytes docs/manual-html/pnl-manual63x.png | Bin 1558 -> 1561 bytes docs/manual-html/pnl-manual64x.png | Bin 2172 -> 2178 bytes docs/manual-html/pnl-manual65x.png | Bin 2123 -> 2139 bytes docs/manual-html/pnl-manual69x.png | Bin 237 -> 237 bytes docs/manual-html/pnl-manual7x.png | Bin 399 -> 430 bytes docs/manual-html/pnl-manualli1.html | 164 +- docs/manual-html/pnl-manualli2.html | 1501 +-- docs/manual-html/pnl-manualse1.html | 869 +- docs/manual-html/pnl-manualse10.html | 130 +- docs/manual-html/pnl-manualse11.html | 252 +- docs/manual-html/pnl-manualse12.html | 331 +- docs/manual-html/pnl-manualse13.html | 1643 ++- docs/manual-html/pnl-manualse14.html | 715 +- docs/manual-html/pnl-manualse15.html | 144 +- docs/manual-html/pnl-manualse16.html | 77 +- docs/manual-html/pnl-manualse2.html | 1020 +- docs/manual-html/pnl-manualse3.html | 634 +- docs/manual-html/pnl-manualse4.html | 7693 +++++++++--- docs/manual-html/pnl-manualse5.html | 156 +- docs/manual-html/pnl-manualse6.html | 1214 +- docs/manual-html/pnl-manualse7.html | 1720 ++- docs/manual-html/pnl-manualse8.html | 266 +- docs/manual-html/pnl-manualse9.html | 307 +- docs/pnl-manual.pdf | Bin 686458 -> 756069 bytes src/pnl.def | 5 +- 57 files changed, 13915 insertions(+), 20780 deletions(-) diff --git a/docs/manual-html/pnl-manual.css b/docs/manual-html/pnl-manual.css index be6adbef..bcf0a325 100644 --- a/docs/manual-html/pnl-manual.css +++ b/docs/manual-html/pnl-manual.css @@ -10,6 +10,19 @@ .lmsy8-{font-size:72%;} .lmsy6-{font-size:54%;} .lmex-10{font-size:90%;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmtt-10x-x-109{ font-family: monospace,monospace;} +.ec-lmtt-10x-x-109{ font-family: monospace,monospace;} +.ec-lmtt-10x-x-109{ font-family: monospace,monospace;} .ec-lmtt-10x-x-109{ font-family: monospace,monospace;} .ec-lmtt-10x-x-109{ font-family: monospace,monospace;} .ec-lmtt-10x-x-109{ font-family: monospace,monospace;} @@ -22,13 +35,36 @@ .lmmi-10x-x-109{} .lmsy-10x-x-109{} .msbm-10x-x-109{} -.ec-lmri-10x-x-109{ font-style:italic;} -.ec-lmri-10x-x-109{ font-style:italic;} -.ec-lmri-10x-x-109{ font-style:italic;} -.ec-lmri-10x-x-109{ font-style:italic;} -.ec-lmri-10x-x-109{ font-style:italic;} -.ec-lmri-10x-x-109{ font-style:italic;} -.ec-lmri-10x-x-109{ font-style:italic;} +.ec-lmtt-10{font-size:90%; font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtt-10{ font-family: monospace,monospace;} +.ec-lmtk-10{font-size:90%; font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtk-10{ font-family: monospace,monospace;} +.ec-lmtti-10{font-size:90%; font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} +.ec-lmtti-10{ font-family: monospace,monospace; font-style:italic;} .ec-lmbx-10x-x-109{ font-weight: bold;} .ec-lmbx-10x-x-109{ font-weight: bold;} .ec-lmbx-10x-x-109{ font-weight: bold;} @@ -36,6 +72,12 @@ .ec-lmbx-10x-x-109{ font-weight: bold;} .ec-lmbx-10x-x-109{ font-weight: bold;} .ec-lmbx-10x-x-109{ font-weight: bold;} +.ec-lmbx-10x-x-109{ font-weight: bold;} +.ec-lmbx-10x-x-109{ font-weight: bold;} +.ec-lmbx-10x-x-109{ font-weight: bold;} +.ec-lmbxi-10x-x-109{ font-style:italic; 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white-space: nowrap; } table.boxedverbatim {width:100%;} pre.boxedverbatim{border:solid 1px; padding-right:1em;} @@ -184,6 +227,12 @@ table.multline, table.multline-star {width:100%;} td.gather {text-align:center; } table.gather {width:100%;} div.gather-star {text-align:center;} +code.lstinline{font-family:monospace,monospace;} +pre.listings{font-family: monospace,monospace; white-space: pre-wrap; margin-top:0.5em; margin-bottom:0.5em; } +.lstlisting .label{margin-right:0.5em; } +pre.lstlisting{font-family: monospace,monospace; white-space: pre-wrap; margin-top:0.5em; margin-bottom:0.5em; } +pre.lstinputlisting{ font-family: monospace,monospace; white-space: pre-wrap; } +.lstinputlisting .label{margin-right:0.5em;} #TBL-2-1{border-left: 1px solid black;} #TBL-2-1{border-right:1px solid black;} #TBL-2-2{border-right:1px solid black;} diff --git a/docs/manual-html/pnl-manual.html b/docs/manual-html/pnl-manual.html index c6e80649..eeaa5eea 100644 --- a/docs/manual-html/pnl-manual.html +++ b/docs/manual-html/pnl-manual.html @@ -24,15686 +24,151 @@

Pnl Manual


February 2, 2023
+class="ec-lmr-12">November 4, 2023 -

Contents

Contents -
 1 Introduction -
 2 Objects -
 3 Mathematical framework -
 4 Linear Algebra -
 5 Cumulative distribution Functions -
 6 Random Number Generators -
 7 Function bases and regression -
 8 Numerical integration -
 9 Fast Fourier Transform -
 10 Inverse Laplace Transform -
 11 Ordinary differential equations -
 12 Optimization -
 13 Root finding -
 14 Special functions -
 15 Some bindings -
 16 Financial functions -
 Index -
 Index +Contents +
1 Introduction +
1.1 What is Pnl +
1.2 A few helpful conventions +
1.3 Using Pnl +
2 Objects +
2.1 The top-level object +
2.2 List object +
2.3 Array object +
3 Mathematical framework +
3.1 General tools +
3.2 Complex numbers +
4 Linear Algebra +
4.1 Vectors +
4.2 Compact Vectors +
4.3 Matrices +
4.4 Tridiagonal Matrices +
4.5 Band Matrices +
4.6 Sparse Matrices +
4.7 Hyper Matrices +
4.8 Iterative Solvers +
5 Cumulative distribution Functions +
6 Random Number Generators +
6.1 The rng interface +
6.2 The rand interface (deprecated) +
7 Function bases and regression +
7.1 Overview +
7.2 Functions +
8 Numerical integration +
8.1 Overview + + + +
8.2 Functions +
9 Fast Fourier Transform +
9.1 Overview +
9.2 Functions +
10 Inverse Laplace Transform +
11 Ordinary differential equations +
11.1 Overview +
11.2 Functions +
12 Optimization +
12.1 Linear constrained optimization (linear programming) +
12.2 Nonlinear constrained optimization +
13 Root finding +
13.1 Overview +
13.2 Functions +
14 Special functions +
14.1 Real Bessel functions +
14.2 Complex Bessel functions +
14.3 Error functions +
14.4 Gamma functions +
14.5 Digamma function +
14.6 Incomplete Gamma functions +
14.7 Exponential integrals +
14.8 Hypergeometric functions +
15 Some bindings +
15.1 MPI bindings +
15.2 The save/load interface +
16 Financial functions +
Index +
Index
-

-

-

1 Introduction

- - - -

-

-

1.1 What is Pnl

-

Pnl is a scientific library written in C and distributed under the Gnu Lesser General Public Licence -(LGPL). This manual is divided into four parts.

- -

-

-

1.2 A few helpful conventions

- -

-

-

1.3 Using Pnl

-

In this section, we assume that the library is installed in the directory $HOME/pnl-xxx. -

Once installed, the library can be found in the $HOME/pnl-xxx/lib directory and the header files in -the $HOME/pnl-xxx/include directory. -

-

-
1.3.1 Compiling and Linking
-

The header files of the library are installed in a root pnl directory and should always be included -with this pnl/ prefix. So, for instance to use random number generators you should -include - - - -

-
-#include <pnl/pnl_random.h>
-
-

-

-

Compiling and linking by hand. -If gcc or llvm is used, you should pass the following options

- -

This does not work straight away on all OS especially if the library is not installed in a standard -directory namely /usr/ or /usr/local/ for which you need a privileged writing access. On some -systems, you may need to add to the linker flags the dependencies of the library, which can become -very tedious. Therefore, we provide a second automatic mechanism which takes care of the -dependencies on its own. -

-

Compiling and linking using an automatic Makefile. -This mechanism only works under Unix (it has been tested under various Linux distributions and -Mac OS X). -

First, you need to create a new directory wherever you want, put in all your code and create a -Makefile as below -

To define your target just add the executable name, say my-exec, to the BINS list and create an -entry my_exec_SRC carrying the list of source files needed to create your executable. Note that if -dashes ’-’ may appear in an executable name, the name of the associated variable holding the list of -source files is obtained by replacing dashes with underscores ’_’ and adding the _SRC -suffix. -

Assume you want to create two binaries : my-exec based on mixed C and C++ code (file1.c and -file2.cpp) and mybinary based on poo1.cxx and poo2.cpp. You can use the following -Makefile. - - - -

-
-## Flags passed to the linker
-LDFLAGS=
-
-## Flags passed to the compiler
-CFLAGS=
-
-## list of executables to create
-BINS=my-exec mybinary
-
-my_exec_SRC=file1.c file2.cpp
-# optional flags for compiling and linking
-my_exec_CFLAGS=
-my_exec_CXXFLAGS=
-my_exec_LDFLAGS=
-
-mybinary_SRC=poo1.cxx poo2.cpp
-# optional flags for compiling and linking
-mybinary_CFLAGS=
-mybinary_CXXFLAGS=
-mybinary_LDFLAGS=
-
-
-## This line must be the last one
-include full_path_to_pnl_build/CMakeuser.incl
-
-

Let us comment a little the different variables

- -

An example of such a Makefile can be found in pnl-xxx/perso. -

Warning: if a file appears in the source list of several binairies, the flags used to compile this file -are determined by the ones of the first binary involving this file. In the following example main.cpp -will always be compiled with the flag -O3 even for generating bin2 - - - -

-
-BINS=bin1 bin2
-
-bin1_SRC=main.cpp poo1.c
-my_exec_CXXFLAGS=-O3
-
-bin2_SRC=main.cpp poo2.c
-mybinary_CXXFLAGS=-g -O0
-
-## This line must be the last one
-include full_path_to_pnl_build/CMakeuser.incl
-
-

-

-

Compiling and linking using CMake. -If you already use CMake for your new project, just add the following to your toplevel -CMakeLists.txt - - - -

-
-find_package(Pnl REQUIRED)
-set(LIBS ${LIBS} ${PNL_LIBRARIES})
-include_directories(${PNL_INCLUDE_DIRS})
-# Deactivate PNL debugging stuff on Release builds
-if(${CMAKE_BUILD_TYPE} STREQUAL "Release")
-    add_definitions(-DPNL_RANGE_CHECK_OFF)
-endif()
-
-

Then, call cmake with the following extra flag - - - -

-
--DCMAKE_PREFIX_PATH=path/to/build-dir
-
-

or add the variable CMAKE_BUILD_TYPE to the GUI.
-

Just in case, we give an example of a complete although elementary CMakeLists.txt -

-

-

-
1.3.2 Inline Functions and getters
-

If it is supported by your compiler, getter and setter functions are declared as inline functions. This -is automatically detected when running CMake. By default, setter and getter functions check that -the required access is valid, basically it boils down to checking whether the index of the access is -within an acceptable range. These extra tests can become very expensive when getter and setter -functions are intensively called. -

Thus, it is possible to alter this default behaviour by defining the macro PNL_RANGE_CHECK_OFF. -This macro is automatically defined when the library is compiled in Release mode, ie. with --DCMAKE_BUILD_TYPE=Release passed to CMake. -

-

-

2 Objects

-

-

-

2.1 The top-level object

-

The PnlObject structure is used to simulate some inheritance between the ojbects of Pnl. It must be -the first element of all the objects existing in Pnl so that casting any object to a PnlObject is -legal - - - - -

-
-typedef unsigned int PnlType;
-
-typedef void (DestroyFunc) (void **);
-typedef PnlObject* (CopyFunc) (PnlObject *);
-typedef PnlObject* (NewFunc) (PnlObject *);
-typedef void (CloneFunc) (PnlObject *dest, const PnlObject *src);
-struct _PnlObject
-{
-  PnlType type; /*!< a unique integer id */
-  const char *label; /*!< a string identifier (for the moment not useful) */
-  PnlType parent_type; /*!< the identifier of the parent object is any,
-                          otherwise parent_type=id */
-  int nref; /*!< number of references on the object */
-  DestroyFunc *destroy; /*!< frees an object */
-  NewFunc     *constructor; /*!< New function */
-  CopyFunc    *copy; /*!< Copy function */
-  CloneFunc   *clone; /*!< Clone function */
-};
-
-

-

Here is the list of all the types actually defined

- - - -


- - - -
- - - - - - - - - - - - - - - - - - - - - - - -


PnlType Description


PNL_TYPE_VECTOR general vectors
PNL_TYPE_VECTOR_DOUBLE real vectors
PNL_TYPE_VECTOR_INT integer vectors
PNL_TYPE_VECTOR_COMPLEX complex vectors
PNL_TYPE_MATRIX general matrices
PNL_TYPE_MATRIX_DOUBLE real matrices
PNL_TYPE_MATRIX_INT integer matrices
PNL_TYPE_MATRIX_COMPLEX complex matrices
PNL_TYPE_TRIDIAG_MATRIX general tridiagonal matrices
PNL_TYPE_TRIDIAG_MATRIX_DOUBLEreal tridiagonal matrices
PNL_TYPE_BAND_MATRIX general band matrices
PNL_TYPE_BAND_MATRIX_DOUBLE real band matrices
PNL_TYPE_SP_MATRIX sparse general matrices
PNL_TYPE_SP_MATRIX_DOUBLE sparse real matrices
PNL_TYPE_SP_MATRIX_INT sparse integer matrices
PNL_TYPE_SP_MATRIX_COMPLEX sparse complex matrices
PNL_TYPE_HMATRIX general hyper matrices
PNL_TYPE_HMATRIX_DOUBLE real hyper matrices
PNL_TYPE_HMATRIX_INT integer hyper matrices
PNL_TYPE_HMATRIX_COMPLEX complex hyper matrices
PNL_TYPE_BASIS bases
PNL_TYPE_RNG random number generators
PNL_TYPE_LIST doubly linked list
PNL_TYPE_ARRAY array
-
Table 1: PnlTypes
- - - -

-
-

We provide several macros for manipulating PnlObejcts.

- - -

2.2 List object

-

This section describes functions for creating an manipulating lists. Lists are internally stored as -doubly linked lists. -

The structures and functions related to lists are declared in pnl/pnl_list.h. - - - - - -

-
-typedef struct _PnlCell PnlCell;
-struct _PnlCell
-{
-  struct _PnlCell *prev;  /*!< previous cell or 0 */
-  struct _PnlCell *next;  /*!< next cell or 0 */
-  PnlObject *self;       /*!< stored object */
-};
-
-
-typedef struct _PnlList PnlList;
-struct _PnlList
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlList pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  PnlCell *first; /*!< first element of the list */
-  PnlCell *last; /*!< last element of the list */
-  PnlCell *curcell; /*!< last accessed element,
-                         if never accessed is NULL */
-  int icurcell; /*!< index of the last accessed element,
-                     if never accessed is NULLINT */
-  int len; /*!< length of the list */
-};
-
-

-

Important note: Lists only store addresses of objects. So when an object is inserted into a -list, only its address is stored into the list. This implies that you must not free any -objects inserted into a list. The deallocation is automatically handled by the function -pnl_list_free. -

- -

-

-

2.3 Array object

-

This section describes functions for creating and manipulating arrays of PnlObjects. -

The structures and functions related to arrays are declared in pnl/pnl_array.h. - - - - -

-
-typedef struct _PnlArray PnlArray;
-struct _PnlArray
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlArray pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int size;
-  PnlObject **array;
-  int mem_size;
-};
-
-

-

Important note: Arrays only store addresses of objects. So when an object is inserted into an -array, only its address is stored into the array. This implies that you must not free any -objects inserted into a array. The deallocation is automatically handled by the function -pnl_array_free. -

- -

-

-

3 Mathematical framework

-

-

-

3.1 General tools

-

The macros and functions of this paragraph are defined in pnl/pnl_mathtools.h. -

-

-
3.1.1 Constants
-

A few mathematical constants are provided by the library. Most of them are actually -already defined in math.h, values.h or limits.h and a few others have been added. - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - -
M_E

e1
M_LOG2E

log 2e
M_LOG10E

log 10e
M_LN2

log e2
M_LN10

log e10
M_PI

π
M_2PI

2π
M_PI_2

π∕2
M_PI_4

π∕4
M_1_PI

1∕π
M_2_PI

2∕π
M_2_SQRTPI

2√ --
-  π
M_SQRT2PI

sqrt2π
M_SQRT2

√ --
-  2
M_EULER

γ = lim n→∞(∑             )
-   nk=1k1- ln(n)
M_SQRT1_2

1√2--
M_1_SQRT2PI

1√2-π-
M_SQRT2_PI

∘ ----
-  2∕π
INT_MAX

2147483647
MAX_INT

INT_MAX
DBL_MAX

1.79769313486231470e + 308
DOUBLE_MAX

DBL_MAX
DBL_EPSILON

2.2204460492503131e - 16
PNL_NEGINF

-∞
PNL_POSINF

+
PNL_INF

+
NAN

Not a Number
-

-

-
3.1.2 A few macros
- -

-

-
3.1.3 Classifying a floating–point number
- -

-

-
3.1.4 Rouding a floating–point number
- -

-

-
3.1.5 Some standard mathematical functions
- -

-

-
3.1.6 Comparison of floating–point numbers
- - - - -

-

-

3.2 Complex numbers

-

-

-
3.2.1 Overview
-

The complex type and related functions are defined in the header pnl/pnl_complex.h.
-

The first native implementation of complex numbers in the C language appeared in C99, which is -unfortunately not available on all platforms. For this reason, we provide here an implementation of -complex numbers. - - - -

-
-typedef struct {
-    double r; /*!< real part */
-    double i; /*!< imaginary part */
-} dcomplex;
-
-

-

-

-
3.2.2 Constants
-

- - -
CZERO

0 as a complex number
CONE

1 as a complex number
CI

I the unit complex number
-

-

-
3.2.3 Functions
- -

Most algebraic operations on complex numbers are implemented using the following naming for the -functions

- -

For example C_op_idamcb is id(   -)
- a- b. So functions are :

- -

-

-

4 Linear Algebra

-

-

-

4.1 Vectors

-

-

-
4.1.1 Overview
-

The structures and functions related to vectors are declared in pnl/pnl_vector.h. -

Vectors are declared for several basic types : double, int, and dcomplex. In the following -declarations, BASE must be replaced by one the previous types and the corresponding vector -structures are respectively named PnlVect, PnlVectInt, PnlVectComplex - - - -

-
-typedef struct _PnlVect {
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlVect pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int size; /*!< size of the vector */
-  int mem_size; /*!< size of the memory block allocated for array */
-  double *array; /*!< pointer to store the data */
-  int owner; /*!< 1 if the object owns its array member, 0 otherwise */
-} PnlVect;
-
-typedef struct _PnlVectInt {
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlVectInt pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int size; /*!< size of the vector */
-  int mem_size; /*!< size of the memory block allocated for array */
-  int *array; /*!< pointer to store the data */
-  int owner; /*!< 1 if the object owns its array member, 0 otherwise */
-} PnlVectInt;
-
-typedef struct _PnlVectComplex {
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlVectComplex pointer to be cast
-   * to a PnlObject
-   */
-  PnlObject object;
-  int size; /*!< size of the vector */
-  int mem_size; /*!< size of the memory block allocated for array */
-  dcomplex *array; /*!< pointer to store the data */
-  int owner; /*!< 1 if the object owns its array member, 0 otherwise */
-} PnlVectComplex;
-
-

size is the size of the vector, array is a pointer containing the data and owner is an integer to know -if the vector owns its array pointer (owner=1) or shares it with another structure (owner=0). -mem_size is the number of elements the vector can hold at most. -

- - - -

-
4.1.2 Functions
-

General functions -These functions exist for all types of vector no matter what the basic type is. The following -conventions are used to name functions operating on vectors. Here is the table of prefixes used for -the different basic types. -

-
-

-

-
- - -
type prefix BASE



double pnl_vect double



int pnl_vect_int int



dcomplexpnl_vect_complexdcomplex
-

In this paragraph, we present the functions operating on PnlVect which exist for all types. To -deduce the prototypes of these functions for other basic types, one must replace pnl_vect and -double according the above table. -

-

Constructors and destructors -There are no special functions to access the size of a vector, instead the field size should be -accessed directly. -

- -

Resizing vectors

- -

Accessing elements -If it is supported by the compiler, the following functions are declared inline. To speed up these -functions, you can define the macro PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an -explanation. -

Accessing elements of a vector is faster using the following macros

- - - - - -

Printing vector

- -

Applying external operation to vectors -

- -

Element wise operations -

- -

Scalar products and norms

- -

Comparison functions -

- - - - -

Ordering functions The following functions are not defined for PnlVectComplex because there is -no total ordering on Complex numbers -

- -

Misc -

- -

Complex vector functions -

- -

There exist functions to directly access the real or imaginary parts of an element of a complex -vector. These functions also have inlined versions that are used if the variable HAVE_INLINE was -declared at compilation time. -

- -

Equivalently to these functions, there exist macros. When the compiler is able to handle inline -code, there is no gain in using macros instead of inlined functions at least in principle. -

- -

-

-

4.2 Compact Vectors

-

-

-
4.2.1 Short description
- - - - -
-typedef struct PnlVectCompact {
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlVectCompact pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int size; /* size of the vector */
-  double val; /* single value */
-  double *array; /* Pointer to double values */
-  char convert; /* ’a’, ’d’ : array, double */
-} PnlVectCompact;
-
-

-

-

-
4.2.2 Functions
- -

-

-

4.3 Matrices

-

-

-
4.3.1 Overview
-

The structures and functions related to matrices are declared in pnl/pnl_matrix.h. - - - - - - -

-
-typedef struct _PnlMat{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlMat pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int m; /*!< nb rows */
-  int n; /*!< nb columns */
-  int mn; /*!< product m*n */
-  int mem_size; /*!< size of the memory block allocated for array */
-  double *array; /*!< pointer to store the data row-wise */
-  int owner; /*!< 1 if the object owns its array member, 0 otherwise */
-} PnlMat;
-
-typedef struct _PnlMatInt{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlMatInt pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int m; /*!< nb rows */
-  int n; /*!< nb columns */
-  int mn; /*!< product m*n */
-  int mem_size; /*!< size of the memory block allocated for array */
-  int *array; /*!< pointer to store the data row-wise */
-  int owner; /*!< 1 if the object owns its array member, 0 otherwise */
-} PnlMatInt;
-
-typedef struct _PnlMatComplex{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlMatComplex pointer to be cast
-   * to a PnlObject
-   */
-  PnlObject object;
-  int m; /*!< nb rows */
-  int n; /*!< nb columns */
-  int mn; /*!< product m*n */
-  int mem_size; /*!< size of the memory block allocated for array */
-  dcomplex *array; /*!< pointer to store the data row-wise */
-  int owner; /*!< 1 if the object owns its array member, 0 otherwise */
-} PnlMatComplex;
-                                                                                    
-
-                                                                                    
-
-

m is the number of rows, n is the number of columns. array is a pointer containing the data of -the matrix stored line wise, The element (i, j) of the matrix is array[i*m+j]. owner -is an integer to know if the matrix owns its array pointer (owner=1) or shares it with -another structure (owner=0). mem_size is the number of elements the matrix can hold at -most. -

The following operations are implemented on matrices and vectors. alpha -and beta are numbers, A and B are matrices and x and y are vectors. -

- - - - - - -
pnl_mat_axpy B := alpha * A + B
pnl_mat_scalar_prod x’ A y
pnl_mat_dgemm C := alpha * op (A) * op (B) + beta * C
pnl_mat_mult_vect_transpose_inplacey = A’ * x
pnl_mat_mult_vect_inplace y = A * x
pnl_mat_lAxpby y := lambda * A * x + beta * y
pnl_mat_dgemv y := alpha * op (A) * x + beta * y
pnl_mat_dger A := alpha x * y’ + A
-

-

-
4.3.2 Generic Functions
-

These functions exist for all types of matrices no matter what the basic type is. The following -conventions are used to name functions operating on matrices. Here is the table of prefixes used for -the different basic types. -

-
-

-

-
- - -
type prefix BASE



double pnl_mat double



int pnl_mat_int int



dcomplexpnl_mat_complexdcomplex
-

In this paragraph we present the functions operating on PnlMat which exist for all types. To deduce -the prototypes of these functions for other basic types, one must replace pnl_mat and double -according the above table. -

-

Constructors and destructors -There are no special functions to access the sizes of a matrix, instead the fields m, n and mn give -direct access to the number of rows, columns and the size of the matrix. -

- -

Accessing elements. -If it is supported by the compiler, the following functions are declared inline. To speed up these -functions, you can define the macro PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an -explanation. -

Accessing elements of a matrix is faster using the following macros

- - -

Printing Matrices -

- - - - -

Applying external operations -

- -

Element wise operations -

- -

Comparison functions -

- - - - -

Ordering operations -

- -

Standard matrix operations -

- -

Linear systems and matrix decompositions -The following functions are designed to solve linear system of the from A x = b where A is a matrix -and b is a vector except in the functions pnl_mat_syslin_mat, pnl_mat_lu_syslin_mat and -pnl_mat_chol_syslin_mat which expect the right hand side member to be a matrix too. Whenever -the vector b is not needed once the system is solved, you should consider using “inplace” -functions. -

All the functions described in this paragraph return OK if the computations have been carried out -successfully and FAIL otherwise. -

- -

The following functions are designed to invert matrices. The authors provide these functions -although they cannot find good reasons to use them. Note that to solve a linear system, one must -used the syslin functions and not invert the system matrix because it is much longer. -

- - - - -

-

-
4.3.3 Functions specific to base type double
-

Linear systems and matrix decompositions -The following functions are designed to solve linear system of the from A x = b where A is a matrix -and b is a vector except in the functions pnl_mat_syslin_mat, pnl_mat_lu_syslin_mat and -pnl_mat_chol_syslin_mat which expect the right hand side member to be a matrix too. Whenever -the vector b is not needed once the system is solved, you should consider using “inplace” -functions. -

All the functions described in this paragraph return OK if the computations have been carried out -successfully and FAIL otherwise. -

- -

-

-
4.3.4 Functions specific to base type dcomplex
- -

-

-
4.3.5 Permutations
- - - - -
-typedef PnlVectInt PnlPermutation;
-
-

-

The PnlPermutation type is actually nothing else than a vector of integers, i.e. a PnlVectInt. It is -used to store the partial pivoting with row interchanges transformation needed in the LU -decomposition. We use the Blas convention for storing permutations. Consider a PnlPermutation p -generated by a LU decomposition of a matrix A : to compute the decomposition, row i of A was -interchanged with row p(i). -

- -

-

-

4.4 Tridiagonal Matrices

-

-

-
4.4.1 Overview
-

The structures and functions related to tridiagonal matrices are declared in pnl/pnl_tridiag_matrix.h. -

We only store the three main diagonals as three vectors. - - - - -

-
-typedef struct PnlTridiagMat{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlTridiagMat pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int size; /*!< number of rows, the matrix must be square */
-  double *D; /*!< diagonal elements */
-  double *DU; /*!< upper diagonal elements */
-  double *DL; /*!< lower diagonal elements */
-} PnlTridiagMat;
-
-

size is the size of the matrix, D is an array of size size containing the diagonal terms. DU, DL are -two arrays of size size-1 containing respectively the upper diagonal (Mi,i+1) and the lower diagonal -(Mi-1,i). - - - - -

-
-typedef struct PnlTridiagMatLU{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlTridiagMatLU pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int size; /*!< number of rows, the matrix must be square */
-  double *D; /*!< diagonal elements */
-  double *DU; /*!< upper diagonal elements */
-  double *DU2; /*!< second upper diagonal elements */
-  double *DL; /*!< lower diagonal elements */
-  int *ipiv; /*!< Permutation: row i has been interchanged with row ipiv(i) */
-};
-
-

This type is used to store the LU decomposition of a tridiagonal matrix. -

-

-
4.4.2 Functions
-

Constructors and destructors -

- - - - -

Accessing elements. -If it is supported by the compiler, the following functions are declared inline. To speed up these -functions, you can use the macro constant PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an -explanation.

- -

Printing Matrix -

- -

Algebra operations -

- -

Element-wise operations -

- -

Standard matrix operations & Linear systems -

- -

-

-

4.5 Band Matrices

-

-

-
4.5.1 Overview
- - - - -
-typedef struct
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlBandMat pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int m; /*!< nb rows */
-  int n; /*!< nb columns */
-  int nu; /*!< nb of upperdiagonals */
-  int nl; /*!< nb of lowerdiagonals */
-  int m_band; /*!< nb rows of the band storage */
-  int n_band; /*!< nb columns of the band storage */
-  double *array;  /*!< a block to store the bands */
-} PnlBandMat;
-
-

-

The structures and functions related to band matrices are declared in pnl/pnl_band_matrix.h. -

-

-
4.5.2 Functions
-

Constructors and destructors -

- -

Accessing elements. -If it is supported by the compiler, the following functions are declared inline. To speed up these -functions, you can use the macro constant PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an -explanation.

- -

Element wise operations -

- -

Standard matrix operations & Linear system -

- -

-

-

4.6 Sparse Matrices

- - - -

-

-
4.6.1 Short description
-

The structures and functions related to matrices are declared in pnl/pnl_sp_matrix.h. - - - - - - -

-
-typedef struct _PnlSpMat
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows a PnlSpMat pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int m; /*!< number of rows */
-  int n; /*!< number of columns */
-  int nz; /*!< number of non-zero elements */
-  int *J; /*!< column indices, vector of size nzmax */
-  int *I; /*!< row offset integer vector,
-            array[I[i]] is the first element of row i.
-            Vector of size (m+1) */
-  double *array; /*!< pointer to store the data of size nzmax*/
-  int nzmax; /*!< size of the memory block allocated for array */
-} PnlSpMat;
-
-typedef struct _PnlSpMatInt
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows a PnlSpMat pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int m; /*!< number of rows */
-  int n; /*!< number of columns */
-  int nz; /*!< number of non-zero elements */
-  int *J; /*!< column indices, vector of size nzmax */
-  int *I; /*!< row offset integer vector,
-            array[I[i]] is the first element of row i.
-            Vector of size (m+1) */
-  int *array; /*!< pointer to store the data of size nzmax */
-  int nzmax; /*!< size of the memory block allocated for array */
-} PnlSpMatInt;
-
-typedef struct _PnlSpMatComplex
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows a PnlSpMat pointer to be cast to a PnlObject
-   */
-                                                                                    
-
-                                                                                    
-  PnlObject object;
-  int m; /*!< number of rows */
-  int n; /*!< number of columns */
-  int nz; /*!< number of non-zero elements */
-  int *J; /*!< column indices, vector of size nzmax */
-  int *I; /*!< row offset integer vector,
-            array[I[i]] is the first element of row i.
-            Vector of size (m+1) */
-  dcomplex *array; /*!< pointer to store the data of size nzmax */
-  int nzmax; /*!< size of the memory block allocated for array */
-} PnlSpMatComplex;
-
-

-

The non zero elements of row i are stored in array between the indices I[i] and I[i+1]-1. The array J -contains the column indices of every element of array.
-

Sparse matrices are defined using the internal template approach and can be used for integer, float -or complex base data according to the following table

-
-

-

-
- - -
base typeprefix type



double pnl_sp_mat PnlSpMat



int pnl_sp_mat_int PnlSpMatInt



dcomplex pnl_sp_mat_complexPnlSpMatComplex
-

-

-
4.6.2 Functions
-

Constructors and destructors -

- -

Accessing elements -

- -

Applying external operations -

- -

Standard matrix operations -

- -

Comparison functions -

- - - - -

-

-

4.7 Hyper Matrices

-

-

-
4.7.1 Short description
-

The Hyper matrix types and related functions are defined in the header pnl/pnl_matrix.h. - - - - - - -

-
-typedef struct PnlHmat{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlHmat pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int ndim; /*!< nb dimensions */
-  int *dims; /*!< pointer to store the values of the ndim dimensions */
-  int mn; /*!< product dim_1 *...*dim_ndim */
-  int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1]
-                with pdims[ndim - 1] = 1 */
-  double *array; /*!< pointer to store */
-} PnlHmat;
-
-typedef struct PnlHmatInt{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlHmatInt pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int ndim; /*!< nb dimensions */
-  int *dims; /*!< pointer to store the value of the ndim dimensions */
-  int mn; /*!< product dim_1 *...*dim_ndim */
-  int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1]
-                with pdims[ndim - 1] = 1 */
-  int *array; /*!< pointer to store */
-} PnlHmatInt;
-
-typedef struct PnlHmatComplex{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlHmatComplex pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int ndim; /*!< nb dimensions */
-  int *dims; /*!< pointer to store the value of the ndim dimensions */
-  int mn; /*!< product dim_1 *...*dim_ndim */
-  int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1]
-                with pdims[ndim - 1] = 1 */
-  dcomplex *array; /*!< pointer to store */
-} PnlHmatComplex;
-
- - - -

ndim is the number of dimensions, dim is an array to store the size of each dimension and nm -contains the product of the sizes of each dimension. array is an array of size mn containing the data. -The integer array pdims is used to create the one–to–one map between the natural indexing and the -linear indexing used in array. -

-

-
4.7.2 Functions
-

These functions exist for all types of hypermatrices no matter what the basic type is. The following -conventions are used to name functions operating on hypermatrices. Here is the table of prefixes -used for the different basic types. -

-
-

-

-
- - -
base typeprefix type



double pnl_hmat PnlHmat



int pnl_hmat_int PnlHmatInt



dcomplex pnl_hmat_complexPnlHmatComplex
-

In this paragraph, we present the functions operating on PnlHmat which exist for all types. To -deduce the prototypes of these functions for other basic types, one must replace pnl_hmat and -double according the above table. -

-

Constructors and destructors -

- -

Accessing elements -

- -

Printing hypermatrices -

- -

Term by term operations -

- -

-

-

4.8 Iterative Solvers

- - - -

-

-
4.8.1 Overview
-

The structures and functions related to solvers are declared in pnl/pnl_linalgsolver.h. - - - - - - - -

-
-typedef struct _PnlIterationBase PnlIterationBase;
-typedef struct _PnlCgSolver PnlCgSolver;
-typedef struct _PnlBicgSolver PnlBicgSolver;
-typedef struct _PnlGmresSolver PnlGmresSolver;
-
-struct _PnlIterationBase
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlVectXXX pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int iteration;
-  int max_iter;
-  double normb;
-  double tol_;
-  double resid;
-  int error;
-  /* char *  err_msg; */
-};
-
-/* When you repeatedly use iterative solvers, do not malloc each time */
-struct _PnlCgSolver
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlCgSolver  pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  PnlVect * r;
-  PnlVect * z;
-  PnlVect * p;
-  PnlVect * q;
-  double rho;
-  double oldrho;
-  double beta;
-  double alpha;
-  PnlIterationBase * iter;
-} ;
-
-struct _PnlBicgSolver
-{
-                                                                                    
-
-                                                                                    
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlBicgSolver pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  double rho_1, rho_2, alpha, beta, omega;
-  PnlVect * p;
-  PnlVect * phat;
-  PnlVect * s;
-  PnlVect * shat;
-  PnlVect * t;
-  PnlVect * v;
-  PnlVect * r;
-  PnlVect *  rtilde;
-  PnlIterationBase * iter;
-} ;
-
-struct _PnlGmresSolver
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlGmresSolver pointer to be cast to a PnlObject
-   */
-  PnlObject object;
-  int restart;
-  double beta;
-  PnlVect * s;
-  PnlVect * cs;
-  PnlVect * sn;
-  PnlVect * w;
-  PnlVect * r;
-  PnlMat * H;
-  PnlVect * v[MAX_RESTART];
-  PnlIterationBase *iter;
-  PnlIterationBase *iter_inner;
-} ;
-
-

-

A Left preconditioner solves the problem : -

-
-PM x = P b,
-
-

and whereas right preconditioner solves -

-
-M  Py = b,    P y = x.
-
-

More information is given in Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd -ed. ed.). SIAM. ISBN 0898715342. OCLC 51266114. The reader will find in this book -some discussion about right or/and left preconditioner and a description of the following -algorithms. -

These algorithms, we implemented with a left preconditioner. Right preconditioner can be easily -computed changing matrix vector multiplication operator from M x to M PR x and solving -PRy = x at the end of algorithm. -

-

-
4.8.2 Functions
-

Three methods are implemented : Conjugate Gradient, BICGstab and GMRES with restart. For -each of them a structure is created to store temporary vectors used in the algorithm. In some cases, -we have to apply iterative methods more than once : for example to solve at each time step a -discrete form of an elliptic problem come from parabolic problem. In the cases, do not call the -constructor and destructor at each time, but instead use the initialization and solve -procedures. -

Formally we have, - - - -

-
-Create iterative method
-For each time step
-  Initialisation of iterative method
-  Solve linear system link to elliptic problem
-end for
-free iterative method
-
-

-

In these functions, we don’t use any particular matrix structure. We give the matrix vector -multiplication as a parameter of the solver. -

-

Conjugate Gradient method -Only available for symmetric and positive matrices.

- -

BICG stab -

- -

GMRES with restart -See Saad, Yousef (2003) for a discussion about the restart parameter. For GMRES we need to -store at the p-th iteration p vectors of the same size of the right and side. It could be -very expensive in term of memory allocation. So GMRES with restart algorithm stop if -p = restart and restarts the algorithm with the previously computed solution as initial -guess. - - - -

Note that if restart equals m, we have a classical GMRES algorithm. -

- -

In the next paragraph, we write all the solvers for PnlMat . This will be done as follows: construct -an application matrix vector. - - - -

-
-static void pnl_mat_mult_vect_applied(const void *mat, const PnlVect *vec,
-                                      const double a , const double b,
-                                      PnlVect *lhs)
-{pnl_mat_lAxpby(a, (PnlMat*)mat, vec, b, lhs);}
-
-

and give it as the parameter of the iterative method - - - -

-
-int pnl_mat_cg_solver_solve(const PnlMat * Matrix, const PnlMat * PC,
-                            PnlVect * x, const PnlVect *b, PnlCgSolver * Solver)
-{ return pnl_cg_solver_solve(pnl_mat_mult_vect_applied,
-                             Matrix, pnl_mat_mult_vect_applied,
-                             PC, x, b, Solver);}
-
-

-

In practice, we cannot define all iterative methods for all structures. With this implementation, the -user can easily :

- -

Iterative algorithms for PnlMat -

- - - - -

-

-

5 Cumulative distribution Functions

-

The functions related to this chapter are declared in pnl/pnl_cdf.h. -

For various distribution functions, we provide functions named pnl_cdf_xxx where xxx is -the abbreviation of the distribution name. All these functions are based on the same -prototype -

-
-
-               ∫ x
-p = 1- q;  p =    density(u)du
-
-
-

-

- - -

-

-

6 Random Number Generators

-

The functionalities described in this chapter are declared in pnl/pnl_random.h. -

Random number generators should be called through the new rng interface based on the -PnlRng object. This interface uses reentrant functions and is suitable for multi-threaded -applications. -

The older rand interface is kept for compatibility purposes only and should not be used in new code. -

- - - -


- - - -
- - - - - - - - - - - - -
Random generatorindex Type Info




KNUTH PNL_RNG_KNUTH pseudo
MRGK3 PNL_RNG_MRGK3 pseudo
MRGK5 PNL_RNG_MRGK5 pseudo
SHUFL PNL_RNG_SHUFL pseudo
L’ECUYER PNL_RNG_L_ECUYER pseudo
TAUSWORTHE PNL_RNG_TAUSWORTHE pseudo
MERSENNE PNL_RNG_MERSENNE pseudo
SQRT PNL_RNG_SQRT quasi
HALTON PNL_RNG_HALTON quasi
FAURE PNL_RNG_FAURE quasi
SOBOL_I4 PNL_RNG_SOBOL_I4 quasi uses 32 bit intergers
SOBOL_I8 PNL_RNG_SOBOL2_I8 quasi uses 64 bit intergers
NIEDERREITER PNL_RNG_NIEDERREITERquasi
-
Table 2: Indices of the random generators
- - - -

-
-

6.1 The rng interface

-

It is possible to create several random number generators each with its own state variable so that -they can evolve independently in a shared memory environment. These generators are suitable for -use in multi-threaded programs. - - - - -

-
-typedef struct _PnlRng PnlRng;
-struct _PnlRng
-{
-  PnlObject object;
-  int type; /*!< generator type *
-  void (*Compute)(PnlRng *g, double *sample); /*!< the function to compute the
-                                                next number in the sequence */
-  int rand_or_quasi; /*!< can be PNL_MC or PNL_QMC */
-  int dimension; /*!< dimension of the space in which we draw the samples */
-  int counter; /*!< counter = number of samples already drawn */
-  int has_gauss; /*!< Is a gaussian deviate available? */
-  double gauss; /*!< If has_gauss==1, gauss a gaussian sample */
-  int size_state; /*!< size in bytes of the state variable */
-  void *state; /*!< state of the random generator */
-};
-
-

-

- -

Some auxiliary functions internally used (to be used with caution)

- -

The following functions return one sample from the specified distribution.

- -

The following functions take an already existing PnlVect *as first argument and fill each entry of -the vector with a sample from the specified distribution. All the entries are independent. The -difference between n-samples from a distribution in dimension 1, and one sample from the same -distribution in dimension n only matters when using a quasi random number generator. -

- -

The following functions take an already existing PnlMat *as first argument and fill each entry of the -matrix with a sample from the specified distribution. All the entries are independent. On -return, the matrix M is of size samples x dimension. The rows of M are independent and -identically distributed. Each row is a sample from the given law in dimension dimension. -

- -

Some examples - - - -

-
-#include <stdlib.h>
-#include "pnl/pnl_random.h"
-
-int main ()
-{
-  int i, M;
-  PnlRng *rng = pnl_rng_create(PNL_RNG_MERSENNE);
-  PnlVect *v = pnl_vect_new();
-  M = 10000;
-
-  /* rng must be initialized. When sseed=0, a default
-     value depending on the generator is used */
-     pnl_rng_sseed(rng, 0);
-
-  for (i=0 ; i<M ; i++)
-  {
-    /* Simulates a normal random vector in R^{10} */
-    pnl_vect_rng_normal(v, 10, rng);
-    /* Do something with v */
-  }
-
-  pnl_vect_free(&v);
-  pnl_rng_free(&rng); /* Frees the generator */
-  exit(0);
-}
-
-

- - - -

-
-#include <stdlib.h>
-#include <time.h>
-#include "pnl/pnl_random.h"
-
-int main ()
-{
-  int i, M;
-  double E;
-  PnlRng *rng = pnl_rng_create(PNL_RNG_MERSENNE);
-  M = 10000;
-
-  /* rng must be initialized. */
-  pnl_rng_sseed(rng, time (NULL));
 
-  for (i=0 ; i<M ; i++)
-  {
-    /* Simulates an exponential random variable */
-    E = pnl_rng_exp(1, rng);
-    /* Do something with E */
-  }
 
-  pnl_rng_free(&rng); /* Frees the generator */
-  exit(0);
-}
-
-

-

-

-

6.2 The rand interface (deprecated)

-

Note: For backward compatibility with older versions of the PNL, we still provide the old rand -interface to random number generation although we strongly encourage users to use the new rng -interface (see section 6.1). -

Every generator is identified by an integer valued macro. One must NOT refer to a generator using -directly the value of the macro PNL_RNG_XXX because there is no warranty that the order used -to store the generators will remain the same in future releases. Instead, one should call generators -directly using their macro names. -

The initial seeds of all the generators are fixed by the function pnl_rand_init but you can change it -by calling pnl_rand_sseed. -

Before starting to use random number generators, you must initialize them by calling -

- - - - - -

Once a generator is chosen, there are several functions available in the library to draw samples -according to a given law. -

The following functions return one sample from a specified law.

- -

The following functions take an already existing PnlVect * as its first argument and fill each entry -of the vector with a sample from the specified law. All the entries are independent. The difference -between n-samples from a distribution in dimension 1, and one sample from the same -distribution in dimension n only matters when using a Quasi random number generator. -

- -

The following functions take an already existing PnlMat * as first argument and fill each entry of -the vector with a sample from the specified law. All the entries are in-dependant. On return, -the matrix M is of size samples x dimension. The rows of M are independently and -identically distributed. Each row is a sample from the given law in dimension dimension. -

- -

Because of the use of Quasi random number generators, you may need to draw a set of samples at -once because they represent one sample from a multi-dimensional distribution. The following -function enables to draw one sample from the dimension-dimensional standard normal -distribution and store it so that you can access the elements individually afterwards. -

- - - - -

-

-

7 Function bases and regression

-

-

-

7.1 Overview

-

To use these functionalities, you should include pnl/pnl_basis.h. - - - - -

-
-struct _PnlBasis
-{
-  /**
-   * Must be the first element in order for the object mechanism to work
-   * properly. This allows any PnlBasis pointer to be cast to a PnlObject
-   */
-  PnlObject     object;
-  /** The basis type */
-  int           id;
-  /** The string to label the basis */
-  const char   *label;
-  /** The number of variates */
-  int           nb_variates;
-  /** The total number of elements in the basis */
-  int           nb_func;
-  /** The tensor matrix */
-  PnlMatInt    *T;
-  /** The sparse Tensor matrix */
-  PnlSpMatInt  *SpT;
-  /** The number of functions in the tensor #T */
-  int           len_T;
-  /** Compute the i-th element of the one dimensional basis.  As a convention, (*f)(x, 0) MUST be equal to 1 */
-  double      (*f)(double x, int i, int dim, void *params);
-  /** Compute the first derivative of i-th element of the one dimensional basis */
-  double      (*Df)(double x, int i, int dim, void *params);
-  /** Compute the second derivative of the i-th element of the one dimensional basis */
-  double      (*D2f)(double x, int i, int dim, void *params);
-  /** PNL_TRUE if the basis is reduced */
-  int           isreduced;
-  /** The center of the domain */
-  double       *center;
-  /** The inverse of the scaling factor to map the domain to [-1, 1]^nb_variates */
-  double       *scale;
-  /** An array of additional functions */
-  PnlRnFuncR   *func_list;
-  /** The number of functions in #func_list */
-  int           len_func_list;
-  /** Extra parameters to pass to basis functions */
-  void         *params;
-  /** Size of params in bytes to be passed to malloc */
-  size_t        params_size;
-};
-                                                                                    
-
-                                                                                    
-
-

-

A PnlBasis is a family of multivariate functions with real values. Tow different kinds of functions -can be stored in these families: tensor functions — originally, this was the only possibility — and -standard multivariate function typed as PnlRnFuncR. -

-

Tensor functions. -Tensors functions are built as a tensor product of one dimensional elements. Hence, we only need a -tensor matrix T to describe a multi-dimensional basis in terms of the one dimensional one. These -tensors functions can be easily evaluated and differentiated twice, see pnl_basis_eval, -pnl_basis_eval_vect, pnl_basis_eval_D, pnl_basis_eval_D_vect, pnl_basis_eval_D2, -pnl_basis_eval_D2_vect, pnl_basis_eval_derivs, pnl_basis_eval_derivs_vect. -

The two tensors T and SpT do actually store the same information — T(i,j) is the degree w.r.t the -j-th variable in the i-th function. Originally, we were only using the dense representation T, which is -far more convenient to use when building the basis but it slows down the evaluation of -the basis by a great deal. To overcome this lack of efficiency, a sparse storage has been -added. -

-
- - - -


- - - -
- - -
PNL_BASIS_CANONICAL

for the Canonical polynomials
PNL_BASIS_HERMITE

for the Hermite polynomials
PNL_BASIS_TCHEBYCHEV

for the Tchebychev polynomials
-
Table 3: Names of the bases. See also function pnl_basis_type_register to register more -basis types.
- - - -

-
-

The Hermite polynomials are defined by -

-
-
-                   n
-H (x) = (- 1)n ex22-d-e- x22-.
- n               dxn
-
-
-

If G is a real valued standard normal random variable, E[Hn(G)Hm(G)] = n!1{n=m}. -
-

-

Standard multivariate functions. -These functions are supposed to be PnlRnFuncR. -

To make this toolbox more complete, it is now possible to add some extra functions, which -are not tensor product. They are stored using an independent mechanism in func_list. -These additional functions are only taken into account by the methods pnl_basis_i, -pnl_basis_i_vect, pnl_basis_eval and pnl_basis_eval_vect. Note in particular that it -is not possible to differentiate these functions and that they are not sensitive to the -isreduced attribute. To add an extra function to an existing PnlBasis, call the function -pnl_basis_add_function. -

-

-

7.2 Functions

- -

Functional regression based on a least square approach often leads to ill conditioned linear systems. -One way of improving the stability of the system is to use centered and renormalised -polynomials so that the original domain of interest D (a subset of d) is mapped to [-1,1]d. -If the domain D is rectangular and writes [a,b] where a,b d, the mapping is done -by -

-
- - - -
-          (               )
-            xi --(bi +-ai)∕2
-x ∈ D ↦- →     (bi - ai)∕2
-                            i=1,⋅⋅⋅,d
-
-
(1)
-

- -

Note that this renormlization does not apply to the extra functions by pnl_basis_add_function but -only to the functions defined by the tensor T. -

- - - - -

The following functions are provided for compatibility purposes but are marked as deprecated. Use -the functions with the _vect extension.

- - - - - - - -

-

-

8 Numerical integration

-

-

-

8.1 Overview

-

To use these functionalities, you should include pnl/pnl_integration.h. -

Numerical integration methods are designed to numerically evaluate the integral over a finite or -non finite interval (resp. over a square) of real valued functions defined on (resp. on -2). - - - -

-
-typedef struct {
-  double (*function) (double x, void *params);
-  void *params;
-} PnlFunc;
-
-typedef struct {
-  double (*function) (double x, double y, void *params);
-  void *params;
-} PnlFunc2D;
-
-

-

We provide the following two macros to evaluate a PnlFunc or PnlFunc2D at a given -point - - - -

-
-#define PNL_EVAL_FUNC(F, x) (*((F)->function))(x, (F)->params)
-#define PNL_EVAL_FUNC2D(F, x, y) (*((F)->function))(x, y, (F)->params)
-
-

-

-

-

8.2 Functions

- - - - -

-

-

9 Fast Fourier Transform

-

-

-

9.1 Overview

-

This toolbox uses C version of the Fortran FFTPack library available on http://www.netlib.org/fftpack. -
-

The forward Fourier transform of a vector c is defined by -

-
-
-     N∑- 1   -ijk2π∕N
-zj =     cke       ,  j = 0,⋅⋅⋅ ,N - 1
-     k=0
-
-
-

The inverse Fourier transform enables to recover c from z and is defined by -

-
-
-    -1 N∑- 1   ijk2π∕N
-ck = N      cke      ,  j = 0,⋅⋅⋅ ,N - 1
-       k=0
-
-
-

Note that the inverse Fourier transform is scaled by -1
-N, such that the inverse Fourier transform -applies to the Fourier transform just yields the original vector. -

The coefficients of the Fourier transform of a real function satisfy the following relation -

- - - -
-
-     -----
-zk = zN-k,
-
-
(2)
-

where N is the number of discretization points. -

A few remarks on the FFT of real functions and its inverse transform:

- -

FFTPack storage - -The functions use the fftpack storage convention for half-complex sequences. In this convention, the -half-complex transform of a real sequence is stored with frequencies in increasing order, -starting from zero, with the real and imaginary parts of each frequency in neighboring -locations. -

The storage scheme is best shown by some examples. The table below shows the output for an -odd-length sequence, n = 5. The two columns give the correspondence between the 5 values in the -half-complex sequence (stored in a PnlVect V ) and the values (PnlVectComplex C) that would be -returned if the same real input sequence were passed to pnl_fft as a complex sequence (with -imaginary parts set to 0), -

-
- - - -
-C(0) = V(0) + i0,
-C(1) = V(1) + iV (2),
-C(2) = V(3) + iV (4), -----
-C(3) = V(3) - iV (4) = C (2),
-C(4) = V(1) + iV (2) = C-(1)
-
-
(3)
-

-

The elements of index greater than N∕2 of the complex array, as C(3) and C(4) are filled in using -the symmetry condition. -

The next table shows the output for an even-length sequence, n = 6. In the even case, there are two -values which are purely real, -

-
-
-C(0) = V(0) + i0,
-
-C(1) = V(1) + iV (2),
-C(2) = V(3) + iV (4),---
-C(3) = V(5) - i0 = C (0),---
-C(4) = V(3) - iV (4) = C-(2),
-C(5) = V(1) + iV (2) = C (1)
-
-
(4)
-

-

-

-

9.2 Functions

-

To use the following functions, you should include pnl/pnl_fft.h. - - - -

-

-
9.2.1 Direct call functions
-

All FFT functions need some extra memory to perform their computations. This is automatically -handled by all the functions but you can these repeatedly, for instance inside a Monte Carlo loop, -you should allocate a workspace once and for all and use the same at every iteration. In this case, -use the functions defined in Section 9.2.2. -

- -

-

-
9.2.2 Function with workspace
- -

-

-

10 Inverse Laplace Transform

-

For a real valued function f such that tf(t)e-σct is integrable over +, we can define its Laplace -transform -

-
-
-       ∫
- ^       ∞      -λt
-f(λ) =  0  f(t)e   dt    for λ ∈ ℂ with Re (λ ) ≥ σc.
-
-
- - - -

-

To use the following functions, you should include pnl/pnl_laplace.h. - - - - -

-
-typedef struct
-{
-  dcomplex (*F) (dcomplex x, void *params);
-  void *params;
-} PnlCmplxFunc;
- 
-
-

-

- -

-

-

11 Ordinary differential equations

-

-

-

11.1 Overview

-

To use these functionalities, you should include pnl/pnl_integration.h. -

These functions are designed for numerically solving n-dimensional first order ordinary differential -equation of the general form -

-
-
-dyi(t) = fi(t,y1(t),⋅⋅⋅ ,yn(t))
-dt
-
-
-

The system of equations is defined by the following structure - - - -

-
-typedef struct
-{
-  void (*F) (int neqn, double t, const double *y, double *yp, void *params);
-  int neqn;
-  void *params;
-} PnlODEFunc ;
-
-

-

- -

We provide the following macro to evaluate a PnlODEFunc at a given point - - - -

-
-#define PNL_EVAL_ODEFUNC(Fstruct, t, y, yp) \
-    (*((Fstruct)->F))((Fstruct)->neqn, t, y, yp, (Fstruct)->params)
-
-

-

-

-

11.2 Functions

- -

-

-

12 Optimization

-

To use the functions described in this section, you should include pnl/pnl_optim.h. -

-

-

12.1 Linear constrained optimization (linear programming)

-

-

-
12.1.1 Overview
-

Consider the minimization problem - - - -

-
min x CT x -
s.t. Aineq x Bineq -
Aeq x = Beq -
xmin x xmax
-

-

-
12.1.2 Functions
-

To solve such a linear problem, we provide a wrapper to the LPSolve library ( -http://lpsolve.sourceforge.net).

- -

-

-

12.2 Nonlinear constrained optimization

-

-

-
12.2.1 Overview
-

A standard Constrained Nonlinear Optimization problem can be written as: - - - -

-
-
-     (
-     |{ min  f(x)
-(O )   cI(x) ≥ 0
-     |( cE (x ) = 0
-
-
-

-

where the function f : n is the objective function, cI : n mI are the inequality -constraints and cE : n mE are the equality constraints. These functions are supposed to be -smooth. -

In general, the inequality constraints are of the form cI(x) = (g(x), x- l, u - x). The vector l and -u are the lower and upper bounds on the variables x and g(x) and the non linear inequality -constraints. -

Under some conditions, if x n is a solution of problem (O), then there exist a vector -λ = (λIE) mI × mE, such that the well known Karush-Kuhn-Tucker (KKT) optimality -conditions are satisfied: -

-
-
-    (
-    ||  ∇ ℓ(x,λI,λE) = ∇f (x)- ∇cI (x)λI - ∇cE (x)λE = 0
-    ||||                     cE (x ) = 0
-    |{                      cI(x) ≥ 0
-(P )|                       λI ≥ 0
-    ||||               cI(x)λI = 0, i = 1...m
-    ||(                i    i             I
-
-
-

-

l is known as the Lagrangian of the problem (O), λI and λE as the dual variables while x is the -primal variable. - - - -

-

-
12.2.2 Functions
-

To solve an inequality constrained optimization problem, ie mE = 0, we provide the following -function.

- -

-

-

13 Root finding

-

-

-

13.1 Overview

-

To provide a uniformed framework to root finding functions, we use several structures for storing -different kind of functions. The pointer params is used to store the extra parameters. These new -types come with dedicated macros starting in PNL_EVAL to evaluate the function and their Jacobian. - - - - -

-
-/*
- * f: R --> R
- * The function  pointer returns f(x)
- *
-typedef struct {
-  double (*F) (double x, void *params);
-  void *params;
-} PnlFunc ;
-#define PNL_EVAL_FUNC(Fstruct, x) (*((Fstruct)->F))(x, (Fstruct)->params)
-
-

- - - - -

-
-/*
- * f: R^2 --> R
- * The function pointer returns f(x)
- *
-typedef struct {
-  double (*F) (double x, double y, void *params);
-  void *params;
-} PnlFunc2D ;
-#define PNL_EVAL_FUNC2D(Fstruct, x, y) (*((Fstruct)->F))(x, y, (Fstruct)->params)
-
-

- - - - -

-
-/*
- * f: R --> R
- * The function pointer computes f(x) and Df(x) and stores them in fx
- * and dfx respectively
- *
-typedef struct {
-  void (*F) (double x, double *fx, double *dfx, void *params);
-  void *params;
-} PnlFuncDFunc ;
-#define PNL_EVAL_FUNC_FDF(Fstruct, x, fx, dfx) (*((Fstruct)->F))(x, fx, dfx, (Fstruct)->params)
-
-

- - - - -

-
-/*
- * f: R^n --> R
- * The function pointer returns f(x)
- *
-typedef struct {
-  double (*F) (const PnlVect *x, void *params);
-  void *params;
-} PnlRnFuncR ;
-#define PNL_EVAL_RNFUNCR(Fstruct, x) (*((Fstruct)->F))(x, (Fstruct)->params)
-
-

- - - - - -

-
-/*
- * f: R^n --> R^m
- * The function pointer computes the vector f(x) and stores it in
- * fx (vector of size m)
- *
-typedef struct {
-  void (*F) (const PnlVect *x, PnlVect *fx, void *params);
-  void *params;
-} PnlRnFuncRm ;
-#define PNL_EVAL_RNFUNCRM(Fstruct, x, fx) (*((Fstruct)->F))(x, fx, (Fstruct)->params)
-
-/*
- * Synonymous of PnlRnFuncRm for f:R^n --> R^n
- *
-typedef PnlRnFuncRm PnlRnFuncRn;
-#define PNL_EVAL_RNFUNCRN  PNL_EVAL_RNFUNCRM
-
-

- - - - - -

-
-/*
- * f: R^n --> R^m
- * The function pointer computes the vector f(x) and stores it in fx
- * (vector of size m)
- * The Dfunction pointer computes the matrix Df(x) and stores it in dfx
- * (matrix of size m x n)
- *
-typedef struct {
-  void (*F) (const PnlVect *x, PnlVect *fx, void *params);
-  void (*DF) (const PnlVect *x, PnlMat *dfx, void *params);
-  void (*FDF) (const PnlVect *x, PnlVect *fx, PnlMat *dfx, void *params);
-  void *params;
-} PnlRnFuncRmDFunc ;
-#define PNL_EVAL_RNFUNCRM_DF(Fstruct, x, dfx) \
-    (*((Fstruct)->Dfunction))(x, dfx, (Fstruct)->params)
-#define PNL_EVAL_RNFUNCRM_FDF(Fstruct, x, fx, dfx) \
-    (*((Fstruct)->F))(x, fx, dfx, (Fstruct)->params)
-#define PNL_EVAL_RNFUNCRM_F_DF(Fstruct, x, fx, dfx)    \
-      if ( (Fstruct)->FDF != NULL )                    \
-        {                                              \
-          PNL_EVAL_RNFUNCRN_FDF (Fstruct, x, fx, dfx); \
-        }                                              \
-      else                                             \
-        {                                              \
-          PNL_EVAL_RNFUNCRN (Fstruct, x, fx);          \
-          PNL_EVAL_RNFUNCRN_DF (Fstruct, x, dfx);      \
-        }
-/*
- * Synonymous of PnlRnFuncRmDFunc for f:R^n --> R^m
- *
-typedef PnlRnFuncRmDFunc PnlRnFuncRnDFunc;
-#define PNL_EVAL_RNFUNCRN_DF PNL_EVAL_RNFUNCRM_DF
-#define PNL_EVAL_RNFUNCRN_FDF PNL_EVAL_RNFUNCRM_FDF
-#define PNL_EVAL_RNFUNCRN_F_DF PNL_EVAL_RNFUNCRM_F_DF
-
-

-

-

-

13.2 Functions

-

To use the following functions, you should include pnl/pnl_root.h. - - - -

-

Real valued functions of a real argument -

- -

Vector valued functions with several arguments -

- -

We provide two wrappers for calling minpack routines.

- -

-

-

14 Special functions

-

The special function approximations are defined in the header pnl/pnl_specfun.h.
-

Most of these functions rely on the Cephes library which uses its own error mechanism which can be -activated or deactivated using the two following functions

- -

-

-

14.1 Real Bessel functions

- -

-

-

14.2 Complex Bessel functions

- -

-

-

14.3 Error functions

- - - - -

-

-

14.4 Gamma functions

-

For x > 0, the Gamma Function is defined by -

-
-
-       ∫
-Γ (x ) =  ∞ e-u ux-1du.
-        0
-
-
-

-

- - - - -

-

-

14.5 Digamma function

-

For x > 0, the digamma function ψ is defined as the logarithmic derivative of the Gamma function -Γ -

-
-
-       -d-          Γ-′(x)
-ψ (x ) = dx logΓ (x ) = Γ (x).
-
-
-

The function ψ admits the following integral representation -

-
-
-       ∫ ∞ ( - u     -xu  )
-ψ(x) =       e---  -e-----  .
-        0     u    1- e- u
-
-
-

-

- - - - -

-

-

14.6 Incomplete Gamma functions

-

For a and x > 0, the Incomplete Gamma Function is defined by -

-
-
-         ∫
-           ∞  -u  a-1
-Γ (a,x ) = x  e   u   du.
-
-
-

A relation similar to the one existing for the standard Gamma function holds -

-
-
-          - xa-e-x+-Γ (a-+-1,x)
-Γ (a,x) =          a         .
-
-
-

- - - -

-
Γ(a) = - 0ua-1 e-u du -
P(a,x) = Γ (a) - Γ (a,x)
--------------
-    Γ (a) =   1
------
-Γ (a) - 0xua-1 e-u du -
Q(a,x) = 1 - P(a,x) = Γ (a,x)
- Γ (a) = --1--
-Γ (a ) - xe-u ua-1du.
- -

-

-

14.7 Exponential integrals

-

For x > 0 and n , the function En is defined by -

-
-
-        ∫ ∞
-En (x ) =    e-xu u-ndu
-         1
-
-
-

-

This function is linked to the Incomplete Gamma function by -

-
-
-        ∫ ∞                             ∫ ∞
-En (x) =    e- xu(xu )-nxn-1d(xu) = xn-1     e-tt-ndt = xn-1Γ (1 - n,x) ,
-         x                               x
-
-
-

from which we can deduce -

-
-
-nE    (x) = e- x- xE  (x ).
-   n+1             n
-
-
-

For n > 1, the series expansion is given by -

-
-
-                      [                                         ]
-         n-1             --1--   --x--   ---x2---   --x3----
-En(x) = x   Γ (1 - n)+  -1 - n + 2 - n - 2(3-  n) + 6(4 - n) - ... .
-
-
-

The asymptotic behaviour is given by -

-
-
-            [                     ]
-E  (x ) = e--x 1 - n-+ n(n-+-1) + ....
- n       x       x      x2
-
-
-

The special case n = 1 gives -

-
-
-        ∫ ∞  e- u
-E1 (x) =     ---du,   |Arg (x )| ≥ π.
-         x    u
-
-
-

For any complex number x with positive real part, this can be written -

-
-
-         ∫    -ux
-E  (x) =  ∞  e---du,   ℜ(x) ≥ 0.
-  1       1   u
-
-
-

By integrating the Taylor expansion of e-t ∕t, and extracting the logarithmic singularity, we can -derive the following series representation for E1(x), -

-
-
-                   ∑∞ (- 1)kxk
-E1(x) = - γ - lnx -   --k-k!--    |Arg(x)| < π.
-                   k=1
-
-
-

The function E1 is linked to the exponential integral Ei -

-
-
-        ∫              ∫
-Ei (x) =  x  eudu =  -  ∞  e-u-du  ∀x ⁄= 0.
-         - ∞ u          -x  u
-
-
-

The above definition can be used for positive values of x, but the integral has to be -understood in terms of its Cauchy principal value, due to the singularity of the integrand at -zero. -

-
- - - -
-Ei(- x) = - E (x ),  ℜ (x) ≥ 0.
-             1
-
-
-

We deduce, -

-
-
-                  ∑∞ -xk-
-Ei (x ) = γ + lnx +   k k!,  x > 0.
-                  k=1
-
-
-

For x -

-
-
-         {
-            - Ei (- x) - iπ x < 0,
-Γ (0,x) =   - Ei (- x) x > 0.
-
-
-

-

- -

-

-

14.8 Hypergeometric functions

- -

-

-

15 Some bindings

-

-

-

15.1 MPI bindings

- - - -

-

-
15.1.1 Overview
-

We provide some bindings for the MPI library to natively handle PnlObjects. -

The functionnalities described in this chapter are declared in pnl/pnl_mpi.h. -

-

-
15.1.2 Functions
-

All the following functions return an error code as an integer value. This returned value should be -tested against MPI_SUCCESS to check that no error occurred. -

- -

For more expect users, we provide the following nonblocking functions.

- -

-

-

15.2 The save/load interface

-

The interface is only accessible when the MPI bindings are compiled since it is based on the -Packing/Unpacking facilities of MPI. -

The functionnalities described in this chapter are declared in pnl/pnl_mpi.h.

- - - - -

-

-

16 Financial functions

-

The financial functions are defined in the header pnl/pnl_finance.h.
-

- - -

Practitioners do not speak in terms of option prices, but rather compare prices in terms of their -implied Black & Scholes volatilities. So this parameter is very useful in practice. Here, we propose -two functions to compute σimpl : the first one is for one up-let, maturity, strike, option price. the -second function is for a list of strikes and maturities, a matrix of prices (with strikes varying -row-wise). -

- - - - - -

Index

-
ABS, 1
-

-C_op_amcb, 2
-C_op_amib, 3
-C_op_apcb, 4
-C_op_apib, 5
-C_op_damb, 6
-C_op_damcb, 7
-C_op_damib, 8
-C_op_dapb, 9
-C_op_dapcb, 10
-C_op_dapib, 11
-C_op_idamb, 12
-C_op_idamcb, 13
-C_op_idapb, 14
-C_op_idapcb, 15
-Cabs, 16
-Cadd, 17
-Carg, 18
-Ccos, 19
-Ccosh, 20
-Ccotan, 21
-Ccotanh, 22
-Cdiv, 23
-Cexp, 24
-CI, 25
-CIexp, 26
-Cimag, 27
-Cinv, 28
-Clgamma, 29
-Clog, 30
-Cminus, 31
-CMPLX, 32
-Cmul, 33
-Complex, 34
-Complex_polar, 35
-CONE, 36
-Conj, 37
-Cpow, 38
-Cpow_real, 39
-Cprintf, 40
- - - -CRadd, 41
-CRdiv, 42
-Creal, 43
-CRmul, 44
-CRsub, 45
-Csin, 46
-Csinh, 47
-Csqr_norm, 48
-Csqrt, 49
-Csub, 50
-Ctan, 51
-Ctanh, 52
-Ctgamma, 53
-CUB, 54
-CZERO, 55
-

-DBL_EPSILON, 56
-DBL_MAX, 57
-DOUBLE_MAX, 58
-

-GET, 59
-GET_COMPLEX, 60
-GET_IMAG, 61
-GET_INT, 62
-GET_REAL, 63
-

-INT_MAX, 64
-

-LET, 65
-LET_COMPLEX, 66
-LET_IMAG, 67
-LET_INT, 68
-LET_REAL, 69
-

-M_1_PI, 70
-M_1_SQRT2PI, 71
-M_2_PI, 72
-M_2_SQRTPI, 73
-M_2PI, 74
-M_E, 75
-M_EULER, 76
-M_LN10, 77
- - -M_LN2, 78
-M_LOG10E, 79
-M_LOG2E, 80
-M_PI, 81
-M_PI_2, 82
-M_PI_4, 83
-M_SQRT1_2, 84
-M_SQRT2, 85
-M_SQRT2_PI, 86
-M_SQRT2PI, 87
-MAX, 88
-MAX_INT, 89
-MGET, 90
-MGET_COMPLEX, 91
-MGET_INT, 92
-MIN, 93
-MLET, 94
-MLET_COMPLEX, 95
-MLET_INT, 96
-

-NAN, 97
-

-pnl_acosh, 98
-pnl_activate_mtherr, 99
-PNL_ALTERNATE, 100
-pnl_array_clone, 101
-pnl_array_copy, 102
-pnl_array_create, 103
-pnl_array_free, 104
-pnl_array_get, 105
-pnl_array_new, 106
-pnl_array_print, 107
-pnl_array_resize, 108
-pnl_array_set, 109
-pnl_asinh, 110
-pnl_atanh, 111
-pnl_band_mat_clone, 112
-pnl_band_mat_copy, 113
-pnl_band_mat_create, 114
-pnl_band_mat_create_from_mat, 115
-pnl_band_mat_div_band_mat_term, 116
-pnl_band_mat_div_scalar, 117
-pnl_band_mat_free, 118
-pnl_band_mat_get, 119
-pnl_band_mat_inv_term, 120
- - - -pnl_band_mat_lAxpby, 121
-pnl_band_mat_lget, 122
-pnl_band_mat_lu, 123
-pnl_band_mat_lu_syslin, 124
-pnl_band_mat_lu_syslin_inplace, 125
-pnl_band_mat_map, 126
-pnl_band_mat_map_band_mat_inplace, 127
-pnl_band_mat_map_inplace, 128
-pnl_band_mat_minus_band_mat, 129
-pnl_band_mat_minus_scalar, 130
-pnl_band_mat_mult_band_mat_term, 131
-pnl_band_mat_mult_scalar, 132
-pnl_band_mat_mult_vect_inplace, 133
-pnl_band_mat_new, 134
-PNL_BAND_MAT_OBJECT, 135
-pnl_band_mat_plus_band_mat, 136
-pnl_band_mat_plus_scalar, 137
-pnl_band_mat_print_as_full, 138
-pnl_band_mat_resize, 139
-pnl_band_mat_set, 140
-pnl_band_mat_set_all, 141
-pnl_band_mat_syslin, 142
-pnl_band_mat_syslin_inplace, 143
-pnl_band_mat_to_mat, 144
-pnl_basis_add_elt, 145
-pnl_basis_add_function, 146
-PNL_BASIS_CANONICAL, 147
-pnl_basis_clone, 148
-pnl_basis_copy, 149
-pnl_basis_create, 150
-pnl_basis_create_from_degree, 151
-pnl_basis_create_from_hyperbolic_degree, 152
-pnl_basis_create_from_prod_degree, 153
-pnl_basis_create_from_tensor, 154
-pnl_basis_create_local, 155
-pnl_basis_del_elt, 156
-pnl_basis_del_elt_i, 157
-pnl_basis_eval, 158
-pnl_basis_eval_D, 159
-pnl_basis_eval_D2, 160
-pnl_basis_eval_D2_vect, 161
-pnl_basis_eval_D_vect, 162
-pnl_basis_eval_derivs, 163
-pnl_basis_eval_derivs_vect, 164
-pnl_basis_eval_vect, 165
-pnl_basis_fit_ls, 166
-pnl_basis_free, 167
- - -PNL_BASIS_HERMITE, 168
-pnl_basis_i, 169
-pnl_basis_i_D, 170
-pnl_basis_i_D2, 171
-pnl_basis_i_D2_vect, 172
-pnl_basis_i_D_vect, 173
-pnl_basis_i_vect, 174
-pnl_basis_ik, 175
-pnl_basis_ik_vect, 176
-pnl_basis_new, 177
-PNL_BASIS_OBJECT, 178
-pnl_basis_print, 179
-pnl_basis_set_domain, 180
-pnl_basis_set_from_tensor, 181
-pnl_basis_set_reduced, 182
-PNL_BASIS_TCHEBYCHEV, 183
-pnl_basis_type_register, 184
-pnl_bessel_h1, 185
-pnl_bessel_h1_scaled, 186
-pnl_bessel_h2, 187
-pnl_bessel_h2_scaled, 188
-pnl_bessel_i, 189
-pnl_bessel_i_scaled, 190
-pnl_bessel_j, 191
-pnl_bessel_j_scaled, 192
-pnl_bessel_k, 193
-pnl_bessel_k_scaled, 194
-pnl_bessel_rati, 195
-pnl_bessel_y, 196
-pnl_bessel_y_scaled, 197
-pnl_bicg_solver_create, 198
-pnl_bicg_solver_free, 199
-pnl_bicg_solver_initialisation, 200
-pnl_bicg_solver_new, 201
-pnl_bicg_solver_solve, 202
-pnl_bs_call, 203
-pnl_bs_call_put, 204
-pnl_bs_gamma, 205
-pnl_bs_implicit_vol, 206
-pnl_bs_matrix_implicit_vol, 207
-pnl_bs_put, 208
-pnl_bs_vega, 209
-pnl_cdf2nor, 210
-pnl_cdf_bet, 211
-pnl_cdf_bin, 212
-pnl_cdf_chi, 213
-pnl_cdf_chn, 214
- - -pnl_cdf_f, 215
-pnl_cdf_fnc, 216
-pnl_cdf_gam, 217
-pnl_cdf_nbn, 218
-pnl_cdf_nor, 219
-pnl_cdf_poi, 220
-pnl_cdf_t, 221
-pnl_cdfbchi2n, 222
-pnl_cdfchi2n, 223
-pnl_cdfnor, 224
-pnl_cell_free, 225
-pnl_cell_new, 226
-pnl_cf_call_bs, 227
-pnl_cf_put_bs, 228
-pnl_cg_solver_create, 229
-pnl_cg_solver_free, 230
-pnl_cg_solver_initialisation, 231
-pnl_cg_solver_new, 232
-pnl_cg_solver_solve, 233
-pnl_complex_bessel_h1, 234
-pnl_complex_bessel_h1_scaled, 235
-pnl_complex_bessel_h2, 236
-pnl_complex_bessel_h2_scaled, 237
-pnl_complex_bessel_i, 238
-pnl_complex_bessel_i_scaled, 239
-pnl_complex_bessel_j, 240
-pnl_complex_bessel_j_scaled, 241
-pnl_complex_bessel_k, 242
-pnl_complex_bessel_k_scaled, 243
-pnl_complex_bessel_rati, 244
-pnl_complex_bessel_y, 245
-pnl_complex_bessel_y_scaled, 246
-pnl_complex_isequal, 247
-pnl_complex_isequal_abs, 248
-pnl_complex_isequal_rel, 249
-pnl_cosm1, 250
-pnl_deactivate_mtherr, 251
-pnl_expm1, 252
-pnl_fact, 253
-pnl_fft, 254
-pnl_fft2, 255
-pnl_fft2d, 256
-pnl_fft2d_inplace, 257
-pnl_fft_alloc_wspace, 258
-pnl_fft_inplace, 259
-pnl_fft_inplace_with_wspace, 260
-PNL_GET_PARENT_TYPE, 261
- - -PNL_GET_TYPE, 262
-PNL_GET_TYPENAME, 263
-pnl_gmres_solver_create, 264
-pnl_gmres_solver_free, 265
-pnl_gmres_solver_initialisation, 266
-pnl_gmres_solver_new, 267
-pnl_gmres_solver_solve, 268
-pnl_hmat_clone, 269
-pnl_hmat_copy, 270
-pnl_hmat_create, 271
-pnl_hmat_create_from_ptr, 272
-pnl_hmat_create_from_scalar, 273
-pnl_hmat_free, 274
-pnl_hmat_get, 275
-pnl_hmat_lget, 276
-pnl_hmat_mult_scalar, 277
-pnl_hmat_new, 278
-PNL_HMAT_OBJECT, 279
-pnl_hmat_plus_hmat, 280
-pnl_hmat_print, 281
-pnl_hmat_resize, 282
-pnl_hmat_set, 283
-pnl_ifft, 284
-pnl_ifft2, 285
-pnl_ifft2d, 286
-pnl_ifft2d_inplace, 287
-pnl_ifft_inplace, 288
-pnl_ifft_inplace_with_wspace, 289
-pnl_ilap_cdf_euler, 290
-pnl_ilap_euler, 291
-pnl_ilap_fft, 292
-pnl_ilap_gs, 293
-pnl_ilap_gs_basic, 294
-PNL_INF, 295
-pnl_integration, 296
-pnl_integration_2d, 297
-pnl_integration_GK, 298
-pnl_integration_GK2D, 299
-pnl_integration_qag, 300
-pnl_integration_qagp, 301
-pnl_integration_qng, 302
-pnl_integration_qng_2d, 303
-pnl_inv_cdfnor, 304
-pnl_iround, 305
-PNL_IS_EVEN, 306
-PNL_IS_ODD, 307
-pnl_isequal, 308
- - -pnl_isequal_abs, 309
-pnl_isequal_rel, 310
-pnl_isfinite, 311
-pnl_isinf, 312
-pnl_isnan, 313
-pnl_itrunc, 314
-pnl_lgamma, 315
-PNL_LIST_ARRAY, 316
-pnl_list_clone, 317
-pnl_list_concat, 318
-pnl_list_copy, 319
-pnl_list_free, 320
-pnl_list_get, 321
-pnl_list_insert_first, 322
-pnl_list_insert_last, 323
-pnl_list_new, 324
-PNL_LIST_OBJECT, 325
-pnl_list_print, 326
-pnl_list_remove_first, 327
-pnl_list_remove_i, 328
-pnl_list_remove_last, 329
-pnl_list_resize, 330
-pnl_log1p, 331
-pnl_lround, 332
-pnl_ltrunc, 333
-pnl_mat_add_row, 334
-pnl_mat_axpy, 335
-pnl_mat_bicg_solver_solve, 336
-pnl_mat_cg_solver_solve, 337
-pnl_mat_chol, 338
-pnl_mat_chol_syslin, 339
-pnl_mat_chol_syslin_inplace, 340
-pnl_mat_chol_syslin_mat, 341
-pnl_mat_clone, 342
-pnl_mat_col_permute, 343
-pnl_mat_complex_create_from_mat, 344
-pnl_mat_copy, 345
-pnl_mat_create, 346
-pnl_mat_create_diag, 347
-pnl_mat_create_diag_from_ptr, 348
-pnl_mat_create_from_file, 349
-pnl_mat_create_from_list, 350
-pnl_mat_create_from_ptr, 351
-pnl_mat_create_from_scalar, 352
-pnl_mat_create_from_sp_mat, 353
-pnl_mat_create_from_zero, 354
-pnl_mat_cross, 355
- - -pnl_mat_cumprod, 356
-pnl_mat_cumsum, 357
-pnl_mat_del_row, 358
-pnl_mat_dgemm, 359
-pnl_mat_dgemv, 360
-pnl_mat_dger, 361
-pnl_mat_div_mat_term, 362
-pnl_mat_div_scalar, 363
-pnl_mat_eigen, 364
-pnl_mat_exp, 365
-pnl_mat_extract_subblock, 366
-pnl_mat_find, 367
-pnl_mat_fprint, 368
-pnl_mat_fprint_nsp, 369
-pnl_mat_free, 370
-pnl_mat_get, 371
-pnl_mat_get_col, 372
-pnl_mat_get_row, 373
-pnl_mat_gmres_solver_solve, 374
-pnl_mat_inverse, 375
-pnl_mat_inverse_with_chol, 376
-pnl_mat_isequal, 377
-pnl_mat_isequal_abs, 378
-pnl_mat_isequal_rel, 379
-pnl_mat_kron_mat, 380
-pnl_mat_kron_mat_inplace, 381
-pnl_mat_lAxpby, 382
-pnl_mat_lget, 383
-pnl_mat_log, 384
-pnl_mat_lower_inverse, 385
-pnl_mat_lower_syslin, 386
-pnl_mat_ls, 387
-pnl_mat_ls_mat, 388
-pnl_mat_lu, 389
-pnl_mat_lu_syslin, 390
-pnl_mat_lu_syslin_inplace, 391
-pnl_mat_lu_syslin_mat, 392
-pnl_mat_map, 393
-pnl_mat_map_inplace, 394
-pnl_mat_map_mat, 395
-pnl_mat_map_mat_inplace, 396
-pnl_mat_max, 397
-pnl_mat_max_index, 398
-pnl_mat_min, 399
-pnl_mat_min_index, 400
-pnl_mat_minmax, 401
-pnl_mat_minmax_index, 402
- - -pnl_mat_minus_mat, 403
-pnl_mat_minus_scalar, 404
-pnl_mat_mult_mat, 405
-pnl_mat_mult_mat_inplace, 406
-pnl_mat_mult_mat_term, 407
-pnl_mat_mult_scalar, 408
-pnl_mat_mult_vect, 409
-pnl_mat_mult_vect_inplace, 410
-pnl_mat_mult_vect_transpose, 411
-pnl_mat_mult_vect_transpose_inplace, 412
-pnl_mat_new, 413
-PNL_MAT_OBJECT, 414
-pnl_mat_pchol, 415
-pnl_mat_plus_mat, 416
-pnl_mat_plus_scalar, 417
-pnl_mat_print, 418
-pnl_mat_print_nsp, 419
-pnl_mat_prod, 420
-pnl_mat_prod_vect, 421
-pnl_mat_qr, 422
-pnl_mat_qr_syslin, 423
-pnl_mat_qsort, 424
-pnl_mat_qsort_index, 425
-pnl_mat_rand_normal, 426
-pnl_mat_rand_uni, 427
-pnl_mat_rand_uni2, 428
-pnl_mat_resize, 429
-pnl_mat_rng_bernoulli, 430
-pnl_mat_rng_normal, 431
-pnl_mat_rng_poisson, 432
-pnl_mat_rng_uni, 433
-pnl_mat_rng_uni2, 434
-pnl_mat_row_permute, 435
-pnl_mat_scalar_prod, 436
-pnl_mat_set, 437
-pnl_mat_set_all, 438
-pnl_mat_set_col, 439
-pnl_mat_set_col_from_ptr, 440
-pnl_mat_set_diag, 441
-pnl_mat_set_from_ptr, 442
-pnl_mat_set_id, 443
-pnl_mat_set_row, 444
-pnl_mat_set_row_from_ptr, 445
-pnl_mat_set_subblock, 446
-pnl_mat_set_zero, 447
-pnl_mat_sq_transpose, 448
-pnl_mat_sum, 449
- - -pnl_mat_sum_vect, 450
-pnl_mat_swap_rows, 451
-pnl_mat_syslin, 452
-pnl_mat_syslin_inplace, 453
-pnl_mat_syslin_mat, 454
-pnl_mat_tr, 455
-pnl_mat_trace, 456
-pnl_mat_transpose, 457
-pnl_mat_upper_inverse, 458
-pnl_mat_upper_syslin, 459
-pnl_mat_wrap_array, 460
-pnl_mat_wrap_hmat, 461
-pnl_mat_wrap_mat_rows, 462
-pnl_mat_wrap_vect, 463
-pnl_multiroot_newton, 464
-pnl_nan, 465
-PNL_NEGINF, 466
-pnl_neginf, 467
-pnl_normal_density, 468
-PNL_OBJECT, 469
-pnl_object_create, 470
-pnl_object_load, 471
-pnl_object_load_into_list, 472
-pnl_object_mpi_bcast, 473
-pnl_object_mpi_irecv, 474
-pnl_object_mpi_isend, 475
-pnl_object_mpi_pack, 476
-pnl_object_mpi_pack_size, 477
-pnl_object_mpi_recv, 478
-pnl_object_mpi_reduce, 479
-pnl_object_mpi_send, 480
-pnl_object_mpi_ssend, 481
-pnl_object_mpi_unpack, 482
-pnl_object_save, 483
-pnl_ode_rkf45, 484
-pnl_ode_rkf45_step, 485
-pnl_optim_intpoints_bfgs_solve, 486
-pnl_optim_linprog, 487
-pnl_optim_linprog_sp, 488
-pnl_permutation_create, 489
-pnl_permutation_fprint, 490
-pnl_permutation_free, 491
-pnl_permutation_inverse, 492
-pnl_permutation_new, 493
-pnl_permutation_print, 494
-PNL_POSINF, 495
-pnl_posinf, 496
- - -pnl_pow_i, 497
-pnl_rand_bernoulli, 498
-pnl_rand_bessel, 499
-pnl_rand_chi2, 500
-pnl_rand_exp, 501
-pnl_rand_gamma, 502
-pnl_rand_gauss, 503
-pnl_rand_init, 504
-pnl_rand_name, 505
-pnl_rand_normal, 506
-pnl_rand_or_quasi, 507
-pnl_rand_poisson, 508
-pnl_rand_poisson1, 509
-pnl_rand_sseed, 510
-pnl_rand_uni, 511
-pnl_rand_uni_ab, 512
-pnl_real_fft, 513
-pnl_real_fft2, 514
-pnl_real_fft2d, 515
-pnl_real_fft_inplace, 516
-pnl_real_fft_inplace_with_wspace, 517
-pnl_real_fft_with_wspace, 518
-pnl_real_ifft, 519
-pnl_real_ifft2, 520
-pnl_real_ifft2d, 521
-pnl_real_ifft_inplace, 522
-pnl_real_ifft_inplace_with_wspace, 523
-pnl_real_ifft_with_wspace, 524
-pnl_rng_bernoulli, 525
-pnl_rng_bessel, 526
-pnl_rng_chi2, 527
-pnl_rng_clone, 528
-pnl_rng_copy, 529
-pnl_rng_create, 530
-pnl_rng_create_from_file, 531
-pnl_rng_dblexp, 532
-pnl_rng_dcmt_create_array, 533
-pnl_rng_dcmt_create_array_id, 534
-pnl_rng_dcmt_create_id, 535
-pnl_rng_exp, 536
-pnl_rng_free, 537
-pnl_rng_gamma, 538
-pnl_rng_gauss, 539
-pnl_rng_get_from_id, 540
-pnl_rng_init, 541
-pnl_rng_invgauss, 542
-pnl_rng_lognormal, 543
- - -pnl_rng_ncchi2, 544
-pnl_rng_new, 545
-pnl_rng_normal, 546
-PNL_RNG_OBJECT, 547
-pnl_rng_poisson, 548
-pnl_rng_poisson1, 549
-pnl_rng_save_to_file, 550
-pnl_rng_sdim, 551
-pnl_rng_sseed, 552
-pnl_rng_uni, 553
-pnl_rng_uni_ab, 554
-pnl_root_bisection, 555
-pnl_root_brent, 556
-pnl_root_fsolve, 557
-pnl_root_fsolve_lsq, 558
-pnl_root_newton, 559
-pnl_root_newton_bisection, 560
-pnl_round, 561
-pnl_sf_choose, 562
-pnl_sf_complex_dawson, 563
-pnl_sf_complex_erf, 564
-pnl_sf_complex_erfc, 565
-pnl_sf_complex_erfcx, 566
-pnl_sf_complex_erfi, 567
-pnl_sf_dawson, 568
-pnl_sf_erf, 569
-pnl_sf_erfc, 570
-pnl_sf_erfcx, 571
-pnl_sf_erfi, 572
-pnl_sf_expint_En, 573
-pnl_sf_fact, 574
-pnl_sf_gamma, 575
-pnl_sf_gamma_inc, 576
-pnl_sf_gamma_inc_P, 577
-pnl_sf_gamma_inc_Q, 578
-pnl_sf_hyperg_0F1, 579
-pnl_sf_hyperg_1F1, 580
-pnl_sf_hyperg_2F0, 581
-pnl_sf_hyperg_2F1, 582
-pnl_sf_hyperg_U, 583
-pnl_sf_log_erf, 584
-pnl_sf_log_erfc, 585
-pnl_sf_log_gamma, 586
-pnl_sf_log_gamma_sgn, 587
-pnl_sf_psi, 588
-pnl_sf_w, 589
-pnl_sf_w_im, 590
- - -PNL_SIGN, 591
-pnl_sp_mat_clone, 592
-pnl_sp_mat_copy, 593
-pnl_sp_mat_create, 594
-pnl_sp_mat_create_from_file, 595
-pnl_sp_mat_create_from_mat, 596
-pnl_sp_mat_div_scalar, 597
-pnl_sp_mat_fprint, 598
-pnl_sp_mat_free, 599
-pnl_sp_mat_get, 600
-pnl_sp_mat_isequal, 601
-pnl_sp_mat_isequal_abs, 602
-pnl_sp_mat_isequal_rel, 603
-pnl_sp_mat_kron, 604
-pnl_sp_mat_kron_inplace, 605
-pnl_sp_mat_lAxpby, 606
-pnl_sp_mat_minus_scalar, 607
-pnl_sp_mat_mult_scalar, 608
-pnl_sp_mat_mult_vect, 609
-pnl_sp_mat_new, 610
-PNL_SP_MAT_OBJECT, 611
-pnl_sp_mat_plus_scalar, 612
-pnl_sp_mat_plus_sp_mat, 613
-pnl_sp_mat_plus_sp_mat_inplace, 614
-pnl_sp_mat_print, 615
-pnl_sp_mat_resize, 616
-pnl_sp_mat_set, 617
-pnl_tgamma, 618
-pnl_tridiag_mat_clone, 619
-pnl_tridiag_mat_copy, 620
-pnl_tridiag_mat_create, 621
-pnl_tridiag_mat_create_from_mat, 622
-pnl_tridiag_mat_create_from_ptr, 623
-pnl_tridiag_mat_create_from_scalar, 624
-pnl_tridiag_mat_create_from_two_scalar, 625
-pnl_tridiag_mat_div_scalar, 626
-pnl_tridiag_mat_div_tridiag_mat_term, 627
-pnl_tridiag_mat_fprint, 628
-pnl_tridiag_mat_free, 629
-pnl_tridiag_mat_get, 630
-pnl_tridiag_mat_lAxpby, 631
-pnl_tridiag_mat_lget, 632
-pnl_tridiag_mat_lu_clone, 633
-pnl_tridiag_mat_lu_compute, 634
-pnl_tridiag_mat_lu_copy, 635
-pnl_tridiag_mat_lu_create, 636
-pnl_tridiag_mat_lu_free, 637
- - -pnl_tridiag_mat_lu_new, 638
-pnl_tridiag_mat_lu_resize, 639
-pnl_tridiag_mat_lu_syslin, 640
-pnl_tridiag_mat_lu_syslin_inplace, 641
-pnl_tridiag_mat_map_inplace, 642
-pnl_tridiag_mat_map_tridiag_mat_inplace, 643
-pnl_tridiag_mat_minus_scalar, 644
-pnl_tridiag_mat_minus_tridiag_mat, 645
-pnl_tridiag_mat_mult_scalar, 646
-pnl_tridiag_mat_mult_tridiag_mat_term, 647
-pnl_tridiag_mat_mult_vect, 648
-pnl_tridiag_mat_mult_vect_inplace, 649
-pnl_tridiag_mat_new, 650
-pnl_tridiag_mat_plus_scalar, 651
-pnl_tridiag_mat_plus_tridiag_mat, 652
-pnl_tridiag_mat_print, 653
-pnl_tridiag_mat_resize, 654
-pnl_tridiag_mat_scalar_prod, 655
-pnl_tridiag_mat_set, 656
-pnl_tridiag_mat_syslin, 657
-pnl_tridiag_mat_syslin_inplace, 658
-pnl_tridiag_mat_to_mat, 659
-PNL_TRIDIAGMAT_OBJECT, 660
-pnl_trunc, 661
-pnl_vect_axpby, 662
-pnl_vect_clone, 663
-pnl_vect_compact_copy, 664
-pnl_vect_compact_create, 665
-pnl_vect_compact_create_from_ptr, 666
-pnl_vect_compact_free, 667
-pnl_vect_compact_get, 668
-pnl_vect_compact_new, 669
-pnl_vect_compact_resize, 670
-pnl_vect_compact_set_all, 671
-pnl_vect_compact_set_ptr, 672
-pnl_vect_compact_to_pnl_vect, 673
-pnl_vect_complex_create_from_array, 674
-pnl_vect_complex_get_imag, 675
-pnl_vect_complex_get_real, 676
-pnl_vect_complex_lget_imag, 677
-pnl_vect_complex_lget_real, 678
-pnl_vect_complex_mult_double, 679
-pnl_vect_complex_set_imag, 680
-pnl_vect_complex_set_real, 681
-pnl_vect_complex_split_in_array, 682
-pnl_vect_complex_split_in_vect, 683
-pnl_vect_copy, 684
- - -pnl_vect_create, 685
-pnl_vect_create_from_file, 686
-pnl_vect_create_from_list, 687
-pnl_vect_create_from_mat, 688
-pnl_vect_create_from_ptr, 689
-pnl_vect_create_from_scalar, 690
-pnl_vect_create_from_zero, 691
-pnl_vect_create_submat, 692
-pnl_vect_create_subvect, 693
-pnl_vect_create_subvect_with_ind, 694
-pnl_vect_cross, 695
-pnl_vect_cumprod, 696
-pnl_vect_cumsum, 697
-pnl_vect_dist, 698
-pnl_vect_div_scalar, 699
-pnl_vect_div_vect_term, 700
-pnl_vect_eq_all, 701
-pnl_vect_extract_submat, 702
-pnl_vect_extract_subvect, 703
-pnl_vect_extract_subvect_with_ind, 704
-pnl_vect_find, 705
-pnl_vect_fprint, 706
-pnl_vect_fprint_asrow, 707
-pnl_vect_fprint_nsp, 708
-pnl_vect_free, 709
-pnl_vect_get, 710
-pnl_vect_inv_term, 711
-pnl_vect_isequal, 712
-pnl_vect_isequal_abs, 713
-pnl_vect_isequal_rel, 714
-pnl_vect_lget, 715
-pnl_vect_map, 716
-pnl_vect_map_inplace, 717
-pnl_vect_map_vect, 718
-pnl_vect_map_vect_inplace, 719
-pnl_vect_max, 720
-pnl_vect_max_index, 721
-pnl_vect_min, 722
-pnl_vect_min_index, 723
-pnl_vect_minmax, 724
-pnl_vect_minmax_index, 725
-pnl_vect_minus, 726
-pnl_vect_minus_scalar, 727
-pnl_vect_minus_vect, 728
-pnl_vect_mult_scalar, 729
-pnl_vect_mult_vect_term, 730
-pnl_vect_new, 731
- - -pnl_vect_norm_infty, 732
-pnl_vect_norm_one, 733
-pnl_vect_norm_two, 734
-PNL_VECT_OBJECT, 735
-pnl_vect_permute, 736
-pnl_vect_permute_inplace, 737
-pnl_vect_permute_inverse, 738
-pnl_vect_permute_inverse_inplace, 739
-pnl_vect_plus_scalar, 740
-pnl_vect_plus_vect, 741
-pnl_vect_print, 742
-pnl_vect_print_asrow, 743
-pnl_vect_print_nsp, 744
-pnl_vect_prod, 745
-pnl_vect_qsort, 746
-pnl_vect_qsort_index, 747
-pnl_vect_rand_normal, 748
-pnl_vect_rand_normal_d, 749
-pnl_vect_rand_uni, 750
-pnl_vect_rand_uni_d, 751
-pnl_vect_resize, 752
-pnl_vect_resize_from_ptr, 753
-pnl_vect_resize_from_scalar, 754
-pnl_vect_reverse, 755
-pnl_vect_rng_bernoulli, 756
-pnl_vect_rng_bernoulli_d, 757
-pnl_vect_rng_normal, 758
-pnl_vect_rng_normal_d, 759
-pnl_vect_rng_poisson, 760
-pnl_vect_rng_poisson_d, 761
-pnl_vect_rng_uni, 762
-pnl_vect_rng_uni_d, 763
-pnl_vect_scalar_prod, 764
-pnl_vect_set, 765
-pnl_vect_set_all, 766
-pnl_vect_set_subblock, 767
-pnl_vect_set_zero, 768
-pnl_vect_sum, 769
-pnl_vect_swap_elements, 770
-pnl_vect_wrap_array, 771
-pnl_vect_wrap_hmat, 772
-pnl_vect_wrap_mat, 773
-pnl_vect_wrap_mat_row, 774
-pnl_vect_wrap_subvect, 775
-pnl_vect_wrap_subvect_with_last, 776
-

-RCadd, 777
- - -RCdiv, 778
-RCmul, 779
-RCsub, 780
-

-SQR, 781
-Structs
-    PnlArray, 782
-    PnlBandMat, 783
-    PnlBasis, 784
-    PnlBicgSolver, 785
-    PnlCell, 786
-    PnlCgSolver, 787
-    PnlCmplxFunc, 788
-    PnlFunc, 789
-    PnlFunc2D, 790
-    PnlFuncDFunc, 791
-    PnlGmresSolver, 792
-    PnlHmat, 793
-    PnlHmatComplex, 794
-    PnlHmatInt, 795
-    PnlIterationBase, 796
-    PnlList, 797
-    PnlMat, 798
-    PnlMatComplex, 799
-    PnlMatInt, 800
-    PnlObject, 801
-    PnlODEFunc, 802
-    PnlPermutation, 803
-    PnlRnFuncR, 804
-    PnlRnFuncRm, 805
-    PnlRnFuncRmDFunc, 806
-    PnlRnFuncRn, 807
-    PnlRnFuncRnDFunc, 808
-    PnlRng, 809
-    PnlSpMat, 810
-    PnlSpMatComplex, 811
-    PnlSpMatInt, 812
-    PnlTridiagMat, 813
-    PnlTridiagMatLU, 814
-    PnlVect, 815
-    PnlVectCompact, 816
-    PnlVectComplex, 817
-    PnlVectInt, 818
-

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Index

-
ABS, Index +

+ABS, 1
-

+

C_op_amcb, 2
C_op_amib, 34
Complex_polar, 35
+ + + CONE, 36
Conj, 37
- - - Cpow, 38
Cpow_real, 71
M_2_PI, 72
+ + + M_2_SQRTPI, 73
M_2PI, 74
- - - M_E, 75
M_EULER, pnl_acosh, 98
pnl_activate_mtherr, 99
+href="pnl-manualse14.html#dx16-135002" >99
PNL_ALTERNATE, 100
pnl_array_clone, 101
+href="pnl-manualse2.html#dx4-14017" >101
pnl_array_copy, 102
+href="pnl-manualse2.html#dx4-14016" >102
pnl_array_create, 103
+href="pnl-manualse2.html#dx4-14015" >103
pnl_array_free, 104
+href="pnl-manualse2.html#dx4-14018" >104
pnl_array_get, 105
+href="pnl-manualse2.html#dx4-14020" >105
pnl_array_new, 106
+href="pnl-manualse2.html#dx4-14014" >106
pnl_array_print, 107
+href="pnl-manualse2.html#dx4-14022" >107
pnl_array_resize, 108
+href="pnl-manualse2.html#dx4-14019" >108
pnl_array_set, 109
+href="pnl-manualse2.html#dx4-14021" >109
pnl_asinh, 110
pnl_atanh, 114
pnl_band_mat_create_from_mat, 115
+ + + pnl_band_mat_div_band_mat_term, 116
pnl_band_mat_div_scalar, 117
- - - pnl_band_mat_free, 118
pnl_band_mat_get, pnl_band_mat_new, 134
PNL_BAND_MAT_OBJECT, 135
+href="pnl-manualse2.html#dx4-12026" >135
pnl_band_mat_plus_band_mat, 136
pnl_band_mat_plus_scalar, pnl_band_mat_to_mat, 144
pnl_basis_add_elt, 145
+href="pnl-manualse7.html#dx9-109007" >145
pnl_basis_add_function, 146
+href="pnl-manualse7.html#dx9-108001" >146
PNL_BASIS_CANONICAL, 147
pnl_basis_clone, 148
+href="pnl-manualse7.html#dx9-109002" >148
pnl_basis_copy, 149
+href="pnl-manualse7.html#dx9-109003" >149
pnl_basis_create, 150
+href="pnl-manualse7.html#dx9-106005" >150
pnl_basis_create_from_degree, 151
+href="pnl-manualse7.html#dx9-106006" >151
pnl_basis_create_from_hyperbolic_degree, 152
+href="pnl-manualse7.html#dx9-106008" >152
pnl_basis_create_from_prod_degree, 153
+href="pnl-manualse7.html#dx9-106007" >153
pnl_basis_create_from_tensor, 154
+href="pnl-manualse7.html#dx9-106009" >154
pnl_basis_del_elt, 155
+href="pnl-manualse7.html#dx9-109005" >155
pnl_basis_del_elt_i, 156
+href="pnl-manualse7.html#dx9-109006" >156
pnl_basis_eval, 157
+href="pnl-manualse7.html#dx9-109035" >157
pnl_basis_eval_D, 158
+href="pnl-manualse7.html#dx9-109036" >158
pnl_basis_eval_D2, 159
+href="pnl-manualse7.html#dx9-109037" >159
pnl_basis_eval_D2_vect, 160
+href="pnl-manualse7.html#dx9-109030" >160
pnl_basis_eval_D_vect, 161
+href="pnl-manualse7.html#dx9-109029" >161
pnl_basis_eval_derivs, 162
-pnl_basis_eval_derivs_vect, 163
-pnl_basis_eval_vect, 164
+href="pnl-manualse7.html#dx9-109038" >162
+pnl_basis_eval_derivs_vect, 163
+pnl_basis_eval_vect, 164
pnl_basis_fit_ls, 165
+href="pnl-manualse7.html#dx9-109022" >165
pnl_basis_free, 166
+href="pnl-manualse7.html#dx9-109004" >166
PNL_BASIS_HERMITE, 167
pnl_basis_i, 168
+href="pnl-manualse7.html#dx9-109032" >168
pnl_basis_i_D, 169
+href="pnl-manualse7.html#dx9-109033" >169
pnl_basis_i_D2, 170
+href="pnl-manualse7.html#dx9-109034" >170
pnl_basis_i_D2_vect, 171
+href="pnl-manualse7.html#dx9-109026" >171
pnl_basis_i_D_vect, 172
+href="pnl-manualse7.html#dx9-109025" >172
pnl_basis_i_vect, 173
+href="pnl-manualse7.html#dx9-109024" >173
pnl_basis_ik, 174
+href="pnl-manualse7.html#dx9-109031" >174
pnl_basis_ik_vect, 175
+href="pnl-manualse7.html#dx9-109023" >175
+pnl_basis_local_create, 176
+pnl_basis_local_create_regular, 177
+pnl_basis_local_get_index, 178
pnl_basis_new, 176
+href="pnl-manualse7.html#dx9-109001" >179
PNL_BASIS_OBJECT, 177
+href="pnl-manualse2.html#dx4-12028" >180
pnl_basis_print, 178
+href="pnl-manualse7.html#dx9-109018" >181
pnl_basis_set_domain, 179
+href="pnl-manualse7.html#dx9-109020" >182
pnl_basis_set_from_tensor, 180
+href="pnl-manualse7.html#dx9-106010" >183
pnl_basis_set_reduced, 181
+href="pnl-manualse7.html#dx9-109021" >184
PNL_BASIS_TCHEBYCHEV, 182
+href="pnl-manualse7.html#dx9-106003" >185
pnl_basis_type_register, 183
+href="pnl-manualse7.html#dx9-109008" >186
pnl_bessel_h1, 184
+href="pnl-manualse14.html#dx16-136010" >187
pnl_bessel_h1_scaled, 185
+href="pnl-manualse14.html#dx16-136011" >188
pnl_bessel_h2, 186
+href="pnl-manualse14.html#dx16-136012" >189
pnl_bessel_h2_scaled, 187
+href="pnl-manualse14.html#dx16-136013" >190
pnl_bessel_i, 188
+href="pnl-manualse14.html#dx16-136001" >191
pnl_bessel_i_scaled, 189
+href="pnl-manualse14.html#dx16-136002" >192
pnl_bessel_j, 190
+href="pnl-manualse14.html#dx16-136004" >193
pnl_bessel_j_scaled, 191
+href="pnl-manualse14.html#dx16-136005" >194
pnl_bessel_k, 192
+href="pnl-manualse14.html#dx16-136008" >195
pnl_bessel_k_scaled, 193
+href="pnl-manualse14.html#dx16-136009" >196
pnl_bessel_rati, 194
+href="pnl-manualse14.html#dx16-136003" >197
pnl_bessel_y, 195
+href="pnl-manualse14.html#dx16-136006" >198
pnl_bessel_y_scaled, 196
+href="pnl-manualse14.html#dx16-136007" >199
pnl_bicg_solver_create, 197
+href="pnl-manualse4.html#dx6-97002" >200
pnl_bicg_solver_free, 198
+href="pnl-manualse4.html#dx6-97004" >201
pnl_bicg_solver_initialisation, 199
+href="pnl-manualse4.html#dx6-97003" >202
pnl_bicg_solver_new, 200
+href="pnl-manualse4.html#dx6-97001" >203
pnl_bicg_solver_solve, 201
+href="pnl-manualse4.html#dx6-97005" >204
pnl_bs_call, 202
+href="pnl-manualse16.html#dx18-149003" >205
pnl_bs_call_put, 203
+href="pnl-manualse16.html#dx18-149005" >206
pnl_bs_gamma, 204
+href="pnl-manualse16.html#dx18-149007" >207
pnl_bs_implicit_vol, 205
+href="pnl-manualse16.html#dx18-149008" >208
pnl_bs_matrix_implicit_vol, 206
+href="pnl-manualse16.html#dx18-149009" >209
+ + + pnl_bs_put, 207
+href="pnl-manualse16.html#dx18-149004" >210
pnl_bs_vega, 208
+href="pnl-manualse16.html#dx18-149006" >211
pnl_cdf2nor, 209
+href="pnl-manualse5.html#dx7-100016" >212
pnl_cdf_bet, 210
+href="pnl-manualse5.html#dx7-100001" >213
pnl_cdf_bin, 211
- - - +href="pnl-manualse5.html#dx7-100002" >214
pnl_cdf_chi, 212
+href="pnl-manualse5.html#dx7-100003" >215
pnl_cdf_chn, 213
+href="pnl-manualse5.html#dx7-100004" >216
pnl_cdf_f, 214
+href="pnl-manualse5.html#dx7-100005" >217
pnl_cdf_fnc, 215
+href="pnl-manualse5.html#dx7-100006" >218
pnl_cdf_gam, 216
+href="pnl-manualse5.html#dx7-100007" >219
pnl_cdf_nbn, 217
+href="pnl-manualse5.html#dx7-100008" >220
pnl_cdf_nor, 218
+href="pnl-manualse5.html#dx7-100009" >221
pnl_cdf_poi, 219
+href="pnl-manualse5.html#dx7-100010" >222
pnl_cdf_t, 220
+href="pnl-manualse5.html#dx7-100011" >223
pnl_cdfbchi2n, 221
+href="pnl-manualse5.html#dx7-100013" >224
pnl_cdfchi2n, 222
+href="pnl-manualse5.html#dx7-100012" >225
pnl_cdfnor, 223
+href="pnl-manualse5.html#dx7-100015" >226
pnl_cell_free, 224
+href="pnl-manualse2.html#dx4-13033" >227
pnl_cell_new, 225
+href="pnl-manualse2.html#dx4-13029" >228
pnl_cf_call_bs, 226
+href="pnl-manualse16.html#dx18-149001" >229
pnl_cf_put_bs, 227
+href="pnl-manualse16.html#dx18-149002" >230
pnl_cg_solver_create, 228
+href="pnl-manualse4.html#dx6-96002" >231
pnl_cg_solver_free, 229
+href="pnl-manualse4.html#dx6-96004" >232
pnl_cg_solver_initialisation, 230
+href="pnl-manualse4.html#dx6-96003" >233
pnl_cg_solver_new, 231
+href="pnl-manualse4.html#dx6-96001" >234
pnl_cg_solver_solve, 232
+href="pnl-manualse4.html#dx6-96005" >235
pnl_complex_bessel_h1, 233
+href="pnl-manualse14.html#dx16-137010" >236
pnl_complex_bessel_h1_scaled, 234
+href="pnl-manualse14.html#dx16-137011" >237
pnl_complex_bessel_h2, 235
+href="pnl-manualse14.html#dx16-137012" >238
pnl_complex_bessel_h2_scaled, 236
+href="pnl-manualse14.html#dx16-137013" >239
pnl_complex_bessel_i, 237
+href="pnl-manualse14.html#dx16-137001" >240
pnl_complex_bessel_i_scaled, 238
+href="pnl-manualse14.html#dx16-137002" >241
pnl_complex_bessel_j, 239
+href="pnl-manualse14.html#dx16-137004" >242
pnl_complex_bessel_j_scaled, 240
+href="pnl-manualse14.html#dx16-137005" >243
pnl_complex_bessel_k, 241
+href="pnl-manualse14.html#dx16-137008" >244
pnl_complex_bessel_k_scaled, 242
+href="pnl-manualse14.html#dx16-137009" >245
pnl_complex_bessel_rati, 243
+href="pnl-manualse14.html#dx16-137003" >246
pnl_complex_bessel_y, 244
+href="pnl-manualse14.html#dx16-137006" >247
pnl_complex_bessel_y_scaled, 245
+href="pnl-manualse14.html#dx16-137007" >248
pnl_complex_isequal, 246
+href="pnl-manualse3.html#dx5-26043" >249
pnl_complex_isequal_abs, 247
+href="pnl-manualse3.html#dx5-26041" >250
pnl_complex_isequal_rel, 248
+href="pnl-manualse3.html#dx5-26042" >251
pnl_cosm1, 249
+href="pnl-manualse3.html#dx5-21009" >252
pnl_deactivate_mtherr, 250
+href="pnl-manualse14.html#dx16-135001" >253
pnl_expm1, 251
+href="pnl-manualse3.html#dx5-21008" >254
pnl_fact, 252
+href="pnl-manualse3.html#dx5-21001" >255
pnl_fft, 253
+href="pnl-manualse9.html#dx11-117003" >256
+ + + pnl_fft2, 254
+href="pnl-manualse9.html#dx11-117005" >257
pnl_fft2d, 255
+href="pnl-manualse9.html#dx11-117015" >258
pnl_fft2d_inplace, 256
+href="pnl-manualse9.html#dx11-117013" >259
pnl_fft_alloc_wspace, 257
+href="pnl-manualse9.html#dx11-118001" >260
pnl_fft_inplace, 258
- - - +href="pnl-manualse9.html#dx11-117001" >261
pnl_fft_inplace_with_wspace, 259
+href="pnl-manualse9.html#dx11-118002" >262
PNL_GET_PARENT_TYPE, 260
+href="pnl-manualse2.html#dx4-12034" >263
PNL_GET_TYPE, 261
+href="pnl-manualse2.html#dx4-12033" >264
PNL_GET_TYPENAME, 262
+href="pnl-manualse2.html#dx4-12032" >265
pnl_gmres_solver_create, 263
+href="pnl-manualse4.html#dx6-98002" >266
pnl_gmres_solver_free, 264
+href="pnl-manualse4.html#dx6-98004" >267
pnl_gmres_solver_initialisation, 265
+href="pnl-manualse4.html#dx6-98003" >268
pnl_gmres_solver_new, 266
+href="pnl-manualse4.html#dx6-98001" >269
pnl_gmres_solver_solve, 267
+href="pnl-manualse4.html#dx6-98005" >270
pnl_hmat_clone, 268
+href="pnl-manualse4.html#dx6-89007" >271
pnl_hmat_copy, 269
+href="pnl-manualse4.html#dx6-89006" >272
pnl_hmat_create, 270
+href="pnl-manualse4.html#dx6-89002" >273
pnl_hmat_create_from_ptr, 271
+href="pnl-manualse4.html#dx6-89004" >274
pnl_hmat_create_from_scalar, 272
+href="pnl-manualse4.html#dx6-89003" >275
pnl_hmat_free, 273
+href="pnl-manualse4.html#dx6-89005" >276
pnl_hmat_get, 274
+href="pnl-manualse4.html#dx6-90002" >277
pnl_hmat_lget, 275
+href="pnl-manualse4.html#dx6-90003" >278
pnl_hmat_mult_scalar, 276
+href="pnl-manualse4.html#dx6-92002" >279
pnl_hmat_new, 277
+href="pnl-manualse4.html#dx6-89001" >280
PNL_HMAT_OBJECT, 278
+href="pnl-manualse2.html#dx4-12025" >281
pnl_hmat_plus_hmat, 279
+href="pnl-manualse4.html#dx6-92001" >282
pnl_hmat_print, 280
+href="pnl-manualse4.html#dx6-91001" >283
pnl_hmat_resize, 281
+href="pnl-manualse4.html#dx6-89008" >284
pnl_hmat_set, 282
+href="pnl-manualse4.html#dx6-90001" >285
pnl_ifft, 283
+href="pnl-manualse9.html#dx11-117004" >286
pnl_ifft2, 284
+href="pnl-manualse9.html#dx11-117006" >287
pnl_ifft2d, 285
+href="pnl-manualse9.html#dx11-117016" >288
pnl_ifft2d_inplace, 286
+href="pnl-manualse9.html#dx11-117014" >289
pnl_ifft_inplace, 287
+href="pnl-manualse9.html#dx11-117002" >290
pnl_ifft_inplace_with_wspace, 288
+href="pnl-manualse9.html#dx11-118003" >291
pnl_ilap_cdf_euler, 289
+href="pnl-manualse10.html#dx12-119008" >292
pnl_ilap_euler, 290
+href="pnl-manualse10.html#dx12-119007" >293
pnl_ilap_fft, 291
+href="pnl-manualse10.html#dx12-119009" >294
pnl_ilap_gs, 292
+href="pnl-manualse10.html#dx12-119010" >295
pnl_ilap_gs_basic, 293
+href="pnl-manualse10.html#dx12-119011" >296
PNL_INF, 294
+href="pnl-manualse3.html#dx5-17026" >297
pnl_integration, 295
+href="pnl-manualse8.html#dx10-112001" >298
pnl_integration_2d, 296
+href="pnl-manualse8.html#dx10-112002" >299
pnl_integration_GK, 297
+href="pnl-manualse8.html#dx10-112004" >300
pnl_integration_GK2D, 298
+href="pnl-manualse8.html#dx10-112006" >301
pnl_integration_qag, 299
+href="pnl-manualse8.html#dx10-112007" >302
pnl_integration_qagp, 300
+href="pnl-manualse8.html#dx10-112008" >303
+ + + pnl_integration_qng, 301
+href="pnl-manualse8.html#dx10-112003" >304
pnl_integration_qng_2d, 302
+href="pnl-manualse8.html#dx10-112005" >305
pnl_inv_cdfnor, 303
+href="pnl-manualse5.html#dx7-100017" >306
pnl_iround, 304
+href="pnl-manualse3.html#dx5-20005" >307
PNL_IS_EVEN, 305
- - - +href="pnl-manualse3.html#dx5-18002" >308
PNL_IS_ODD, 306
+href="pnl-manualse3.html#dx5-18001" >309
pnl_isequal, 307
+href="pnl-manualse3.html#dx5-22003" >310
pnl_isequal_abs, 308
+href="pnl-manualse3.html#dx5-22002" >311
pnl_isequal_rel, 309
+href="pnl-manualse3.html#dx5-22001" >312
pnl_isfinite, 310
+href="pnl-manualse3.html#dx5-19006" >313
pnl_isinf, 311
+href="pnl-manualse3.html#dx5-19005" >314
pnl_isnan, 312
+href="pnl-manualse3.html#dx5-19004" >315
pnl_itrunc, 313
+href="pnl-manualse3.html#dx5-20001" >316
pnl_lgamma, 314
+href="pnl-manualse3.html#dx5-21002" >317
PNL_LIST_ARRAY, 315
+href="pnl-manualse2.html#dx4-12031" >318
pnl_list_clone, 316
+href="pnl-manualse2.html#dx4-13031" >319
pnl_list_concat, 317
+href="pnl-manualse2.html#dx4-13040" >320
pnl_list_copy, 318
+href="pnl-manualse2.html#dx4-13030" >321
pnl_list_free, 319
+href="pnl-manualse2.html#dx4-13032" >322
pnl_list_get, 320
+href="pnl-manualse2.html#dx4-13034" >323
pnl_list_insert_first, 321
+href="pnl-manualse2.html#dx4-13035" >324
pnl_list_insert_last, 322
+href="pnl-manualse2.html#dx4-13036" >325
pnl_list_new, 323
+href="pnl-manualse2.html#dx4-13028" >326
PNL_LIST_OBJECT, 324
+href="pnl-manualse2.html#dx4-12030" >327
pnl_list_print, 325
+href="pnl-manualse2.html#dx4-13042" >328
pnl_list_remove_first, 326
+href="pnl-manualse2.html#dx4-13038" >329
pnl_list_remove_i, 327
+href="pnl-manualse2.html#dx4-13039" >330
pnl_list_remove_last, 328
+href="pnl-manualse2.html#dx4-13037" >331
pnl_list_resize, 329
+href="pnl-manualse2.html#dx4-13041" >332
pnl_log1p, 330
+href="pnl-manualse3.html#dx5-21007" >333
pnl_lround, 331
+href="pnl-manualse3.html#dx5-20006" >334
pnl_ltrunc, 332
+href="pnl-manualse3.html#dx5-20002" >335
pnl_mat_add_row, 333
+href="pnl-manualse4.html#dx6-50024" >336
pnl_mat_axpy, 334
+href="pnl-manualse4.html#dx6-56007" >337
pnl_mat_bicg_solver_solve, 335
+href="pnl-manualse4.html#dx6-99002" >338
pnl_mat_cg_solver_solve, 336
+href="pnl-manualse4.html#dx6-99001" >339
pnl_mat_chol, 337
+href="pnl-manualse4.html#dx6-57001" >340
pnl_mat_chol_syslin, 338
+href="pnl-manualse4.html#dx6-57006" >341
pnl_mat_chol_syslin_inplace, 339
+href="pnl-manualse4.html#dx6-57007" >342
pnl_mat_chol_syslin_mat, 340
+href="pnl-manualse4.html#dx6-57013" >343
pnl_mat_clone, 341
+href="pnl-manualse4.html#dx6-49014" >344
pnl_mat_col_permute, 342
+href="pnl-manualse4.html#dx6-61011" >345
pnl_mat_complex_create_from_mat, 343
+href="pnl-manualse4.html#dx6-60001" >346
pnl_mat_copy, 344
+href="pnl-manualse4.html#dx6-49007" >347
pnl_mat_create, 345
+href="pnl-manualse4.html#dx6-49002" >348
pnl_mat_create_diag, 346
+href="pnl-manualse4.html#dx6-49009" >349
pnl_mat_create_diag_from_ptr, 347
+href="pnl-manualse4.html#dx6-49008" >350
+ + + pnl_mat_create_from_file, 348
+href="pnl-manualse4.html#dx6-49010" >351
pnl_mat_create_from_list, 349
+href="pnl-manualse4.html#dx6-49006" >352
pnl_mat_create_from_ptr, 350
+href="pnl-manualse4.html#dx6-49005" >353
pnl_mat_create_from_scalar, 351
+href="pnl-manualse4.html#dx6-49003" >354
pnl_mat_create_from_sp_mat, 352
- - - +href="pnl-manualse4.html#dx6-81007" >355
pnl_mat_create_from_zero, 353
+href="pnl-manualse4.html#dx6-49004" >356
pnl_mat_cross, 354
+href="pnl-manualse4.html#dx6-56013" >357
pnl_mat_cumprod, 355
+href="pnl-manualse4.html#dx6-53014" >358
pnl_mat_cumsum, 356
+href="pnl-manualse4.html#dx6-53011" >359
pnl_mat_del_row, 357
+href="pnl-manualse4.html#dx6-50025" >360
pnl_mat_dgemm, 358
+href="pnl-manualse4.html#dx6-56016" >361
pnl_mat_dgemv, 359
+href="pnl-manualse4.html#dx6-56015" >362
pnl_mat_dger, 360
+href="pnl-manualse4.html#dx6-56008" >363
pnl_mat_div_mat_term, 361
+href="pnl-manualse4.html#dx6-53002" >364
pnl_mat_div_scalar, 362
+href="pnl-manualse4.html#dx6-52004" >365
pnl_mat_eigen, 363
+href="pnl-manualse4.html#dx6-56022" >366
pnl_mat_exp, 364
+href="pnl-manualse4.html#dx6-56020" >367
pnl_mat_extract_subblock, 365
+href="pnl-manualse4.html#dx6-49018" >368
pnl_mat_find, 366
+href="pnl-manualse4.html#dx6-55009" >369
pnl_mat_fprint, 367
+href="pnl-manualse4.html#dx6-51002" >370
+pnl_mat_fprint_csv, 371
pnl_mat_fprint_nsp, 368
+href="pnl-manualse4.html#dx6-51006" >372
pnl_mat_free, 369
+href="pnl-manualse4.html#dx6-49011" >373
pnl_mat_get, 370
+href="pnl-manualse4.html#dx6-50008" >374
pnl_mat_get_col, 371
+href="pnl-manualse4.html#dx6-50016" >375
pnl_mat_get_row, 372
+href="pnl-manualse4.html#dx6-50015" >376
pnl_mat_gmres_solver_solve, 373
+href="pnl-manualse4.html#dx6-99003" >377
pnl_mat_inverse, 374
+href="pnl-manualse4.html#dx6-57017" >378
pnl_mat_inverse_with_chol, 375
+href="pnl-manualse4.html#dx6-57018" >379
pnl_mat_isequal, 376
+href="pnl-manualse4.html#dx6-54001" >380
pnl_mat_isequal_abs, 377
+href="pnl-manualse4.html#dx6-54002" >381
pnl_mat_isequal_rel, 378
+href="pnl-manualse4.html#dx6-54003" >382
pnl_mat_kron_mat, 379
+href="pnl-manualse4.html#dx6-53004" >383
pnl_mat_kron_mat_inplace, 380
+href="pnl-manualse4.html#dx6-53003" >384
pnl_mat_lAxpby, 381
+href="pnl-manualse4.html#dx6-56014" >385
pnl_mat_lget, 382
+href="pnl-manualse4.html#dx6-50009" >386
pnl_mat_log, 383
+href="pnl-manualse4.html#dx6-56021" >387
pnl_mat_lower_inverse, 384
+href="pnl-manualse4.html#dx6-57016" >388
pnl_mat_lower_syslin, 385
+href="pnl-manualse4.html#dx6-57005" >389
pnl_mat_ls, 386
+href="pnl-manualse4.html#dx6-59003" >390
pnl_mat_ls_mat, 387
+href="pnl-manualse4.html#dx6-59004" >391
pnl_mat_lu, 388
+href="pnl-manualse4.html#dx6-57003" >392
pnl_mat_lu_syslin, 389
+href="pnl-manualse4.html#dx6-57008" >393
pnl_mat_lu_syslin_inplace, 390
+href="pnl-manualse4.html#dx6-57009" >394
pnl_mat_lu_syslin_mat, 391
+href="pnl-manualse4.html#dx6-57014" >395
pnl_mat_map, 392
+href="pnl-manualse4.html#dx6-53006" >396
pnl_mat_map_inplace, 393
+href="pnl-manualse4.html#dx6-53005" >397
+ + + pnl_mat_map_mat, 394
+href="pnl-manualse4.html#dx6-53008" >398
pnl_mat_map_mat_inplace, 395
+href="pnl-manualse4.html#dx6-53007" >399
pnl_mat_max, 396
+href="pnl-manualse4.html#dx6-55001" >400
pnl_mat_max_index, 397
+href="pnl-manualse4.html#dx6-55005" >401
pnl_mat_min, 398
+href="pnl-manualse4.html#dx6-55002" >402
pnl_mat_min_index, 399
- - - +href="pnl-manualse4.html#dx6-55004" >403
pnl_mat_minmax, 400
+href="pnl-manualse4.html#dx6-55003" >404
pnl_mat_minmax_index, 401
+href="pnl-manualse4.html#dx6-55006" >405
pnl_mat_minus_mat, 402
+href="pnl-manualse4.html#dx6-56002" >406
pnl_mat_minus_scalar, 403
+href="pnl-manualse4.html#dx6-52002" >407
pnl_mat_mult_mat, 404
+href="pnl-manualse4.html#dx6-56017" >408
pnl_mat_mult_mat_inplace, 405
+href="pnl-manualse4.html#dx6-56018" >409
pnl_mat_mult_mat_term, 406
+href="pnl-manualse4.html#dx6-53001" >410
pnl_mat_mult_scalar, 407
+href="pnl-manualse4.html#dx6-52003" >411
pnl_mat_mult_vect, 408
+href="pnl-manualse4.html#dx6-56009" >412
pnl_mat_mult_vect_inplace, 409
+href="pnl-manualse4.html#dx6-56010" >413
pnl_mat_mult_vect_transpose, 410
+href="pnl-manualse4.html#dx6-56011" >414
pnl_mat_mult_vect_transpose_inplace, 411
+href="pnl-manualse4.html#dx6-56012" >415
pnl_mat_new, 412
+href="pnl-manualse4.html#dx6-49001" >416
PNL_MAT_OBJECT, 413
+href="pnl-manualse2.html#dx4-12023" >417
pnl_mat_pchol, 414
+href="pnl-manualse4.html#dx6-57002" >418
pnl_mat_plus_mat, 415
+href="pnl-manualse4.html#dx6-56001" >419
pnl_mat_plus_scalar, 416
+href="pnl-manualse4.html#dx6-52001" >420
pnl_mat_print, 417
+href="pnl-manualse4.html#dx6-51001" >421
+pnl_mat_print_csv, 422
pnl_mat_print_nsp, 418
+href="pnl-manualse4.html#dx6-51005" >423
pnl_mat_prod, 419
+href="pnl-manualse4.html#dx6-53012" >424
pnl_mat_prod_vect, 420
+href="pnl-manualse4.html#dx6-53013" >425
pnl_mat_qr, 421
+href="pnl-manualse4.html#dx6-59001" >426
pnl_mat_qr_syslin, 422
+href="pnl-manualse4.html#dx6-59002" >427
pnl_mat_qsort, 423
+href="pnl-manualse4.html#dx6-55007" >428
pnl_mat_qsort_index, 424
+href="pnl-manualse4.html#dx6-55008" >429
pnl_mat_rand_normal, 425
+href="pnl-manualse6.html#dx8-103021" >430
pnl_mat_rand_uni, 426
+href="pnl-manualse6.html#dx8-103019" >431
pnl_mat_rand_uni2, 427
+href="pnl-manualse6.html#dx8-103020" >432
pnl_mat_resize, 428
+href="pnl-manualse4.html#dx6-49015" >433
pnl_mat_rng_bernoulli, 429
+href="pnl-manualse6.html#dx8-102055" >434
pnl_mat_rng_normal, 430
+href="pnl-manualse6.html#dx8-102054" >435
pnl_mat_rng_poisson, 431
+href="pnl-manualse6.html#dx8-102056" >436
pnl_mat_rng_uni, 432
+href="pnl-manualse6.html#dx8-102052" >437
pnl_mat_rng_uni2, 433
+href="pnl-manualse6.html#dx8-102053" >438
pnl_mat_row_permute, 434
+href="pnl-manualse4.html#dx6-61012" >439
pnl_mat_scalar_prod, 435
+href="pnl-manualse4.html#dx6-56019" >440
pnl_mat_set, 436
+href="pnl-manualse4.html#dx6-50007" >441
pnl_mat_set_all, 437
+href="pnl-manualse4.html#dx6-50010" >442
pnl_mat_set_col, 438
+href="pnl-manualse4.html#dx6-50020" >443
pnl_mat_set_col_from_ptr, 439
+href="pnl-manualse4.html#dx6-50021" >444
+ + + pnl_mat_set_diag, 440
+href="pnl-manualse4.html#dx6-50013" >445
pnl_mat_set_from_ptr, 441
+href="pnl-manualse4.html#dx6-50014" >446
pnl_mat_set_id, 442
+href="pnl-manualse4.html#dx6-50012" >447
pnl_mat_set_row, 443
+href="pnl-manualse4.html#dx6-50022" >448
pnl_mat_set_row_from_ptr, 444
+href="pnl-manualse4.html#dx6-50023" >449
pnl_mat_set_subblock, 445
+href="pnl-manualse4.html#dx6-49019" >450
pnl_mat_set_zero, 446
- - - +href="pnl-manualse4.html#dx6-50011" >451
pnl_mat_sq_transpose, 447
+href="pnl-manualse4.html#dx6-56003" >452
pnl_mat_sum, 448
+href="pnl-manualse4.html#dx6-53009" >453
pnl_mat_sum_vect, 449
+href="pnl-manualse4.html#dx6-53010" >454
pnl_mat_swap_rows, 450
+href="pnl-manualse4.html#dx6-50019" >455
pnl_mat_syslin, 451
+href="pnl-manualse4.html#dx6-57010" >456
pnl_mat_syslin_inplace, 452
+href="pnl-manualse4.html#dx6-57011" >457
pnl_mat_syslin_mat, 453
+href="pnl-manualse4.html#dx6-57012" >458
pnl_mat_tr, 454
+href="pnl-manualse4.html#dx6-56005" >459
pnl_mat_trace, 455
+href="pnl-manualse4.html#dx6-56006" >460
pnl_mat_transpose, 456
+href="pnl-manualse4.html#dx6-56004" >461
pnl_mat_upper_inverse, 457
+href="pnl-manualse4.html#dx6-57015" >462
pnl_mat_upper_syslin, 458
+href="pnl-manualse4.html#dx6-57004" >463
pnl_mat_wrap_array, 459
+href="pnl-manualse4.html#dx6-49012" >464
pnl_mat_wrap_hmat, 460
+href="pnl-manualse4.html#dx6-90004" >465
pnl_mat_wrap_mat_rows, 461
+href="pnl-manualse4.html#dx6-50018" >466
pnl_mat_wrap_vect, 462
+href="pnl-manualse4.html#dx6-49013" >467
pnl_multiroot_newton, 463
+href="pnl-manualse13.html#dx15-134001" >468
pnl_nan, 464
+href="pnl-manualse3.html#dx5-19001" >469
PNL_NEGINF, 465
+href="pnl-manualse3.html#dx5-17024" >470
pnl_neginf, 466
+href="pnl-manualse3.html#dx5-19003" >471
pnl_normal_density, 467
+href="pnl-manualse5.html#dx7-100014" >472
PNL_OBJECT, 468
+href="pnl-manualse2.html#dx4-12021" >473
pnl_object_create, 469
+href="pnl-manualse2.html#dx4-12035" >474
pnl_object_load, 470
+href="pnl-manualse15.html#dx17-148004" >475
pnl_object_load_into_list, 471
+href="pnl-manualse15.html#dx17-148005" >476
pnl_object_mpi_bcast, 472
+href="pnl-manualse15.html#dx17-147007" >477
pnl_object_mpi_irecv, 473
+href="pnl-manualse15.html#dx17-147010" >478
pnl_object_mpi_isend, 474
+href="pnl-manualse15.html#dx17-147009" >479
pnl_object_mpi_pack, 475
+href="pnl-manualse15.html#dx17-147002" >480
pnl_object_mpi_pack_size, 476
+href="pnl-manualse15.html#dx17-147001" >481
pnl_object_mpi_recv, 477
+href="pnl-manualse15.html#dx17-147006" >482
pnl_object_mpi_reduce, 478
+href="pnl-manualse15.html#dx17-147008" >483
pnl_object_mpi_send, 479
+href="pnl-manualse15.html#dx17-147004" >484
pnl_object_mpi_ssend, 480
+href="pnl-manualse15.html#dx17-147005" >485
pnl_object_mpi_unpack, 481
+href="pnl-manualse15.html#dx17-147003" >486
pnl_object_save, 482
+href="pnl-manualse15.html#dx17-148003" >487
pnl_ode_rkf45, 483
+href="pnl-manualse11.html#dx13-122001" >488
pnl_ode_rkf45_step, 484
+href="pnl-manualse11.html#dx13-122002" >489
pnl_optim_intpoints_bfgs_solve, 485
+href="pnl-manualse12.html#dx14-129001" >490
pnl_optim_linprog, 486
+href="pnl-manualse12.html#dx14-126001" >491
+ + + pnl_optim_linprog_sp, 487
+href="pnl-manualse12.html#dx14-126002" >492
pnl_permutation_create, 488
+href="pnl-manualse4.html#dx6-61004" >493
pnl_permutation_fprint, 489
+href="pnl-manualse4.html#dx6-61013" >494
pnl_permutation_free, 490
+href="pnl-manualse4.html#dx6-61005" >495
pnl_permutation_inverse, 491
+href="pnl-manualse4.html#dx6-61006" >496
pnl_permutation_new, 492
+href="pnl-manualse4.html#dx6-61003" >497
pnl_permutation_print, 493
- - - +href="pnl-manualse4.html#dx6-61014" >498
PNL_POSINF, 494
+href="pnl-manualse3.html#dx5-17025" >499
pnl_posinf, 495
+href="pnl-manualse3.html#dx5-19002" >500
pnl_pow_i, 496
+href="pnl-manualse3.html#dx5-21010" >501
pnl_rand_bernoulli, 497
+href="pnl-manualse6.html#dx8-103005" >502
pnl_rand_bessel, 498
+href="pnl-manualse6.html#dx8-103014" >503
pnl_rand_chi2, 499
+href="pnl-manualse6.html#dx8-103013" >504
pnl_rand_exp, 500
+href="pnl-manualse6.html#dx8-103007" >505
pnl_rand_gamma, 501
+href="pnl-manualse6.html#dx8-103012" >506
pnl_rand_gauss, 502
+href="pnl-manualse6.html#dx8-103022" >507
pnl_rand_init, 503
+href="pnl-manualse6.html#dx8-103001" >508
pnl_rand_name, 504
+href="pnl-manualse6.html#dx8-103004" >509
pnl_rand_normal, 505
+href="pnl-manualse6.html#dx8-103010" >510
pnl_rand_or_quasi, 506
+href="pnl-manualse6.html#dx8-103002" >511
pnl_rand_poisson, 507
+href="pnl-manualse6.html#dx8-103006" >512
pnl_rand_poisson1, 508
+href="pnl-manualse6.html#dx8-103011" >513
pnl_rand_sseed, 509
+href="pnl-manualse6.html#dx8-103003" >514
pnl_rand_uni, 510
+href="pnl-manualse6.html#dx8-103008" >515
pnl_rand_uni_ab, 511
+href="pnl-manualse6.html#dx8-103009" >516
pnl_real_fft, 512
+href="pnl-manualse9.html#dx11-117007" >517
pnl_real_fft2, 513
+href="pnl-manualse9.html#dx11-117011" >518
pnl_real_fft2d, 514
+href="pnl-manualse9.html#dx11-117017" >519
pnl_real_fft_inplace, 515
+href="pnl-manualse9.html#dx11-117009" >520
pnl_real_fft_inplace_with_wspace, 516
+href="pnl-manualse9.html#dx11-118004" >521
pnl_real_fft_with_wspace, 517
+href="pnl-manualse9.html#dx11-118006" >522
pnl_real_ifft, 518
+href="pnl-manualse9.html#dx11-117008" >523
pnl_real_ifft2, 519
+href="pnl-manualse9.html#dx11-117012" >524
pnl_real_ifft2d, 520
+href="pnl-manualse9.html#dx11-117018" >525
pnl_real_ifft_inplace, 521
+href="pnl-manualse9.html#dx11-117010" >526
pnl_real_ifft_inplace_with_wspace, 522
+href="pnl-manualse9.html#dx11-118005" >527
pnl_real_ifft_with_wspace, 523
+href="pnl-manualse9.html#dx11-118007" >528
pnl_rng_bernoulli, 524
+href="pnl-manualse6.html#dx8-102029" >529
pnl_rng_bessel, 525
+href="pnl-manualse6.html#dx8-102042" >530
pnl_rng_chi2, 526
+href="pnl-manualse6.html#dx8-102040" >531
pnl_rng_clone, 527
+href="pnl-manualse6.html#dx8-102022" >532
pnl_rng_copy, 528
+href="pnl-manualse6.html#dx8-102021" >533
pnl_rng_create, 529
+href="pnl-manualse6.html#dx8-102018" >534
pnl_rng_create_from_file, 530
+href="pnl-manualse15.html#dx17-148001" >535
pnl_rng_dblexp, 531
+href="pnl-manualse6.html#dx8-102032" >536
pnl_rng_dcmt_create_array, 532
+href="pnl-manualse6.html#dx8-102025" >537
pnl_rng_dcmt_create_array_id, 533
+href="pnl-manualse6.html#dx8-102024" >538
+ + + pnl_rng_dcmt_create_id, 534
+href="pnl-manualse6.html#dx8-102023" >539
pnl_rng_exp, 535
+href="pnl-manualse6.html#dx8-102031" >540
pnl_rng_free, 536
+href="pnl-manualse6.html#dx8-102017" >541
pnl_rng_gamma, 537
+href="pnl-manualse6.html#dx8-102039" >542
pnl_rng_gauss, 538
+href="pnl-manualse6.html#dx8-102043" >543
pnl_rng_get_from_id, 539
+href="pnl-manualse6.html#dx8-102028" >544
pnl_rng_init, 540
- - - +href="pnl-manualse6.html#dx8-102027" >545
pnl_rng_invgauss, 541
+href="pnl-manualse6.html#dx8-102037" >546
pnl_rng_lognormal, 542
+href="pnl-manualse6.html#dx8-102036" >547
pnl_rng_ncchi2, 543
+href="pnl-manualse6.html#dx8-102041" >548
pnl_rng_new, 544
+href="pnl-manualse6.html#dx8-102026" >549
pnl_rng_normal, 545
+href="pnl-manualse6.html#dx8-102035" >550
PNL_RNG_OBJECT, 546
+href="pnl-manualse2.html#dx4-12029" >551
pnl_rng_poisson, 547
+href="pnl-manualse6.html#dx8-102030" >552
pnl_rng_poisson1, 548
+href="pnl-manualse6.html#dx8-102038" >553
pnl_rng_save_to_file, 549
+href="pnl-manualse15.html#dx17-148002" >554
pnl_rng_sdim, 550
+href="pnl-manualse6.html#dx8-102020" >555
pnl_rng_sseed, 551
+href="pnl-manualse6.html#dx8-102019" >556
pnl_rng_uni, 552
+href="pnl-manualse6.html#dx8-102033" >557
pnl_rng_uni_ab, 553
+href="pnl-manualse6.html#dx8-102034" >558
pnl_root_bisection, 554
+href="pnl-manualse13.html#dx15-133004" >559
pnl_root_brent, 555
+href="pnl-manualse13.html#dx15-133001" >560
pnl_root_fsolve, 556
+href="pnl-manualse13.html#dx15-134002" >561
pnl_root_fsolve_lsq, 557
+href="pnl-manualse13.html#dx15-134003" >562
pnl_root_newton, 558
+href="pnl-manualse13.html#dx15-133003" >563
pnl_root_newton_bisection, 559
+href="pnl-manualse13.html#dx15-133002" >564
pnl_round, 560
+href="pnl-manualse3.html#dx5-20004" >565
pnl_sf_choose, 561
+href="pnl-manualse14.html#dx16-139005" >566
pnl_sf_complex_dawson, 562
+href="pnl-manualse14.html#dx16-138012" >567
pnl_sf_complex_erf, 563
+href="pnl-manualse14.html#dx16-138002" >568
pnl_sf_complex_erfc, 564
+href="pnl-manualse14.html#dx16-138004" >569
pnl_sf_complex_erfcx, 565
+href="pnl-manualse14.html#dx16-138006" >570
pnl_sf_complex_erfi, 566
+href="pnl-manualse14.html#dx16-138010" >571
pnl_sf_dawson, 567
+href="pnl-manualse14.html#dx16-138011" >572
pnl_sf_erf, 568
+href="pnl-manualse14.html#dx16-138001" >573
pnl_sf_erfc, 569
+href="pnl-manualse14.html#dx16-138003" >574
pnl_sf_erfcx, 570
+href="pnl-manualse14.html#dx16-138005" >575
pnl_sf_erfi, 571
+href="pnl-manualse14.html#dx16-138009" >576
pnl_sf_expint_En, 572
+href="pnl-manualse14.html#dx16-142001" >577
pnl_sf_fact, 573
+href="pnl-manualse14.html#dx16-139001" >578
pnl_sf_gamma, 574
+href="pnl-manualse14.html#dx16-139002" >579
pnl_sf_gamma_inc, 575
+href="pnl-manualse14.html#dx16-141001" >580
pnl_sf_gamma_inc_P, 576
+href="pnl-manualse14.html#dx16-141002" >581
pnl_sf_gamma_inc_Q, 577
+href="pnl-manualse14.html#dx16-141003" >582
pnl_sf_hyperg_0F1, 578
+href="pnl-manualse14.html#dx16-143004" >583
pnl_sf_hyperg_1F1, 579
+href="pnl-manualse14.html#dx16-143002" >584
pnl_sf_hyperg_2F0, 580
+href="pnl-manualse14.html#dx16-143003" >585
+ + + pnl_sf_hyperg_2F1, 581
+href="pnl-manualse14.html#dx16-143001" >586
pnl_sf_hyperg_U, 582
+href="pnl-manualse14.html#dx16-143005" >587
pnl_sf_log_erf, 583
+href="pnl-manualse14.html#dx16-138013" >588
pnl_sf_log_erfc, 584
+href="pnl-manualse14.html#dx16-138014" >589
pnl_sf_log_gamma, 585
+href="pnl-manualse14.html#dx16-139003" >590
pnl_sf_log_gamma_sgn, 586
+href="pnl-manualse14.html#dx16-139004" >591
pnl_sf_psi, 587
- - - +href="pnl-manualse14.html#dx16-140001" >592
pnl_sf_w, 588
+href="pnl-manualse14.html#dx16-138007" >593
pnl_sf_w_im, 589
+href="pnl-manualse14.html#dx16-138008" >594
PNL_SIGN, 590
+href="pnl-manualse3.html#dx5-18007" >595
pnl_sp_mat_clone, 591
+href="pnl-manualse4.html#dx6-81003" >596
pnl_sp_mat_copy, 592
+href="pnl-manualse4.html#dx6-81004" >597
pnl_sp_mat_create, 593
+href="pnl-manualse4.html#dx6-81002" >598
pnl_sp_mat_create_from_file, 594
+href="pnl-manualse4.html#dx6-81009" >599
pnl_sp_mat_create_from_mat, 595
+href="pnl-manualse4.html#dx6-81008" >600
pnl_sp_mat_div_scalar, 596
+href="pnl-manualse4.html#dx6-83004" >601
pnl_sp_mat_fprint, 597
+href="pnl-manualse4.html#dx6-84001" >602
pnl_sp_mat_free, 598
+href="pnl-manualse4.html#dx6-81005" >603
pnl_sp_mat_get, 599
+href="pnl-manualse4.html#dx6-82002" >604
pnl_sp_mat_isequal, 600
+href="pnl-manualse4.html#dx6-85001" >605
pnl_sp_mat_isequal_abs, 601
+href="pnl-manualse4.html#dx6-85002" >606
pnl_sp_mat_isequal_rel, 602
+href="pnl-manualse4.html#dx6-85003" >607
pnl_sp_mat_kron, 603
+href="pnl-manualse4.html#dx6-84008" >608
pnl_sp_mat_kron_inplace, 604
+href="pnl-manualse4.html#dx6-84007" >609
pnl_sp_mat_lAxpby, 605
+href="pnl-manualse4.html#dx6-84004" >610
pnl_sp_mat_minus_scalar, 606
+href="pnl-manualse4.html#dx6-83002" >611
pnl_sp_mat_mult_scalar, 607
+href="pnl-manualse4.html#dx6-83003" >612
pnl_sp_mat_mult_vect, 608
+href="pnl-manualse4.html#dx6-84003" >613
pnl_sp_mat_new, 609
+href="pnl-manualse4.html#dx6-81001" >614
PNL_SP_MAT_OBJECT, 610
+href="pnl-manualse2.html#dx4-12024" >615
pnl_sp_mat_plus_scalar, 611
+href="pnl-manualse4.html#dx6-83001" >616
+pnl_sp_mat_plus_sp_mat, 617
pnl_sp_mat_plus_sp_mat_inplace, 612
+href="pnl-manualse4.html#dx6-84005" >618
pnl_sp_mat_print, 613
+href="pnl-manualse4.html#dx6-84002" >619
pnl_sp_mat_resize, 614
+href="pnl-manualse4.html#dx6-81006" >620
pnl_sp_mat_set, 615
-pnl_sp_mat_sp_mat, 616
+href="pnl-manualse4.html#dx6-82001" >621
pnl_tgamma, 617
+href="pnl-manualse3.html#dx5-21003" >622
pnl_tridiag_mat_clone, 618
+href="pnl-manualse4.html#dx6-65009" >623
pnl_tridiag_mat_copy, 619
+href="pnl-manualse4.html#dx6-65008" >624
pnl_tridiag_mat_create, 620
+href="pnl-manualse4.html#dx6-65002" >625
pnl_tridiag_mat_create_from_mat, 621
+href="pnl-manualse4.html#dx6-65006" >626
pnl_tridiag_mat_create_from_ptr, 622
+href="pnl-manualse4.html#dx6-65005" >627
pnl_tridiag_mat_create_from_scalar, 623
+href="pnl-manualse4.html#dx6-65003" >628
pnl_tridiag_mat_create_from_two_scalar, 624
+href="pnl-manualse4.html#dx6-65004" >629
pnl_tridiag_mat_div_scalar, 625
+href="pnl-manualse4.html#dx6-68006" >630
pnl_tridiag_mat_div_tridiag_mat_term, 626
+href="pnl-manualse4.html#dx6-69002" >631
pnl_tridiag_mat_fprint, 627
+href="pnl-manualse4.html#dx6-67001" >632
+ + + pnl_tridiag_mat_free, 628
+href="pnl-manualse4.html#dx6-65010" >633
pnl_tridiag_mat_get, 629
+href="pnl-manualse4.html#dx6-66002" >634
pnl_tridiag_mat_lAxpby, 630
+href="pnl-manualse4.html#dx6-70003" >635
pnl_tridiag_mat_lget, 631
+href="pnl-manualse4.html#dx6-66003" >636
pnl_tridiag_mat_lu_clone, 632
+href="pnl-manualse4.html#dx6-70010" >637
pnl_tridiag_mat_lu_compute, 633
+href="pnl-manualse4.html#dx6-70013" >638
pnl_tridiag_mat_lu_copy, 634
- - - +href="pnl-manualse4.html#dx6-70009" >639
pnl_tridiag_mat_lu_create, 635
+href="pnl-manualse4.html#dx6-70008" >640
pnl_tridiag_mat_lu_free, 636
+href="pnl-manualse4.html#dx6-70011" >641
pnl_tridiag_mat_lu_new, 637
+href="pnl-manualse4.html#dx6-70007" >642
pnl_tridiag_mat_lu_resize, 638
+href="pnl-manualse4.html#dx6-70012" >643
pnl_tridiag_mat_lu_syslin, 639
+href="pnl-manualse4.html#dx6-70015" >644
pnl_tridiag_mat_lu_syslin_inplace, 640
+href="pnl-manualse4.html#dx6-70014" >645
pnl_tridiag_mat_map_inplace, 641
+href="pnl-manualse4.html#dx6-69003" >646
pnl_tridiag_mat_map_tridiag_mat_inplace, 642
+href="pnl-manualse4.html#dx6-69004" >647
pnl_tridiag_mat_minus_scalar, 643
+href="pnl-manualse4.html#dx6-68004" >648
pnl_tridiag_mat_minus_tridiag_mat, 644
+href="pnl-manualse4.html#dx6-68002" >649
pnl_tridiag_mat_mult_scalar, 645
+href="pnl-manualse4.html#dx6-68005" >650
pnl_tridiag_mat_mult_tridiag_mat_term, 646
+href="pnl-manualse4.html#dx6-69001" >651
pnl_tridiag_mat_mult_vect, 647
+href="pnl-manualse4.html#dx6-70002" >652
pnl_tridiag_mat_mult_vect_inplace, 648
+href="pnl-manualse4.html#dx6-70001" >653
pnl_tridiag_mat_new, 649
+href="pnl-manualse4.html#dx6-65001" >654
pnl_tridiag_mat_plus_scalar, 650
+href="pnl-manualse4.html#dx6-68003" >655
pnl_tridiag_mat_plus_tridiag_mat, 651
+href="pnl-manualse4.html#dx6-68001" >656
pnl_tridiag_mat_print, 652
+href="pnl-manualse4.html#dx6-67002" >657
pnl_tridiag_mat_resize, 653
+href="pnl-manualse4.html#dx6-65011" >658
pnl_tridiag_mat_scalar_prod, 654
+href="pnl-manualse4.html#dx6-70004" >659
pnl_tridiag_mat_set, 655
+href="pnl-manualse4.html#dx6-66001" >660
pnl_tridiag_mat_syslin, 656
+href="pnl-manualse4.html#dx6-70006" >661
pnl_tridiag_mat_syslin_inplace, 657
+href="pnl-manualse4.html#dx6-70005" >662
pnl_tridiag_mat_to_mat, 658
+href="pnl-manualse4.html#dx6-65007" >663
PNL_TRIDIAGMAT_OBJECT, 659
+href="pnl-manualse2.html#dx4-12027" >664
pnl_trunc, 660
+href="pnl-manualse3.html#dx5-20003" >665
pnl_vect_axpby, 661
+href="pnl-manualse4.html#dx6-37010" >666
pnl_vect_clone, 662
+href="pnl-manualse4.html#dx6-32010" >667
pnl_vect_compact_copy, 663
+href="pnl-manualse4.html#dx6-45005" >668
pnl_vect_compact_create, 664
+href="pnl-manualse4.html#dx6-45002" >669
pnl_vect_compact_create_from_ptr, 665
+href="pnl-manualse4.html#dx6-45003" >670
pnl_vect_compact_free, 666
+href="pnl-manualse4.html#dx6-45006" >671
pnl_vect_compact_get, 667
+href="pnl-manualse4.html#dx6-45008" >672
pnl_vect_compact_new, 668
+href="pnl-manualse4.html#dx6-45001" >673
pnl_vect_compact_resize, 669
+href="pnl-manualse4.html#dx6-45004" >674
pnl_vect_compact_set_all, 670
+href="pnl-manualse4.html#dx6-45009" >675
pnl_vect_compact_set_ptr, 671
+href="pnl-manualse4.html#dx6-45010" >676
pnl_vect_compact_to_pnl_vect, 672
+href="pnl-manualse4.html#dx6-45007" >677
pnl_vect_complex_create_from_array, 673
+href="pnl-manualse4.html#dx6-42002" >678
pnl_vect_complex_get_imag, 674
+href="pnl-manualse4.html#dx6-42006" >679
+ + + pnl_vect_complex_get_real, 675
+href="pnl-manualse4.html#dx6-42005" >680
pnl_vect_complex_lget_imag, 676
+href="pnl-manualse4.html#dx6-42008" >681
pnl_vect_complex_lget_real, 677
+href="pnl-manualse4.html#dx6-42007" >682
pnl_vect_complex_mult_double, 678
+href="pnl-manualse4.html#dx6-42001" >683
pnl_vect_complex_set_imag, 679
+href="pnl-manualse4.html#dx6-42010" >684
pnl_vect_complex_set_real, 680
+href="pnl-manualse4.html#dx6-42009" >685
pnl_vect_complex_split_in_array, 681
- - - +href="pnl-manualse4.html#dx6-42003" >686
pnl_vect_complex_split_in_vect, 682
+href="pnl-manualse4.html#dx6-42004" >687
pnl_vect_copy, 683
+href="pnl-manualse4.html#dx6-32009" >688
pnl_vect_create, 684
+href="pnl-manualse4.html#dx6-32002" >689
pnl_vect_create_from_file, 685
+href="pnl-manualse4.html#dx6-32008" >690
pnl_vect_create_from_list, 686
+href="pnl-manualse4.html#dx6-32007" >691
pnl_vect_create_from_mat, 687
+href="pnl-manualse4.html#dx6-32006" >692
pnl_vect_create_from_ptr, 688
+href="pnl-manualse4.html#dx6-32005" >693
pnl_vect_create_from_scalar, 689
+href="pnl-manualse4.html#dx6-32004" >694
pnl_vect_create_from_zero, 690
+href="pnl-manualse4.html#dx6-32003" >695
pnl_vect_create_submat, 691
+href="pnl-manualse4.html#dx6-49016" >696
pnl_vect_create_subvect, 692
+href="pnl-manualse4.html#dx6-32013" >697
pnl_vect_create_subvect_with_ind, 693
+href="pnl-manualse4.html#dx6-32011" >698
pnl_vect_cross, 694
+href="pnl-manualse4.html#dx6-38005" >699
pnl_vect_cumprod, 695
+href="pnl-manualse4.html#dx6-37014" >700
pnl_vect_cumsum, 696
+href="pnl-manualse4.html#dx6-37012" >701
pnl_vect_dist, 697
+href="pnl-manualse4.html#dx6-38006" >702
pnl_vect_div_scalar, 698
+href="pnl-manualse4.html#dx6-36005" >703
pnl_vect_div_vect_term, 699
+href="pnl-manualse4.html#dx6-37004" >704
pnl_vect_eq_all, 700
+href="pnl-manualse4.html#dx6-39004" >705
pnl_vect_extract_submat, 701
+href="pnl-manualse4.html#dx6-49017" >706
pnl_vect_extract_subvect, 702
+href="pnl-manualse4.html#dx6-32014" >707
pnl_vect_extract_subvect_with_ind, 703
+href="pnl-manualse4.html#dx6-32012" >708
pnl_vect_find, 704
+href="pnl-manualse4.html#dx6-40009" >709
pnl_vect_fprint, 705
+href="pnl-manualse4.html#dx6-35002" >710
pnl_vect_fprint_asrow, 706
+href="pnl-manualse4.html#dx6-35004" >711
pnl_vect_fprint_nsp, 707
+href="pnl-manualse4.html#dx6-35006" >712
pnl_vect_free, 708
+href="pnl-manualse4.html#dx6-32016" >713
pnl_vect_get, 709
+href="pnl-manualse4.html#dx6-34008" >714
pnl_vect_inv_term, 710
+href="pnl-manualse4.html#dx6-37003" >715
pnl_vect_isequal, 711
+href="pnl-manualse4.html#dx6-39001" >716
pnl_vect_isequal_abs, 712
+href="pnl-manualse4.html#dx6-39002" >717
pnl_vect_isequal_rel, 713
+href="pnl-manualse4.html#dx6-39003" >718
pnl_vect_lget, 714
+href="pnl-manualse4.html#dx6-34009" >719
pnl_vect_map, 715
+href="pnl-manualse4.html#dx6-37006" >720
pnl_vect_map_inplace, 716
+href="pnl-manualse4.html#dx6-37007" >721
pnl_vect_map_vect, 717
+href="pnl-manualse4.html#dx6-37008" >722
pnl_vect_map_vect_inplace, 718
+href="pnl-manualse4.html#dx6-37009" >723
pnl_vect_max, 719
+href="pnl-manualse4.html#dx6-40001" >724
pnl_vect_max_index, 720
+href="pnl-manualse4.html#dx6-40005" >725
pnl_vect_min, 721
+href="pnl-manualse4.html#dx6-40002" >726
+ + + pnl_vect_min_index, 722
+href="pnl-manualse4.html#dx6-40004" >727
pnl_vect_minmax, 723
+href="pnl-manualse4.html#dx6-40003" >728
pnl_vect_minmax_index, 724
+href="pnl-manualse4.html#dx6-40006" >729
pnl_vect_minus, 725
+href="pnl-manualse4.html#dx6-36001" >730
pnl_vect_minus_scalar, 726
+href="pnl-manualse4.html#dx6-36003" >731
pnl_vect_minus_vect, 727
+href="pnl-manualse4.html#dx6-37002" >732
pnl_vect_mult_scalar, 728
- - - +href="pnl-manualse4.html#dx6-36004" >733
pnl_vect_mult_vect_term, 729
+href="pnl-manualse4.html#dx6-37005" >734
pnl_vect_new, 730
+href="pnl-manualse4.html#dx6-32001" >735
pnl_vect_norm_infty, 731
+href="pnl-manualse4.html#dx6-38003" >736
pnl_vect_norm_one, 732
+href="pnl-manualse4.html#dx6-38002" >737
pnl_vect_norm_two, 733
+href="pnl-manualse4.html#dx6-38001" >738
PNL_VECT_OBJECT, 734
+href="pnl-manualse2.html#dx4-12022" >739
pnl_vect_permute, 735
+href="pnl-manualse4.html#dx6-61007" >740
pnl_vect_permute_inplace, 736
+href="pnl-manualse4.html#dx6-61008" >741
pnl_vect_permute_inverse, 737
+href="pnl-manualse4.html#dx6-61009" >742
pnl_vect_permute_inverse_inplace, 738
+href="pnl-manualse4.html#dx6-61010" >743
pnl_vect_plus_scalar, 739
+href="pnl-manualse4.html#dx6-36002" >744
pnl_vect_plus_vect, 740
+href="pnl-manualse4.html#dx6-37001" >745
pnl_vect_print, 741
+href="pnl-manualse4.html#dx6-35001" >746
pnl_vect_print_asrow, 742
+href="pnl-manualse4.html#dx6-35003" >747
pnl_vect_print_nsp, 743
+href="pnl-manualse4.html#dx6-35005" >748
pnl_vect_prod, 744
+href="pnl-manualse4.html#dx6-37013" >749
pnl_vect_qsort, 745
+href="pnl-manualse4.html#dx6-40007" >750
pnl_vect_qsort_index, 746
+href="pnl-manualse4.html#dx6-40008" >751
pnl_vect_rand_normal, 747
+href="pnl-manualse6.html#dx8-103016" >752
pnl_vect_rand_normal_d, 748
+href="pnl-manualse6.html#dx8-103018" >753
pnl_vect_rand_uni, 749
+href="pnl-manualse6.html#dx8-103015" >754
pnl_vect_rand_uni_d, 750
+href="pnl-manualse6.html#dx8-103017" >755
pnl_vect_resize, 751
+href="pnl-manualse4.html#dx6-33001" >756
pnl_vect_resize_from_ptr, 752
+href="pnl-manualse4.html#dx6-33003" >757
+pnl_vect_resize_from_scalar, 758
pnl_vect_reverse, 753
+href="pnl-manualse4.html#dx6-41002" >759
pnl_vect_rng_bernoulli, 754
+href="pnl-manualse6.html#dx8-102044" >760
pnl_vect_rng_bernoulli_d, 755
+href="pnl-manualse6.html#dx8-102045" >761
pnl_vect_rng_normal, 756
+href="pnl-manualse6.html#dx8-102049" >762
pnl_vect_rng_normal_d, 757
+href="pnl-manualse6.html#dx8-102051" >763
pnl_vect_rng_poisson, 758
+href="pnl-manualse6.html#dx8-102046" >764
pnl_vect_rng_poisson_d, 759
+href="pnl-manualse6.html#dx8-102047" >765
pnl_vect_rng_uni, 760
+href="pnl-manualse6.html#dx8-102048" >766
pnl_vect_rng_uni_d, 761
+href="pnl-manualse6.html#dx8-102050" >767
pnl_vect_scalar_prod, 762
+href="pnl-manualse4.html#dx6-38004" >768
pnl_vect_set, 763
+href="pnl-manualse4.html#dx6-34007" >769
pnl_vect_set_all, 764
+href="pnl-manualse4.html#dx6-34010" >770
pnl_vect_set_subblock, 765
+href="pnl-manualse4.html#dx6-32015" >771
pnl_vect_set_zero, 766
+href="pnl-manualse4.html#dx6-34011" >772
pnl_vect_sum, 767
+href="pnl-manualse4.html#dx6-37011" >773
+ + + pnl_vect_swap_elements, 768
+href="pnl-manualse4.html#dx6-41001" >774
pnl_vect_wrap_array, 769
+href="pnl-manualse4.html#dx6-32017" >775
pnl_vect_wrap_hmat, 770
+href="pnl-manualse4.html#dx6-90005" >776
pnl_vect_wrap_mat, 771
+href="pnl-manualse4.html#dx6-32020" >777
pnl_vect_wrap_mat_row, 772
+href="pnl-manualse4.html#dx6-50017" >778
pnl_vect_wrap_subvect, 773
+href="pnl-manualse4.html#dx6-32018" >779
pnl_vect_wrap_subvect_with_last, 774
+href="pnl-manualse4.html#dx6-32019" >780

- - - RCadd, 775
+href="pnl-manualse3.html#dx5-26008" >781
RCdiv, 776
+href="pnl-manualse3.html#dx5-26017" >782
RCmul, 777
+href="pnl-manualse3.html#dx5-26014" >783
RCsub, 778
+href="pnl-manualse3.html#dx5-26011" >784

SQR, 779
+href="pnl-manualse3.html#dx5-18008" >785
Structs
    PnlArray, 780
+href="pnl-manualse2.html#dx4-14001" >786
    PnlBandMat, 781
+href="pnl-manualse4.html#dx6-72001" >787
    PnlBasis, 782
+href="pnl-manualse7.html#dx9-105001" >788
    PnlBicgSolver, 783
+href="pnl-manualse4.html#dx6-94003" >789
    PnlCell, 784
+href="pnl-manualse2.html#dx4-13002" >790
    PnlCgSolver, 785
+href="pnl-manualse4.html#dx6-94002" >791
    PnlCmplxFunc, 786
+href="pnl-manualse10.html#dx12-119001" >792
    PnlFunc, 787
+href="pnl-manualse13.html#dx15-131001" >793
    PnlFunc2D, 788
+href="pnl-manualse13.html#dx15-131011" >794
    PnlFuncDFunc, 789
+href="pnl-manualse13.html#dx15-131021" >795
    PnlGmresSolver, 790
+href="pnl-manualse4.html#dx6-94004" >796
    PnlHmat, 791
+href="pnl-manualse4.html#dx6-87001" >797
    PnlHmatComplex, 792
+href="pnl-manualse4.html#dx6-87003" >798
    PnlHmatInt, 793
+href="pnl-manualse4.html#dx6-87002" >799
    PnlIterationBase, 794
+href="pnl-manualse4.html#dx6-94001" >800
    PnlList, 795
+href="pnl-manualse2.html#dx4-13001" >801
    PnlMat, 796
+href="pnl-manualse4.html#dx6-47001" >802
    PnlMatComplex, 797
+href="pnl-manualse4.html#dx6-47003" >803
    PnlMatInt, 798
+href="pnl-manualse4.html#dx6-47002" >804
    PnlObject, 799
+href="pnl-manualse2.html#dx4-12001" >805
    PnlODEFunc, 800
+href="pnl-manualse11.html#dx13-121001" >806
    PnlPermutation, 801
+href="pnl-manualse4.html#dx6-61001" >807
    PnlRnFuncR, 802
+href="pnl-manualse13.html#dx15-131032" >808
    PnlRnFuncRm, 803
+href="pnl-manualse13.html#dx15-131042" >809
    PnlRnFuncRmDFunc, 804
+href="pnl-manualse13.html#dx15-131060" >810
    PnlRnFuncRn, 805
+href="pnl-manualse13.html#dx15-131043" >811
    PnlRnFuncRnDFunc, 806
+href="pnl-manualse13.html#dx15-131061" >812
    PnlRng, 807
+href="pnl-manualse6.html#dx8-102001" >813
    PnlSpMat, 808
+href="pnl-manualse4.html#dx6-79001" >814
    PnlSpMatComplex, 809
+href="pnl-manualse4.html#dx6-79003" >815
+ + +     PnlSpMatInt, 810
+href="pnl-manualse4.html#dx6-79002" >816
    PnlTridiagMat, 811
+href="pnl-manualse4.html#dx6-63001" >817
    PnlTridiagMatLU, 812
+href="pnl-manualse4.html#dx6-63013" >818
    PnlVect, 813
+href="pnl-manualse4.html#dx6-29001" >819
    PnlVectCompact, 814
+href="pnl-manualse4.html#dx6-44001" >820
    PnlVectComplex, 815
+href="pnl-manualse4.html#dx6-29003" >821
    PnlVectInt, 816
+href="pnl-manualse4.html#dx6-29002" >822

- - - -
-
  • All macro names begin with +

    All macro names begin with PNL_ and are capitalised. -

  • -
  • Differences between
  • +
  • +

    Differences between copy and clone methods. The copy methods take a single argument and return a pointer to an object of the same type which is an independent copy of its argument. - Example: - - - -

    -     PnlVect *v1, *v2; -  
        v1 = pnl_vect_create_from_scalar (5, 2.5); -  
        v2 = pnl_vect_copy (v1); -  
         -
    -

    +

        PnlVect *v1, *v2; 
    +    v1 = pnl_vect_create_from_scalar (5, 2.5); 
    +    v2 = pnl_vect_copy (v1);
    + +

    v1 and v2 are two vectors of size 5 with all their elements equal to 2.5. Note that v2 1.2 v1 and v2 are independent in the sense that a modification to one of them does not affect the other. -

    The clone methods take two arguments and fill the first one with the second one. - Example: - - - -

    -
    -     PnlVect *v1, *v2; -  
        v1 = pnl_vect_create_from_scalar (5, 2.5); -  
        v2 = pnl_vect_new (); -  
        pnl_vect_clone (v2, v1); -  
         -
    -

    The clone methods take two arguments and fill the first one with the second one. Example: +

    +
        PnlVect *v1, *v2; 
    +    v1 = pnl_vect_create_from_scalar (5, 2.5); 
    +    v2 = pnl_vect_new (); 
    +    pnl_vect_clone (v2, v1);
    + +

    v1 and v2 are two vectors of size 5 with all their elements equal to 2.5. Note that v2 1.2 v1 and v2 are independent in the sense that a modification to one of them does not modify the other. + + +

  • -
  • All objects are measured using integers +

    All objects are measured using integers int and not size_t. Hence, iterations over vectors, matrices, …should use an index of type int. -

  • -
  • In fonctions ending in
  • +
  • +

    In fonctions ending in inplace, the output parameter must be different from any of the input - parameters.

  • -

    + parameters.

    +

    1.3 Using Pnl

    -

    In this section, we assume that the library is installed in the directory

    In this section, we assume that the library is installed in the directory $HOME/pnl-xxx. -

    Once installed, the library can be found in the

    Once installed, the library can be found in the $HOME/pnl-xxx/lib directory and the header files in the $HOME/pnl-xxx/include directory. -

    +

    1.3.1 Compiling and Linking
    -

    The header files of the library are installed in a root

    The header files of the library are installed in a root pnl directory and should always be included with this pnl/ prefix. So, for instance to use random number generators you should -include - - - -

    -
    -#include <pnl/pnl_random.h> -
    -

    -

    -

    pnl/ prefix. So, for instance to use random number generators you should include +

    +
    #include <pnl/pnl_random.h>
    +

    Compiling and linking by hand. If gcc or llvm is used, you should pass the following options

      -
    • +

      -I$HOME/pnl-xxx/include for compiling -

    • -
    • +
    • +

      -L$HOME/pnl-xxx/lib -lpnl for linking

    -

    This does not work straight away on all OS especially if the library is not installed in a standard +class="ec-lmtt-10x-x-109"> -lpnl for linking

    +

    This does not work straight away on all OS especially if the library is not installed in a standard directory namely /usr/ or /usr/local/ for which you need a privileged writing access. On some systems, you may need to add to the linker flags the dependencies of the library, which can become very tedious. Therefore, we provide a second automatic mechanism which takes care of the dependencies on its own. + + +

    -

    Compiling and linking using an automatic Makefile. This mechanism only works under Unix (it has been tested under various Linux distributions and Mac OS X). -

    First, you need to create a new directory wherever you want, put in all your code and create a +

    First, you need to create a new directory wherever you want, put in all your code and create a Makefile as below -

    To define your target just add the executable name, say

    To define your target just add the executable name, say my-exec, to the BINS list and create an entry 1.3.1

    Assume you want to create two binaries :

    Assume you want to create two binaries : my-exec based on mixed C and C++ code (file1.c and file2.cpp) and mybinary based on poo1.cxx and poo2.cpp. You can use the following -Makefile. - - - -

    -
    -## Flags passed to the linker - 
    LDFLAGS= - 

    ## Flags passed to the compiler - 
    CFLAGS= - 

    ## list of executables to create - 
    BINS=my-exec mybinary - 

    my_exec_SRC=file1.c file2.cpp - 
    # optional flags for compiling and linking - 
    my_exec_CFLAGS= - 
    my_exec_CXXFLAGS= - 
    my_exec_LDFLAGS= - 

    mybinary_SRC=poo1.cxx poo2.cpp - 
    # optional flags for compiling and linking - 
    mybinary_CFLAGS= - 
    mybinary_CXXFLAGS= - 
    mybinary_LDFLAGS= - 


    ## This line must be the last one - 
    include full_path_to_pnl_build/CMakeuser.incl -
    -

    Let us comment a little the different variables

    +class="ec-lmtt-10x-x-109">poo2.cpp
    . You can use the following Makefile. +

    +
    ## Flags passed to the linker 
    +LDFLAGS= 
    + 
    +## Flags passed to the compiler 
    +CFLAGS= 
    + 
    +## list of executables to create 
    +BINS=my-exec mybinary 
    + 
    +my_exec_SRC=file1.c file2.cpp 
    +# optional flags for compiling and linking 
    +my_exec_CFLAGS= 
    +my_exec_CXXFLAGS= 
    +my_exec_LDFLAGS= 
    + 
    +mybinary_SRC=poo1.cxx poo2.cpp 
    +# optional flags for compiling and linking 
    +mybinary_CFLAGS= 
    +mybinary_CXXFLAGS= 
    +mybinary_LDFLAGS= 
    + 
    + 
    +## This line must be the last one 
    +include full_path_to_pnl_build/CMakeuser.incl
    +

    Let us comment a little the different variables

      -
    • +

      CFLAGS: global flags used for creating objects based on C code -

    • -
    • +
    • +

      CXXFLAGS: global flags used for creating objects based on C++ code -

    • -
    • +
    • +

      LDFLAGS: gobal linker flags. -

    • -
    • +
    • +

      binaryname_CFLAGS: flags used when creating the objects based on C code and required by binaryname -

    • -
    • +
    • +

      binaryname_CXXFLAGS: flags used when creating the objects based on C++ code and required by binaryname - - - -

    • -
    • +
    • +

      binaryname_LDFLAGS: flags used when linking objects for creating binaryname

    -

    An example of such a Makefile can be found in binaryname

    +

    An example of such a Makefile can be found in pnl-xxx/perso. -

    Warning: if a file appears in the source list of several binairies, the flags used to compile this file -are determined by the ones of the first binary involving this file. In the following example main.cpp -will always be compiled with the flag -O3 even for generating bin2 -

    -
    -BINS=bin1 bin2 - 

    bin1_SRC=main.cpp poo1.c - 
    my_exec_CXXFLAGS=-O3 - 

    bin2_SRC=main.cpp poo2.c - 
    mybinary_CXXFLAGS=-g -O0 - 

    ## This line must be the last one - 
    include full_path_to_pnl_build/CMakeuser.incl -
    -

    -

    -

    Warning: if a file appears in the source list of several binairies, the flags used to compile this +file are determined by the ones of the first binary involving this file. In the following +example main.cpp will always be compiled with the flag -O3 even for generating bin2 +

    +
    BINS=bin1 bin2 
    + 
    +bin1_SRC=main.cpp poo1.c 
    +my_exec_CXXFLAGS=-O3 
    + 
    +bin2_SRC=main.cpp poo2.c 
    +mybinary_CXXFLAGS=-g -O0 
    + 
    +## This line must be the last one 
    +include full_path_to_pnl_build/CMakeuser.incl
    +

    Compiling and linking using CMake. If you already use CMake for your new project, just add the following to your toplevel CMakeLists.txt - - - -

    -
    -find_package(Pnl REQUIRED) - 
    set(LIBS ${LIBS} ${PNL_LIBRARIES}) - 
    include_directories(${PNL_INCLUDE_DIRS}) - 
    # Deactivate PNL debugging stuff on Release builds - 
    if(${CMAKE_BUILD_TYPE} STREQUAL "Release") - 
        add_definitions(-DPNL_RANGE_CHECK_OFF) - 
    endif() -
    -

    Then, call cmake with the following extra flag - - - -

    -
    --DCMAKE_PREFIX_PATH=path/to/build-dir -
    -

    or add the variable CMakeLists.txt

    +
    find_package(Pnl REQUIRED) 
    +set(LIBS ${LIBS} ${PNL_LIBRARIES}) 
    +include_directories(${PNL_INCLUDE_DIRS}) 
    +# Deactivate PNL debugging stuff on Release builds 
    +if(${CMAKE_BUILD_TYPE} STREQUAL "Release") 
    +    add_definitions(-DPNL_RANGE_CHECK_OFF) 
    +endif()
    +

    Then, call cmake with the following extra flag

    +
    -DCMAKE_PREFIX_PATH=path/to/build-dir
    +

    or add the variable CMAKE_BUILD_TYPE to the GUI.
    -

    Just in case, we give an example of a complete although elementary

    Just in case, we give an example of a complete although elementary CMakeLists.txt

    -

    +

    1.3.2 Inline Functions and getters
    -

    If it is supported by your compiler, getter and setter functions are declared as inline functions. This +

    If it is supported by your compiler, getter and setter functions are declared as inline functions. This is automatically detected when running CMake. By default, setter and getter functions check that the required access is valid, basically it boils down to checking whether the index of the access is within an acceptable range. These extra tests can become very expensive when getter and setter functions are intensively called. -

    Thus, it is possible to alter this default behaviour by defining the macro

    Thus, it is possible to alter this default behaviour by defining the macro PNL_RANGE_CHECK_OFF. This macro is automatically defined when the library is compiled in Release mode, ie. with + Inverse Laplace Transform - - - + + + @@ -20,11 +20,11 @@ href="#tailpnl-manualse10.html">tail] [up]

    10 Inverse Laplace Transform

    + id="x12-11900010">Inverse Laplace Transform

    For a real valued function f such that tf(t10
    -
    +
     ^     ∫ ∞      -λt
-f(λ) =  0  f(t)e   dt    for λ ∈ ℂ with Re (λ ) ≥ σc.
-
    +src="pnl-manual36x.png" alt=" ∫ ∞ -λt +f^(λ) = f(t)e dt for λ ∈ ℂ with Re (λ ) ≥ σc. + 0 +" class="math-display" /> +

    To use the following functions, you should include pnl/pnl_laplace.h. - - - -

    -
    -typedef struct - 
    { - 
      dcomplex (*F) (dcomplex x, void *params); - 
      void *params; - 
    } PnlCmplxFunc; - 
      -
    -

    + id="dx12-119001">

    + +
    typedef struct 
    +{ 
    +  dcomplex (*F) (dcomplex x, void *params); 
    +  void *params; 
    +} PnlCmplxFunc;
    +href="#x12-119006r10">pnl_ilap_gs gives far more accurate + results.

    diff --git a/docs/manual-html/pnl-manualse11.html b/docs/manual-html/pnl-manualse11.html index 6a3adccf..8e2c6326 100644 --- a/docs/manual-html/pnl-manualse11.html +++ b/docs/manual-html/pnl-manualse11.html @@ -1,13 +1,13 @@ - + Ordinary differential equations - - - + + + @@ -20,11 +20,11 @@ href="#tailpnl-manualse11.html">tail] [up]

    11 Ordinary differential equations

    + id="x13-12000011">Ordinary differential equations

    11.1 Overview

    + id="x13-12100011.1">Overview

    To use these functionalities, you should include pnl/pnl_integration.h.

    These functions are designed for numerically solving 11.1
    -
    +
    dyi
-dt-(t) = fi(t,y1(t),⋅⋅⋅,yn(t))
-
    -

    -The system of equations is defined by the following structure - - - -

    -
    -typedef struct - 
    { - 
      void (*F) (int neqn, double t, const double *y, double *yp, void *params); - 
      int neqn; - 
      void *params; - 
    } PnlODEFunc ; -
    -

    -

    +src="pnl-manual37x.png" alt="dyi(t) = fi(t,y1(t),⋅⋅⋅ ,yn(t)) +dt +" class="math-display" /> + +

    The system of equations is defined by the following structure

    +
    typedef struct 
    +{ 
    +  void (*F) (int neqn, double t, const double *y, double *yp, void *params); 
    +  int neqn; 
    +  void *params; 
    +} PnlODEFunc ;
    + t, y, params). y and yp should be both of size neqn

    We provide the following macro to evaluate a PnlODEFunc at a given point - - - -

    -
    -#define PNL_EVAL_ODEFUNC(Fstruct, t, y, yp) \ - 
        (*((Fstruct)->F))((Fstruct)->neqn, t, y, yp, (Fstruct)->params) -
    -

    -

    +href="#x13-12100011.1">PnlODEFunc at a given point

    +
    #define PNL_EVAL_ODEFUNC(Fstruct, t, y, yp) \ 
    +    (*((Fstruct)->F))((Fstruct)->neqn, t, y, yp, (Fstruct)->params)
    +

    11.2 Functions

    + id="x13-12200011.2">Functions
      -
    • int  +

      int pnl_ode_rkf45 (PnlODEFunc PnlODEFunc *f, double *y, double t, double t_out, double relerr, double abserr, int *flag)
      Description This function computes the solution of the system defined by the PnlODEFunc +href="#x13-12100011.1">PnlODEFunc f at the point t_out. On input, (t,y) should be the initial condition, abserr,relerr are the - maximum absolute and relative errors for local error tests (at each step, abs(local - error) should be less that relerr * abs(y) + abserr). Note that if abserr = 0 or - relerr = 0 on input, an optimal value for these variables is computed inside the - function The function returns an error OK or FAIL. In case of an OK code, the y - contains the solution computed at t_out, in case of a FAIL code, flag should be - examined to determine the reason of the error. Here are the different possible values for - flag + maximum absolute and relative errors for local error tests (at each step, abs(local error) + should be less that relerr * abs(y) + abserr). Note that if abserr = 0 or relerr = 0 on + input, an optimal value for these variables is computed inside the function The + function returns an error OK or FAIL. In case of an OK code, the y contains the + solution computed at t_out, in case of a FAIL code, flag should be examined to + determine the reason of the error. Here are the different possible values for flag +

        -
      • flag = 2 : integration reached t_out, it indicates successful return and is the normal +
      • +

        flag = 2 : integration reached t_out, it indicates successful return and is the normal mode for continuing integration. -

      • -
      • flag = 3 : integration was not completed because relative error tolerance was too +

      • +
      • +

        flag = 3 : integration was not completed because relative error tolerance was too small. relerr has been increased appropriately for continuing. -

      • -
      • flag = 4 : integration was not completed because more than 3000 derivative +

      • +
      • +

        flag = 4 : integration was not completed because more than 3000 derivative evaluations were needed. this is approximately 500 steps. -

      • -
      • flag = 5 : integration was not completed because solution vanished making a pure +

      • +
      • + + + +

        flag = 5 : integration was not completed because solution vanished making a pure relative error test impossible. must use non-zero abserr to continue. using the one-step integration mode for one step is a good way to proceed. -

      • -
      • flag = 6 : integration was not completed because requested accuracy could not be +

      • +
      • +

        flag = 6 : integration was not completed because requested accuracy could not be achieved using smallest allowable stepsize. user must increase the error tolerance before continued integration can be attempted. - - - -

      • -
      • flag = 7 : it is likely that rkf45 is inefficient for solving this problem. too much +

      • +
      • +

        flag = 7 : it is likely that rkf45 is inefficient for solving this problem. too much output is restricting the natural stepsize choice. use the one-step integrator mode. see pnl_ode_rkf45_step. -

      • -
      • flag = 8 : invalid input parameters this indicator occurs if any of the following is - satisfied - neqn <= 0, t=tout, relerr or abserr <= 0.
      +href="#x13-12200011.2">pnl_ode_rkf45_step. +

    • +
    • +

      flag = 8 : invalid input parameters this indicator occurs if any of the following is + satisfied - neqn <= 0, t=tout, relerr or abserr <= 0.

    -
  • int  +

    int pnl_ode_rkf45_step (PnlODEFunc PnlODEFunc *f, double *y, double *t, double t_out, double @@ -155,11 +249,11 @@

    11.2 *iwork, int *flag)
    Description Same as
    pnl_ode_rkf45 but it only computes one step of integration in the +href="#x13-12200011.2">pnl_ode_rkf45 but it only computes one step of integration in the direction of t_out. work and iwork are working arrays of size 3 + 6 * neqn and 5 respectively and should remain untouched between successive calls to the function. On output t holds the point at which integration stopped and y the value of the solution at that - point.

  • + point.

    diff --git a/docs/manual-html/pnl-manualse12.html b/docs/manual-html/pnl-manualse12.html index 18762059..03403b01 100644 --- a/docs/manual-html/pnl-manualse12.html +++ b/docs/manual-html/pnl-manualse12.html @@ -1,13 +1,13 @@ - + Optimization - - - + + + @@ -20,17 +20,17 @@ href="#tailpnl-manualse12.html">tail] [up]

    12 Optimization

    + id="x14-12300012">Optimization

    To use the functions described in this section, you should include pnl/pnl_optim.h.

    12.1 Linear constrained optimization (linear programming)

    + id="x14-12400012.1">Linear constrained optimization (linear programming)

    12.1.1 Overview
    + id="x14-12500012.1.1">Overview

    Consider the minimization problem

    @@ -40,35 +40,35 @@
    12.1.1 x
    CT T x
    s.t. AineqAineq x Bineq Bineq
    AeqAeq x = Beq Beq
    xmin x xmax
    @@ -78,15 +78,16 @@
    12.1.1

    12.1.2 Functions
    + id="x14-12600012.1.2">Functions

    To solve such a linear problem, we provide a wrapper to the LPSolve library -(LPSolve library ( +http://lpsolve.sourceforge.net).

      -
    • int  +

      int pnl_optim_linprog (const PnlVect *C, const 12.1.2 PnlVect *xopt, double *fobj_opt)
      Description This function has the following arguments: +class="newline" /> Description This function has the following arguments:

        -
      • C The coefficients of the linear objective function. -
      • -
      • A_ineq The l.h.s matrix of the inequality constraints. Can be NULL. -
      • -
      • B_ineq The r.h.s vector of the inequality constraints. The lengh of B_ineq must +
      • +

        C The coefficients of the linear objective function. +

      • +
      • +

        A_ineq The l.h.s matrix of the inequality constraints. Can be NULL. +

      • +
      • +

        B_ineq The r.h.s vector of the inequality constraints. The lengh of B_ineq must match the number of rows of A_ineq. -

      • -
      • A_eq The l.h.s matrix of the equality constraints. Can be NULL. -
      • -
      • B_eq The r.h.s vector of the equality constraints. The lengh of B_eq must match +

      • +
      • +

        A_eq The l.h.s matrix of the equality constraints. Can be NULL. +

      • +
      • +

        B_eq The r.h.s vector of the equality constraints. The lengh of B_eq must match the number of rows of A_ineq. -

      • -
      • x_min The lower bound on x. If NULL, it means all the components of x must be +

      • +
      • +

        x_min The lower bound on x. If NULL, it means all the components of x must be non negative. -

      • -
      • x_max The upper bound on x. If NULL, it means +Infinity for all the components. -
      • -
      • debug TRUE or FALSE. If TRUE some debugging details are printed. -
      • -
      • xopt The argmin of the problem. -
      • -
      • fobj_opt The value of the obective funtion at the optimum xopt
      +

    • +
    • +

      x_max The upper bound on x. If NULL, it means +Infinity for all the components. +

    • +
    • +

      debug TRUE or FALSE. If TRUE some debugging details are printed. +

    • +
    • +

      xopt The argmin of the problem. + + + +

    • +
    • +

      fobj_opt The value of the obective funtion at the optimum xopt

    The function returns OK or FAIL.

    -
  • int  +

    int pnl_optim_linprog_sp (const PnlSpMat *C, const 12.1.2 PnlVect *xopt, double *fobj_opt)
    Description This function has the following arguments: - - - +class="newline" /> Description This function has the following arguments:

      -
    • C The coefficients of the linear obejctive function, given as a sparse matrix with a +
    • +

      C The coefficients of the linear obejctive function, given as a sparse matrix with a single column. -

    • -
    • A_ineq The l.h.s matrix of the inequality constraints. Can be NULL. -
    • -
    • B_ineq The r.h.s vector of the inequality constraints. The lengh of B_ineq must +

    • +
    • +

      A_ineq The l.h.s matrix of the inequality constraints. Can be NULL. +

    • +
    • +

      B_ineq The r.h.s vector of the inequality constraints. The lengh of B_ineq must match the number of rows of A_ineq. -

    • -
    • A_eq The l.h.s matrix of the equality constraints. Can be NULL. -
    • -
    • B_eq The r.h.s vector of the equality constraints. The lengh of B_eq must match +

    • +
    • +

      A_eq The l.h.s matrix of the equality constraints. Can be NULL. +

    • +
    • +

      B_eq The r.h.s vector of the equality constraints. The lengh of B_eq must match the number of rows of A_ineq. -

    • -
    • index_min The indices of the variables with a lower bound constraint. The +

    • +
    • +

      index_min The indices of the variables with a lower bound constraint. The corresponding lower bound is given in x_min -

    • -
    • x_min The lower bound on x. If NULL, it means all the components of x must be +

    • +
    • +

      x_min The lower bound on x. If NULL, it means all the components of x must be non negative. Can be NULL. For non specified variables, the default lower bound is 0. -

    • -
    • index_max The indices of the variables with an upper bound constraint. The +

    • +
    • +

      index_max The indices of the variables with an upper bound constraint. The corresponding lower bound is given in x_max. Can be NULL. -

    • -
    • x_max The upper bound on x. If NULL, it means +Infinity for all the components. + + + +

    • +
    • +

      x_max The upper bound on x. If NULL, it means +Infinity for all the components. For non specified variables, the default upper bound is +Infinity.. -

    • -
    • debug TRUE or FALSE. If TRUE some debugging details are printed. -
    • -
    • xopt The argmin of the problem. -
    • -
    • fobj_opt The value of the obective funtion at the optimum xopt
    +

  • +
  • +

    debug TRUE or FALSE. If TRUE some debugging details are printed. +

  • +
  • +

    xopt The argmin of the problem. +

  • +
  • +

    fobj_opt The value of the obective funtion at the optimum xopt

  • The function returns OK or FAIL.

    12.2 Nonlinear constrained optimization

    - - - + id="x14-12700012.2">Nonlinear constrained optimization

    12.2.1 Overview
    + id="x14-12800012.2.1">Overview

    A standard Constrained Nonlinear Optimization problem can be written as:

    -
    +
       (
-   |{  min f(x)
+src=
    + |( cE (x ) = 0 +" class="math-display" /> +

    where the function f : n is the objective function, cI : n mI are the inequality @@ -242,21 +267,24 @@

    12.2.1 : n mE are the equality constraints. These functions are supposed to be smooth. + + +

    In general, the inequality constraints are of the form cI(x) = (g(x),x - l,u - x). The vector l and u -are the lower and upper bounds on the variables . The vector l and +u are the lower and upper bounds on the variables x and g(12.2.1 E) mI × ×mE, such that the well known Karush-Kuhn-Tucker (KKT) optimality @@ -289,18 +317,17 @@

    12.2.1
    -
    +
        (         I  E               I    I     E    E
-    ||||  ∇ ℓ(x,λ ,λ  ) = ∇f (x )E- ∇c (x)λ  - ∇c  (x)λ  = 0
-    |{                     c (x) = 0
-(P )|                     cI(x) ≥ 0
-    ||||                       λI ≥ 0
-    (               cIi(x)λIi = 0,i = 1...mI
-
    - - - +src="pnl-manual40x.png" alt=" ( + ||| ∇ ℓ(x,λI,λE) = ∇f (x)- ∇cI (x)λI - ∇cE (x)λE = 0 + |||| cE (x ) = 0 + { cI(x) ≥ 0 +(P )|| λI ≥ 0 + |||| cI(x)λI = 0, i = 1...m + |( i i I +" class="math-display" /> +

    l is known as the Lagrangian of the problem 12.2.1

    12.2.2 Functions
    + id="x14-12900012.2.2">Functions

    To solve an inequality constrained optimization problem, ie mE = 0, we provide the following function.

      -
    • int  +

      int pnl_optim_intpoints_bfgs_solve (PnlRnFuncRPnlRnFuncR*func, PnlRnFuncRmPnlRnFuncRm*grad_func, PnlRnFuncRmPnlRnFuncRm*nl_constraints, PnlVect *lower_bounds, @@ -344,69 +372,88 @@

      12.2.2 PnlVect *output)
      Description This function has the following arguments: +

      + + +
        -
      • func is the function to minimize +

        func is the function to minimize f. -

      • -
      • grad is the gradient of
      • +
      • +

        grad is the gradient of f. If this gradient is not available, then enter grad=NULL. In this case, finite difference will be used to estimate the gradient. -

      • -
      • nl_constraints is the function
      • +
      • +

        nl_constraints is the function g(x), ie the non linear inequality constraints. -

      • -
      • lower_bounds are the lower bounds on
      • +
      • +

        lower_bounds are the lower bounds on x. Can be NULL if there is no lower bound. -

      • -
      • upper_bounds are the upper bounds on
      • +
      • +

        upper_bounds are the upper bounds on x. Can be NULL if there is no upper bound. -

      • -
      • x_input is the initial point where the algorithm starts. -
      • -
      • tolerance is the precision required in solving (P). -
      • -
      • iter_max is the maximum number of iterations in the algorithm. -
      • -
      • print_algo_steps is a flag to decide to print information. -
      • -
      • x_output is the point where the algorithm stops.
      +

    • +
    • +

      x_input is the initial point where the algorithm starts. +

    • +
    • +

      tolerance is the precision required in solving (P). +

    • +
    • +

      iter_max is the maximum number of iterations in the algorithm. +

    • +
    • +

      print_algo_steps is a flag to decide to print information. +

    • +
    • +

      x_output is the point where the algorithm stops.

    The algorithm returns an int, its value depends on the output status of the algorithm. We have 4 cases:

    +
      +
    • +

      0: Failure: Initial point is not strictly feasible. +

    • +
    • +

      1: Step is too small, we stop the algorithm. -

        -
      • 0: Failure: Initial point is not strictly feasible. -
      • -
      • 1: Step is too small, we stop the algorithm. -
      • -
      • 2: Maximum number of iterations reached. -
      • -
      • 3: A solution has been found up to the required accuracy.
      +

    • +
    • +

      2: Maximum number of iterations reached. +

    • +
    • +

      3: A solution has been found up to the required accuracy.

    The last case is equivalent to the two inequalities:

    -
    -         I                   I    I
-||∇ℓ(x,λ )||∞  = ||∇f (x)- ∇c  (x)λ ||∞ < tolerance
-
    -
    - ||cI(x)λI|| <  tolerance
+<div class= +||∇ ℓ(x,λI)||∞ = ||∇f (x )- ∇cI (x )λI||∞ < tolerance
+ +

    +

    +
    +||cI(x)λI||  < tolerance
          ∞
-
    +" class="math-display" />

    where cI(x)) . * λ + Root finding - - - + + + @@ -20,188 +20,1299 @@ href="#tailpnl-manualse13.html">tail] [up]

    13 Root finding

    + id="x15-13000013">Root finding

    13.1 Overview

    + id="x15-13100013.1">Overview

    To provide a uniformed framework to root finding functions, we use several structures for storing different kind of functions. The pointer params is used to store the extra parameters. These new types come with dedicated macros starting in PNL_EVAL to evaluate the function and their Jacobian. - - - -

    -
    -/* - 
     * f: R --> R - 
     * The function  pointer returns f(x) - 
     * - 
    typedef struct { - 
      double (*F) (double x, void *params); - 
      void *params; - 
    } PnlFunc ; - 
    #define PNL_EVAL_FUNC(Fstruct, x) (*((Fstruct)->F))(x, (Fstruct)->params) -
    -

    + id="dx15-131001">

    +
    /* 
    + * f: R --> R 
    + * The function  pointer returns f(x) 
    + * 
    +typedef struct { 
    +  double (*F) (double x, void *params); 
    +  void *params; 
    +} PnlFunc ; 
    +#define PNL_EVAL_FUNC(Fstruct, x) (*((Fstruct)->F))(x, (Fstruct)->params)
    - - - -

    -
    -/* - 
     * f: R^2 --> R - 
     * The function pointer returns f(x) - 
     * - 
    typedef struct { - 
      double (*F) (double x, double y, void *params); - 
      void *params; - 
    } PnlFunc2D ; - 
    #define PNL_EVAL_FUNC2D(Fstruct, x, y) (*((Fstruct)->F))(x, y, (Fstruct)->params) -
    -

    + id="dx15-131011"> + +

    /* 
    + * f: R^2 --> R 
    + * The function pointer returns f(x) 
    + * 
    +typedef struct { 
    +  double (*F) (double x, double y, void *params); 
    +  void *params; 
    +} PnlFunc2D ; 
    +#define PNL_EVAL_FUNC2D(Fstruct, x, y) (*((Fstruct)->F))(x, y, (Fstruct)->params)
    - - - -

    -
    -/* - 
     * f: R --> R - 
     * The function pointer computes f(x) and Df(x) and stores them in fx - 
     * and dfx respectively - 
     * - 
    typedef struct { - 
      void (*F) (double x, double *fx, double *dfx, void *params); - 
      void *params; - 
    } PnlFuncDFunc ; - 
    #define PNL_EVAL_FUNC_FDF(Fstruct, x, fx, dfx) (*((Fstruct)->F))(x, fx, dfx, (Fstruct)->params) -
    -

    + id="dx15-131021"> + +

    /* 
    + * f: R --> R 
    + * The function pointer computes f(x) and Df(x) and stores them in fx 
    + * and dfx respectively 
    + * 
    +typedef struct { 
    +  void (*F) (double x, double *fx, double *dfx, void *params); 
    +  void *params; 
    +} PnlFuncDFunc ; 
    +#define PNL_EVAL_FUNC_FDF(Fstruct, x, fx, dfx) (*((Fstruct)->F))(x, fx, dfx, (Fstruct)->params)
    - - - -

    -
    -/* - 
     * f: R^n --> R - 
     * The function pointer returns f(x) - 
     * - 
    typedef struct { - 
      double (*F) (const PnlVect *x, void *params); - 
      void *params; - 
    } PnlRnFuncR ; - 
    #define PNL_EVAL_RNFUNCR(Fstruct, x) (*((Fstruct)->F))(x, (Fstruct)->params) -
    -

    + id="dx15-131032"> + +

    /* 
    + * f: R^n --> R 
    + * The function pointer returns f(x) 
    + * 
    +typedef struct { 
    +  double (*F) (const PnlVect *x, void *params); 
    +  void *params; 
    +} PnlRnFuncR ; 
    +#define PNL_EVAL_RNFUNCR(Fstruct, x) (*((Fstruct)->F))(x, (Fstruct)->params)
    + id="dx15-131042"> - - - -

    -
    -/* - 
     * f: R^n --> R^m - 
     * The function pointer computes the vector f(x) and stores it in - 
     * fx (vector of size m) - 
     * - 
    typedef struct { - 
      void (*F) (const PnlVect *x, PnlVect *fx, void *params); - 
      void *params; - 
    } PnlRnFuncRm ; - 
    #define PNL_EVAL_RNFUNCRM(Fstruct, x, fx) (*((Fstruct)->F))(x, fx, (Fstruct)->params) - 

    /* - 
     * Synonymous of PnlRnFuncRm for f:R^n --> R^n - 
     * - 
    typedef PnlRnFuncRm PnlRnFuncRn; - 
    #define PNL_EVAL_RNFUNCRN  PNL_EVAL_RNFUNCRM -
    -

    + id="dx15-131043"> + +

    /* 
    + * f: R^n --> R^m 
    + * The function pointer computes the vector f(x) and stores it in 
    + * fx (vector of size m) 
    + * 
    +typedef struct { 
    +  void (*F) (const PnlVect *x, PnlVect *fx, void *params); 
    +  void *params; 
    +} PnlRnFuncRm ; 
    +#define PNL_EVAL_RNFUNCRM(Fstruct, x, fx) (*((Fstruct)->F))(x, fx, (Fstruct)->params) 
    + 
    +/* 
    + * Synonymous of PnlRnFuncRm for f:R^n --> R^n 
    + * 
    +typedef PnlRnFuncRm PnlRnFuncRn; 
    +#define PNL_EVAL_RNFUNCRN  PNL_EVAL_RNFUNCRM
    + id="dx15-131060"> + id="dx15-131061"> + +
    /* 
    + * f: R^n --> R^m 
    + * The function pointer computes the vector f(x) and stores it in fx 
    + * (vector of size m) 
    + * The Dfunction pointer computes the matrix Df(x) and stores it in dfx 
    + * (matrix of size m x n) 
    + * 
    +typedef struct { 
    +  void (*F) (const PnlVect *x, PnlVect *fx, void *params); 
    +  void (*DF) (const PnlVect *x, PnlMat *dfx, void *params); 
    +  void (*FDF) (const PnlVect *x, PnlVect *fx, PnlMat *dfx, void *params); 
    +  void *params; 
    +} PnlRnFuncRmDFunc ; 
    +#define PNL_EVAL_RNFUNCRM_DF(Fstruct, x, dfx) \ 
    +    (*((Fstruct)->Dfunction))(x, dfx, (Fstruct)->params) 
    +#define PNL_EVAL_RNFUNCRM_FDF(Fstruct, x, fx, dfx) \ 
    +    (*((Fstruct)->F))(x, fx, dfx, (Fstruct)->params) 
    +#define PNL_EVAL_RNFUNCRM_F_DF(Fstruct, x, fx, dfx)    \ 
    +      if ( (Fstruct)->FDF != NULL )                    \ 
    +        {                                              \ 
    +          PNL_EVAL_RNFUNCRN_FDF (Fstruct, x, fx, dfx); \ 
    +        }                                              \ 
    +      else                                             \ 
    +        {                                              \ 
    +          PNL_EVAL_RNFUNCRN (Fstruct, x, fx);          \ 
    +          PNL_EVAL_RNFUNCRN_DF (Fstruct, x, dfx);      \ 
    +        } 
    +/* 
    + * Synonymous of PnlRnFuncRmDFunc for f:R^n --> R^m 
    + * 
    +typedef PnlRnFuncRmDFunc PnlRnFuncRnDFunc; 
    +#define PNL_EVAL_RNFUNCRN_DF PNL_EVAL_RNFUNCRM_DF 
    +#define PNL_EVAL_RNFUNCRN_FDF PNL_EVAL_RNFUNCRM_FDF 
    +#define PNL_EVAL_RNFUNCRN_F_DF PNL_EVAL_RNFUNCRM_F_DF
    -

    -
    -/* - 
     * f: R^n --> R^m - 
     * The function pointer computes the vector f(x) and stores it in fx - 
     * (vector of size m) - 
     * The Dfunction pointer computes the matrix Df(x) and stores it in dfx - 
     * (matrix of size m x n) - 
     * - 
    typedef struct { - 
      void (*F) (const PnlVect *x, PnlVect *fx, void *params); - 
      void (*DF) (const PnlVect *x, PnlMat *dfx, void *params); - 
      void (*FDF) (const PnlVect *x, PnlVect *fx, PnlMat *dfx, void *params); - 
      void *params; - 
    } PnlRnFuncRmDFunc ; - 
    #define PNL_EVAL_RNFUNCRM_DF(Fstruct, x, dfx) \ - 
        (*((Fstruct)->Dfunction))(x, dfx, (Fstruct)->params) - 
    #define PNL_EVAL_RNFUNCRM_FDF(Fstruct, x, fx, dfx) \ - 
        (*((Fstruct)->F))(x, fx, dfx, (Fstruct)->params) - 
    #define PNL_EVAL_RNFUNCRM_F_DF(Fstruct, x, fx, dfx)    \ - 
          if ( (Fstruct)->FDF != NULL )                    \ - 
            {                                              \ - 
              PNL_EVAL_RNFUNCRN_FDF (Fstruct, x, fx, dfx); \ - 
            }                                              \ - 
          else                                             \ - 
            {                                              \ - 
              PNL_EVAL_RNFUNCRN (Fstruct, x, fx);          \ - 
              PNL_EVAL_RNFUNCRN_DF (Fstruct, x, dfx);      \ - 
            } - 
    /* - 
     * Synonymous of PnlRnFuncRmDFunc for f:R^n --> R^m - 
     * - 
    typedef PnlRnFuncRmDFunc PnlRnFuncRnDFunc; - 
    #define PNL_EVAL_RNFUNCRN_DF PNL_EVAL_RNFUNCRM_DF - 
    #define PNL_EVAL_RNFUNCRN_FDF PNL_EVAL_RNFUNCRM_FDF - 
    #define PNL_EVAL_RNFUNCRN_F_DF PNL_EVAL_RNFUNCRM_F_DF -
    -

    -

    +

    13.2 Functions

    + id="x15-13200013.2">Functions

    To use the following functions, you should include pnl/pnl_root.h. - - -

    Real valued functions of a real argument

    On exit, the root is stored in res.

    -
  • int  +

    int pnl_root_bisection (PnlFunc PnlFunc *Func, double xmin, double xmax, double epsrel, double espabs, int N_max, double *res)
    Description Find the root of F between x1 and x2 with the accuracy |x2 - x1| < epsrel * x1 + epsabs or with the maximum number of iterations N_max. On exit, res = (x2 + x1) / - 2.

  • + 2.

    Vector valued functions with several arguments

    On exit, the root is stored in res. Note that the function F must also compute the first derivative of the function. When defining Func, you must either define Func->F and Func->DF or Func->FDF.

    We provide two wrappers for calling minpack routines.

      -
    • int  +

      int pnl_root_fsolve (PnlRnFuncRnDFunc PnlRnFuncRnDFunc *f, PnlVect *x, 13.2 : nn. Note that the number of components of f must be equal to the number of variates of f. This function returns OK or FAIL if something went wrong.
      Parameters  +class="newline" /> Parameters 

        -
      • f is a pointer to a PnlRnFuncRnDFunc used to store the function whose root is to +
      • +

        f is a pointer to a PnlRnFuncRnDFunc used to store the function whose root is to be found. f can also store the Jacobian of the function, if not it is computed using finite differences (see the file examples/minpack_test.c for a usage example). f->FDF can be NULL because it is not used in this function. -

      • -
      • x contains on input the starting point of the search and an approximation of the + + + +

      • +
      • +

        x contains on input the starting point of the search and an approximation of the root of f on output, -

      • -
      • xtol is the precision required on x, if set to 0 a default value is used. -
      • -
      • maxfev is the maximum number of evaluations of the function f before the +

      • +
      • +

        xtol is the precision required on x, if set to 0 a default value is used. +

      • +
      • +

        maxfev is the maximum number of evaluations of the function f before the algorithm returns, if set to 0, a coherent number is determined internally. -

      • -
      • nfev contains on output the number of evaluations of f during the algorithm, -
      • -
      • scale is a vector used to rescale x in a way that each coordinate of the solution is +

      • +
      • +

        nfev contains on output the number of evaluations of f during the algorithm, +

      • +
      • +

        scale is a vector used to rescale x in a way that each coordinate of the solution is approximately of order 1 after rescaling. If on input scale=NULL, a scaling vector is computed internally by the algorithm. -

      • -
      • error_msg is a boolean (TRUE or FALSE) to specify if an error message should +

      • +
      • +

        error_msg is a boolean (TRUE or FALSE) to specify if an error message should be printed when the algorithm stops before having converged. -

      • -
      • On output, fx contains f(x).
      - - - +

    • +
    • +

      On output, fx contains f(x).

    -
  • int  +

    int pnl_root_fsolve_lsq (PnlRnFuncRmDFunc PnlRnFuncRmDFunc *f, PnlVect *x, int m, 13.2 )
    Description Compute the root of x n i13.2
    )
    2, note that there is no reason why m should be equal to n.
    Parameters  +class="newline" /> Parameters 

      -
    • f is a pointer to a PnlRnFuncRmDFunc used to store the function whose root is to +
    • +

      f is a pointer to a PnlRnFuncRmDFunc used to store the function whose root is to be found. f can also store the Jacobian of the function, if not it is computed using finite differences (see the file examples/minpack_test.c for a usage example). f->FDF can be NULL because it is not used in this function. -

    • -
    • x contains on input the starting point of the search and an approximation of the +

    • +
    • + + + +

      x contains on input the starting point of the search and an approximation of the root of f on output, -

    • -
    • m is the number of components of f, -
    • -
    • xtol is the precision required on x, if set to 0 a default value is used. -
    • -
    • ftol is the precision required on f, if set to 0 a default value is used. -
    • -
    • gtol is the precision required on the Jacobian of f, if set to 0 a default value is used. -
    • -
    • maxfev is the maximum number of evaluations of the function f before the +

    • +
    • +

      m is the number of components of f, +

    • +
    • +

      xtol is the precision required on x, if set to 0 a default value is used. +

    • +
    • +

      ftol is the precision required on f, if set to 0 a default value is used. +

    • +
    • +

      gtol is the precision required on the Jacobian of f, if set to 0 a default value is used. +

    • +
    • +

      maxfev is the maximum number of evaluations of the function f before the algorithm returns, if set to 0, a coherent number is determined internally. -

    • -
    • nfev contains on output the number of evaluations of f during the algorithm, -
    • -
    • scale is a vector used to rescale x in a way that each coordinate of the solution is +

    • +
    • +

      nfev contains on output the number of evaluations of f during the algorithm, +

    • +
    • +

      scale is a vector used to rescale x in a way that each coordinate of the solution is approximately of order 1 after rescaling. If on input scale=NULL, a scaling vector is computed internally by the algorithm. -

    • -
    • error_msg is a boolean (TRUE or FALSE) to specify if an error message should +

    • +
    • +

      error_msg is a boolean (TRUE or FALSE) to specify if an error message should be printed when the algorithm stops before having converged. -

    • -
    • On output, fx contains f(x).
    +

  • +
  • +

    On output, fx contains f(x).

  • diff --git a/docs/manual-html/pnl-manualse14.html b/docs/manual-html/pnl-manualse14.html index bc563077..f0bec9ce 100644 --- a/docs/manual-html/pnl-manualse14.html +++ b/docs/manual-html/pnl-manualse14.html @@ -1,13 +1,13 @@ - + Special functions - - - + + + @@ -20,7 +20,7 @@ href="#tailpnl-manualse14.html">tail] [up]

    14 Special functions

    + id="x16-13500014">Special functions

    The special function approximations are defined in the header pnl/pnl_specfun.h.
    @@ -28,36 +28,41 @@

    14 Cephes library which uses its own error mechanism which can be activated or deactivated using the two following functions

      -
    • void  +

      void pnl_deactivate_mtherr ()
      Description Deactivate Cephes error mechanism -

    • -
    • void 
    • +
    • +

      void pnl_activate_mtherr ()
      Description Activate Cephes error mechanism

    +class="newline" /> Description Activate Cephes error mechanism

    14.1 Real Bessel functions

    + id="x16-13600014.1">Real Bessel functions
      -
    • double  +

      double pnl_bessel_i (double v, double x)
      Description Modified Bessel function of the first kind of order v. -

    • -
    • double 
    • +
    • +

      double pnl_bessel_i_scaled (double v, double x)
      Description Modified Bessel function of the first kind of order v divided by e|x|. -

    • -
    • double 
    • +
    • +

      double pnl_bessel_rati (double v, double x)
      Description Ratio of modified Bessel functions of the first kind : I14.1 (x). -

    • -
    • double 
    • +
    • +

      double pnl_bessel_j (double v, double x)
      Description Bessel function of the first kind of order v. -

    • -
    • double pnl_bessel_j_scaled (double v, double x)
      Description Bessel function of the first kind of order v. Same function as - pnl_bessel_j. -
    • -
    • double 
    • +
    • +

      double pnl_bessel_j_scaled (double v, double x)
      Description Bessel function of the first kind of order v. Same function as + pnl_bessel_j. +

    • +
    • +

      double pnl_bessel_y (double v, double x)
      Description Modified Bessel function of the second kind of order v. -

    • -
    • double 
    • +
    • +

      double pnl_bessel_y_scaled (double v, double x)
      Description Modified Bessel function of the second kind of order v. Same function as pnl_bessel_y. -

    • -
    • double pnl_bessel_y. +

    • +
    • +

      double pnl_bessel_k (double v, double x)
      Description Bessel function of the third kind of order v. -

    • -
    • double 
    • +
    • +

      double pnl_bessel_k_scaled (double v, double x)
      Description Bessel function of the third kind of order v multiplied by ex. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_bessel_h1 (double v, double x)
      Description Hankel function of the first kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_bessel_h1_scaled (double v, double x)
      Description Hankel function of the first kind of order v and divided by eIx. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_bessel_h2 (double v, double x)
      Description Hankel function of the second kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_bessel_h2_scaled (double v, double x)
      Description Hankel function of the second kind of order v and multiplied by eIx.

    +class="lmmi-8">Ix.

    + + +

    14.2 Complex Bessel functions

    + id="x16-13700014.2">Complex Bessel functions
      -
    • dcomplex  +

      dcomplex pnl_complex_bessel_i (double v, dcomplex z)
      Description Complex Modified Bessel function of the first kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_i_scaled (double v, dcomplex z)
      Description Complex Modified Bessel function of the first kind of order v divided by 14.2 z)|. - - - -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_rati (double v, dcomplex x)
      Description Ratio of complex modified Bessel functions of the first kind : 14.2 (x). -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_j (double v, dcomplex z)
      Description Complex Bessel function of the first kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_j_scaled (double v, dcomplex z)
      Description Complex Bessel function of the first kind of order v divided by e14.2 z)|. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_y (double v, dcomplex z)
      Description Complex Modified Bessel function of the second kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_y_scaled (double v, dcomplex z)
      Description Complex Modified Bessel function of the second kind of order v divided by 14.2 z)|. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_k (double v, dcomplex z)
      Description Complex Bessel function of the third kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_k_scaled (double v, dcomplex z)
      Description Complex Bessel function of the third kind of order v multiplied by ez. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_h1 (double v, dcomplex z)
      Description Complex Hankel function of the first kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_h1_scaled (double v, dcomplex z)
      Description Complex Hankel function of the first kind of order v and divided by eIz. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_h2 (double v, dcomplex z)
      Description Complex Hankel function of the second kind of order v. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_complex_bessel_h2_scaled (double v, dcomplex z)
      Description Complex Hankel function of the second kind of order v and multiplied by eIz.

    - - - +class="lmmi-8">Iz.

    14.3 Error functions

    + id="x16-13800014.3">Error functions
      -
    • double  +

      double pnl_sf_erf (double x)
      Description Compute the error function 2
@@ -272,26 +301,33 @@ <h4 class=14.3 2 dt. -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_sf_complex_erf (dcomplex z)
      Description Same as pnl_sf_erf for complex arguments. -

    • -
    • double pnl_sf_erf for complex arguments. +

    • +
    • +

      double pnl_sf_erfc (double x)
      Description Compute the complementary error function 1. - erf(x). -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_sf_complex_erfc (dcomplex x)
      Description Same as pnl_sf_erfc for complex arguments. -

    • -
    • double pnl_sf_erfc for complex arguments. +

    • +
    • +

      double pnl_sf_erfcx (double x)
      Description Compute the scaled complementary error function of x, defined by 14.3 (x). -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_sf_complex_erfcx (dcomplex z)
      Description Same as pnl_sf_erfcx for complex arguments. Note that erfcx(-i x) = +href="#x16-13800014.3">pnl_sf_erfcx for complex arguments. Note that erfcx(-i x) = w(x). -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_sf_w (dcomplex z)
      Description Compute e-z2 - erfc(-iziz). -

    • -
    • double 
    • +
    • +

      double pnl_sf_w_im (double x)
      Description Compute 2Dawson2Dawson(x)√ π- -

    • -
    • double  +

    • +
    • +

      double pnl_sf_erfi (double x)
      Description Compute -i erf(i z) -

    • -
    • dcomplex 
    • +
    • +

      dcomplex pnl_sf_complex_erfi (dcomplex z)
      Description Same as pnl_sf_erfi for complex arguments. -

    • -
    • double pnl_sf_erfi for complex arguments. +

    • +
    • +

      double pnl_sf_dawson (double x)
      Description Compute √ --
-  π22 e-x2 - erfi(x). +

    • +
    • +

      dcomplex pnl_sf_complex_dawson (dcomplex z)
      Description Same as pnl_sf_dawson for complex arguments. -

    • -
    • dcomplex pnl_sf_complex_dawson (dcomplex z)
      Description Same as pnl_sf_dawson for complex arguments. -
    • -
    • double 
    • +
    • +

      double pnl_sf_log_erf (double x)
      Description Compute log pnl_sf_erfpnl_sf_erf(x) -

    • -
    • double 
    • +
    • +

      double pnl_sf_log_erfc (double x)
      Description Compute log pnl_sf_erfcpnl_sf_erfc(x)

    +class="rm-lmr-10x-x-109">)

    14.4 Gamma functions

    + id="x16-13900014.4">Gamma functions

    For x > 0, the Gamma Function is defined by

    -
    +
           ∫ ∞
-Γ (x ) =    e-u ux-1du.
+src=
    +" class="math-display" /> +

      -
    • double  +

      double pnl_sf_fact (int n)
      Description Computes factorial of n Γ(n + 1). -

    • -
    • double 
    • +
    • +

      double pnl_sf_gamma (double x)
      Description Computes Γ(14.4 ,x 0 -

    • -
    • double 
    • +
    • +

      double pnl_sf_log_gamma (double x)
      Description Computes log14.4 -

    • int 
    • +
    • +

      int pnl_sf_log_gamma_sgn (double x, double *y, int *sgn)
      14.4 x) (-1 or +1). -

    • -
    • double 
    • +
    • +

      double pnl_sf_choose (int n, int k)
      Description Computes the binomial coefficient (n)
@@ -476,15 +526,15 @@ <h4 class=14.4 for 0 k n in double - precision.

    + precision.

    14.5 Digamma function

    + id="x16-14000014.5">Digamma function

    For x > 0, the digamma function 14.5
    -
    +
            d           Γ ′(x)
-ψ (x ) =---logΓ (x ) =-----.
+src=
    -

    -The function + +

    The function ψ admits the following integral representation

    -
    +
           ∫ ∞ ( - u     -xu  )
-ψ(x) =       e---  -e-----  .
+src=
    -

    +" class="math-display" /> + +

      -
    • double  +

      double pnl_sf_psi (double x)
      Description Return ψ(x).

    +class="rm-lmr-10x-x-109">).

    14.6 Incomplete Gamma functions

    + id="x16-14100014.6">Incomplete Gamma functions

    For a and x > 0, the Incomplete Gamma Function is defined by

    -
    +
             ∫
-           ∞  -u  a-1
-Γ (a,x ) = x  e   u   du.
-
    -

    -A relation similar to the one existing for the standard Gamma function holds +src="pnl-manual51x.png" alt=" ∫ ∞ +Γ (a,x ) = e-u ua-1du. + x +" class="math-display" /> + +

    A relation similar to the one existing for the standard Gamma function holds

    -
    +
    Γ (a,x) = - xa-e-x+-Γ (a-+-1,x).
-                   a
-
    +src="pnl-manual52x.png" alt=" - xa-e-x+-Γ (a-+-1,x) +Γ (a,x) = a . +" class="math-display" /> +

    @@ -591,11 +644,9 @@

    14.6 = Γ (a) - Γ (a,x)
--------------
-    Γ (a) =   1
------
 Γ (a) 14.6 ) = 1 - -P(a,x14.6
      -
    • double  +

      double pnl_sf_gamma_inc (double a, double x)
      Description Computes Γ(a,x), a ,x 0 -

    • -
    • void 
    • +
    • +

      void pnl_sf_gamma_inc_P (double a, double x)
      Description Computes P14.6 ,x 0 -

    • -
    • void 
    • +
    • +

      void pnl_sf_gamma_inc_Q (double a, double x)
      Description Computes Q14.6 0,x 0

    +class="rm-lmr-10x-x-109">0

    14.7 Exponential integrals

    + id="x16-14200014.7">Exponential integrals

    For x > 0 and n , the function En is defined by

    -
    -        ∫ ∞  -xu  -n
-En (x ) =    e    u  du
-         1
-
    +
    +        ∫ ∞
+En (x ) =    e-xu u-ndu
+         1
+
    +

    This function is linked to the Incomplete Gamma function by

    -
    +
            ∫ ∞                             ∫ ∞
 En (x) =    e- xu(xu )-nxn-1d(xu) = xn-1     e-tt-ndt = xn-1Γ (1 - n,x) ,
          x                               x
-
    -

    -from which we can deduce +" class="math-display" /> + +

    from which we can deduce

    -
    +
    nEn+1 (x) = e- x- xEn (x ).
-
    -

    -For + +

    For n > 1, the series expansion is given by

    -
    -                      [                                         ]
-E (x) = xn-1Γ (1 - n)+  ---1-- + --x-- - ---x2---+  --x3----- ... .
- n                       1 - n   2 - n   2(3-  n)   6(4 - n)
-
    -

    -The asymptotic behaviour is given by +

    +                      [                      2          3       ]
+En(x) = xn-1Γ (1 - n)+  ---1-- + --x-- - ---x----+  --x------ ... .
+                         1 - n   2 - n   2(3-  n)   6(4 - n)
+
    + +

    The asymptotic behaviour is given by

    -
    +
             - x[                     ]
 En (x ) = e-- 1 - n-+ n(n-+-1) + ....
          x       x      x2
-
    -

    -The special case + +

    The special case n = 1 gives

    -
    +
            ∫ ∞  e-u
-E1 (x) =     ---du,  |Arg (x )| ≥ π.
-          x   u
-
    -

    -For any complex number + +

    For any complex number x with positive real part, this can be written

    -
    -        ∫ ∞ e-ux
-E1(x) =     -u--du,   ℜ(x) ≥ 0.
-         1
-
    -

    -By integrating the Taylor expansion of +         ∫
+          ∞  e-ux-
+E1 (x) =  1   u  du,   ℜ(x) ≥ 0.
+ + +

    By integrating the Taylor expansion of e-t 14.7
    -
    +
                       ∑∞ (--1)kxk
-E1(x) = - γ - lnx -      kk!      |Arg(x)| < π.
-                   k=1
-
    -

    -The function + +

    The function E1 is linked to the exponential integral Ei

    -
    +
            ∫ x eu        ∫ ∞ e-u
-Ei(x) =     --du =  -     ---du   ∀x ⁄= 0.
-         -∞  u         -x  u
-
    -

    -The above definition can be used for positive values of + +

    The above definition can be used for positive values of x, but the integral has to be understood in terms of its Cauchy principal value, due to the singularity of the integrand at zero.

    -
    -Ei(- x) = - E1 (x ), ℜ (x) ≥ 0.
                                                                                     
 
                                                                                     
-
    -

    -We deduce, +

    +Ei(- x) = - E (x ),  ℜ (x) ≥ 0.
+             1
+
    + +

    We deduce,

    -
    +
                     ∑∞  xk
-Ei(x) = γ + lnx +    ---,  x > 0.
-                 k=1 kk!
-
    -

    -For + +

    For x

    -
    +
             {
             - Ei (- x) - iπ x < 0,
 Γ (0,x) =   - Ei (- x) x > 0.
-
    +" class="math-display" /> +

      -
    • double  + + + +

      double pnl_sf_expint_En (int n, double x)
      Description Computes E_n(x) for n 14.7 0 when n = 0 or 1.

    +class="rm-lmr-10x-x-109">1.

    14.8 Hypergeometric functions

    - - - + id="x16-14300014.8">Hypergeometric functions
      -
    • double  +

      double pnl_sf_hyperg_2F1 (double a, double b, double c, double x)
      Description Compute the Gauss hypergeometric function 2F1(a,b,c,x) for |x| < 1 and for x < -1 when b,a,c,(b-a),(c-a),(c-b) are not integers -

    • -
    • double 
    • +
    • +

      double pnl_sf_hyperg_1F1 (double a, double b, double x)
      Description Compute the hypergeometric function 1F1(a,b,x) -

    • -
    • double 
    • +
    • +

      double pnl_sf_hyperg_2F0 (double a, double b, double x)
      Description Compute the hypergeometric function 2F0(a,b,x) for x<0 using the relation 14.8 a,1 + a - -b,-1
-x). -

    • -
    • double 
    • +
    • +

      double pnl_sf_hyperg_0F1 (double c, double x)
      Description Compute the hypergeometric function 0F1(c,x) -

    • -
    • double 
    • +
    • +

      double pnl_sf_hyperg_U (double a, double b, double x)
      Description Compute the confluent hypergeometric function U(a,b,x) with x > 0

    +class="newline" /> Description Compute the confluent hypergeometric function U(a,b,x) with x > 0

    diff --git a/docs/manual-html/pnl-manualse15.html b/docs/manual-html/pnl-manualse15.html index d0112acb..1cb7f954 100644 --- a/docs/manual-html/pnl-manualse15.html +++ b/docs/manual-html/pnl-manualse15.html @@ -1,13 +1,13 @@ - + Some bindings - - - + + + @@ -20,15 +20,15 @@ href="#tailpnl-manualse15.html">tail] [up]

    15 Some bindings

    + id="x17-14400015">Some bindings

    15.1 MPI bindings

    + id="x17-14500015.1">MPI bindings

    15.1.1 Overview
    + id="x17-14600015.1.1">Overview

    We provide some bindings for the MPI library to natively handle PnlObjects.

    The functionnalities described in this chapter are declared in 15.1.1

    15.1.2 Functions
    + id="x17-14700015.1.2">Functions

    All the following functions return an error code as an integer value. This returned value should be tested against MPI_SUCCESS to check that no error occurred.

    For more expect users, we provide the following nonblocking functions.

    +class="ec-lmri-10x-x-109">MPI_Iprobe).

    15.2 The save/load interface

    + id="x17-14800015.2">The save/load interface

    The interface is only accessible when the MPI bindings are compiled since it is based on the Packing/Unpacking facilities of MPI.

    The functionnalities described in this chapter are declared in pnl/pnl_mpi.h.

      -
    • +

      PnlRng ** pnl_rng_create_from_file (char *str, int n)
      Description Load n rng from the file of name str and returns an array of n PnlRng . -

    • -
    • int 
    • +
    • +

      int pnl_rng_save_to_file (PnlRng **rngtab, int n, char *str)
      Description Save n rng stored in rngtab into the file of name str. -

    • -
    • int 
    • +
    • + + + +

      int pnl_object_save (PnlObject *O, FILE *stream)
      Description Save the object O into the stream stream. stream is typically created by calling fopen with mode="wb". This function can be called several times to save several objects in the same stream. -

    • -
    • PnlObject
    • +
    • +

      PnlObject* pnl_object_load (FILE *stream)
      Description Load an object from the stream stream. stream is typically created by calling fopen with mode="rb". This function can be called several times to load several objects from the same stream. If stream was empty or it did not contain any PnlObjects, the function returns NULL. -

    • -
    • PnlList
    • +
    • +

      PnlList* pnl_object_load_into_list (FILE *stream)
      Description Load as many objects as possible from the stream stream and stores them into a PnlList . stream is typically created by calling fopen with mode="rb". If - stream was empty or it did not contain any PnlObjects, the function returns NULL.

    + stream was empty or it did not contain any PnlObjects, the function returns NULL.

    diff --git a/docs/manual-html/pnl-manualse16.html b/docs/manual-html/pnl-manualse16.html index 2008c659..f012f8ec 100644 --- a/docs/manual-html/pnl-manualse16.html +++ b/docs/manual-html/pnl-manualse16.html @@ -1,13 +1,13 @@ - + Financial functions - - - + + + @@ -20,14 +20,15 @@ href="#tailpnl-manualse16.html">tail] [up]

    16 Financial functions

    + id="x18-14900016">Financial functions

    The financial functions are defined in the header pnl/pnl_finance.h.

      -
    • double  +

      double pnl_bs_call (double s, double k, double T, double r, double divid, double sigma)
      Description Compute the price of a call option with spot s and strike k in the Black-Scholes model with volatility sigma, instantaneous interest rate r, maturity T and dividend rate divid. -

    • -
    • double 
    • +
    • +

      double pnl_bs_put (double s, double k, double T, double r, double divid, double sigma)
      Description Compute the price a put option with spot s and strike k in the Black-Scholes model with volatility sigma, instantaneous interest rate r, maturity T and dividend rate divid. -

    • -
    • double 
    • +
    • +

      double pnl_bs_call_put (int iscall, double s, double k, double T, double r, double divid, double sigma)
      Description Compute the price of a put option if iscall=0 or a call option if iscall=1 with spot s and strike k in the Black-Scholes model with volatility sigma, instantaneous interest rate r, maturity T and dividend rate divid. -

    • -
    • double 
    • +
    • +

      double pnl_bs_vega (double s, double k, double T, double r, double divid, double sigma)
      Description Compute the vega of a put or call option with spot s and strike k in the Black-Scholes model with volatility sigma, instantaneous interest rate r, maturity T and dividend rate divid. -

    • -
    • double 
    • +
    • +

      double pnl_bs_gamma (double s, double k, double T, double r, double divid, double sigma)
      Description Compute the gamma of a put or call option with spot s and strike k in the Black-Scholes model with volatility sigma, instantaneous interest rate r, maturity - T and dividend rate divid.

    + T and dividend rate divid.

    Practitioners do not speak in terms of option prices, but rather compare prices in terms of their implied Black & Scholes volatilities. So this parameter is very useful in practice. Here, we propose two functions to compute 16

    +class="rm-lmr-10x-x-109">0.

    + id="Q1-18-154"> diff --git a/docs/manual-html/pnl-manualse2.html b/docs/manual-html/pnl-manualse2.html index 29bd4b48..ec692354 100644 --- a/docs/manual-html/pnl-manualse2.html +++ b/docs/manual-html/pnl-manualse2.html @@ -1,13 +1,13 @@ - + Objects - - - + + + @@ -30,43 +30,257 @@

    2.1 - - -

    -
    -typedef unsigned int PnlType; - 

    typedef void (DestroyFunc) (void **); - 
    typedef PnlObject* (CopyFunc) (PnlObject *); - 
    typedef PnlObject* (NewFunc) (PnlObject *); - 
    typedef void (CloneFunc) (PnlObject *dest, const PnlObject *src); - 
    struct _PnlObject - 
    { - 
      PnlType type; /*!< a unique integer id */ - 
      const char *label; /*!< a string identifier (for the moment not useful) */ - 
      PnlType parent_type; /*!< the identifier of the parent object is any, - 
                              otherwise parent_type=id */ - 
      int nref; /*!< number of references on the object */ - 
      DestroyFunc *destroy; /*!< frees an object */ - 
      NewFunc     *constructor; /*!< New function */ - 
      CopyFunc    *copy; /*!< Copy function */ - 
      CloneFunc   *clone; /*!< Clone function */ - 
    }; -
    -

    -

    Here is the list of all the types actually defined

    + +
    typedef unsigned int PnlType; 
    + 
    +typedef void (DestroyFunc) (void **); 
    +typedef PnlObject* (CopyFunc) (PnlObject *); 
    +typedef PnlObject* (NewFunc) (PnlObject *); 
    +typedef void (CloneFunc) (PnlObject *dest, const PnlObject *src); 
    +struct _PnlObject 
    +{ 
    +  PnlType type; /*!< a unique integer id */ 
    +  const char *label; /*!< a string identifier (for the moment not useful) */ 
    +  PnlType parent_type; /*!< the identifier of the parent object is any, 
    +                          otherwise parent_type=id */ 
    +  int nref; /*!< number of references on the object */ 
    +  DestroyFunc *destroy; /*!< frees an object */ 
    +  NewFunc     *constructor; /*!< New function */ 
    +  CopyFunc    *copy; /*!< Copy function */ 
    +  CloneFunc   *clone; /*!< Clone function */ 
    +};
    +

    Here is the list of all the types actually defined


    @@ -181,87 +395,102 @@

    2.1

    We provide several macros for manipulating PnlObejcts.

      -
    • +

      PNL_OBJECT (o)
      Description Cast any object into a PnlObject -

    • -
    • +
    • +

      PNL_VECT_OBJECT (o)
      Description Cast any object into a PnlVectObject -

    • -
    • +
    • +

      PNL_MAT_OBJECT (o)
      Description Cast any object into a PnlMatObject -

    • -
    • +
    • +

      PNL_SP_MAT_OBJECT (o)
      Description Cast any object into a PnlSpMatObject -

    • -
    • +
    • +

      PNL_HMAT_OBJECT (o)
      Description Cast any object into a PnlHmatObject -

    • -
    • +
    • +

      PNL_BAND_MAT_OBJECT (o)
      Description Cast any object into a PnlBandMatObject -

    • -
    • +
    • +

      PNL_TRIDIAGMAT_OBJECT (o)
      Description Cast any object into a PnlTridiagMatObject -

    • -
    • +
    • +

      PNL_BASIS_OBJECT (o)
      Description Cast any object into a PnlBasis -

    • -
    • +
    • +

      PNL_RNG_OBJECT (o)
      Description Cast any object into a PnlRng -

    • -
    • +
    • +

      PNL_LIST_OBJECT (o)
      Description Cast any object into a PnlList -

    • -
    • +
    • +

      PNL_LIST_ARRAY (o)
      Description Cast any object into a PnlArray -

    • -
    • +
    • +

      PNL_GET_TYPENAME (o)
      Description Return the name of the type of any object inheriting from PnlObject - - - -

    • -
    • +
    • +

      PNL_GET_TYPE (o)
      Description Return the type of any object inheriting from PnlObject -

    • -
    • +
    • +

      PNL_GET_PARENT_TYPE (o)
      Description Return the parent type of any object inheriting from PnlObject

    +class="newline" /> Description Return the parent type of any object inheriting from PnlObject

      -
    • +

      PnlObject * pnl_object_create (PnlType t)
      Description Create an empty PnlObject of type t which can any of the registered types, see Table 1.

    +href="#x4-120201">1.

    2.2 List object

    This section describes functions for creating an manipulating lists. Lists are internally stored as @@ -272,63 +501,325 @@

    2.2 - - -

    -
    -typedef struct _PnlCell PnlCell; - 
    struct _PnlCell - 
    { - 
      struct _PnlCell *prev;  /*!< previous cell or 0 */ - 
      struct _PnlCell *next;  /*!< next cell or 0 */ - 
      PnlObject *self;       /*!< stored object */ - 
    }; - 


    typedef struct _PnlList PnlList; - 
    struct _PnlList - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlList pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      PnlCell *first; /*!< first element of the list */ - 
      PnlCell *last; /*!< last element of the list */ - 
      PnlCell *curcell; /*!< last accessed element, - 
                             if never accessed is NULL */ - 
      int icurcell; /*!< index of the last accessed element, - 
                         if never accessed is NULLINT */ - 
      int len; /*!< length of the list */ - 
    }; -
    -

    -

    +

    typedef struct _PnlCell PnlCell; 
    +struct _PnlCell 
    +{ 
    +  struct _PnlCell *prev;  /*!< previous cell or 0 */ 
    +  struct _PnlCell *next;  /*!< next cell or 0 */ 
    +  PnlObject *self;       /*!< stored object */ 
    +}; 
    + 
    + 
    +typedef struct _PnlList PnlList; 
    +struct _PnlList 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlList pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  PnlCell *first; /*!< first element of the list */ 
    +  PnlCell *last; /*!< last element of the list */ 
    +  PnlCell *curcell; /*!< last accessed element, 
    +                         if never accessed is NULL */ 
    +  int icurcell; /*!< index of the last accessed element, 
    +                     if never accessed is NULLINT */ 
    +  int len; /*!< length of the list */ 
    +};
    +

    Important note: Lists only store addresses of objects. So when an object is inserted into a list, only its address is stored into the list. This implies that you must not free any objects inserted into a list. The deallocation is automatically handled by the function pnl_list_free. +href="#x4-13027r2.2">pnl_list_free. + + +

    + a print member, we will use it.

    2.3 2.3

    This section describes functions for creating and manipulating arrays of PnlObjects.

    The structures and functions related to arrays are declared in pnl/pnl_array.h. - +

    -
    -typedef struct _PnlArray PnlArray; - 
    struct _PnlArray - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlArray pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; - 
      PnlObject **array; - 
      int mem_size; - 
    }; -
    -

    -

    +

    typedef struct _PnlArray PnlArray; 
    +struct _PnlArray 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlArray pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; 
    +  PnlObject **array; 
    +  int mem_size; 
    +};
    +

    Important note: Arrays only store addresses of objects. So when an object is inserted into an array, only its address is stored into the array. This implies that you must not free any objects inserted into a array. The deallocation is automatically handled by the function pnl_array_free. +href="#x4-14013r2.3">pnl_array_free.

    +href="#x4-120002.1">PnlObject has a field copy.

    diff --git a/docs/manual-html/pnl-manualse3.html b/docs/manual-html/pnl-manualse3.html index 7f1d37e0..01ff3bd4 100644 --- a/docs/manual-html/pnl-manualse3.html +++ b/docs/manual-html/pnl-manualse3.html @@ -1,13 +1,13 @@ - + Mathematical framework - - - + + + @@ -37,7 +37,7 @@

    3.1.1 values.h or limits.h and a few others have been added.

    - - - +class="td11">

    3.1.2 A few macros
    + + +
      -
    • +

      PNL_IS_ODD (int n)
      Description Return 1 if n is odd and 0 otherwise. -

    • -
    • +
    • +

      PNL_IS_EVEN (int n)
      Description Return 1 if n is even and 0 otherwise. -

    • -
    • +
    • +

      PNL_ALTERNATE (int n)
      Description Return 3.1.2 -1)n. -

    • -
    • +
    • +

      MIN (x,y)
      Description Return the minimum of x and y. -

    • -
    • +
    • +

      MAX (x,y)
      Description Return the maximum of x and y. -

    • -
    • +
    • +

      ABS (x)
      Description Return the absolute value of x. -

    • -
    • +
    • +

      PNL_SIGN (x)
      Description Return the sign of x (-1 if x < 0, 0 otheriwse). -

    • -
    • +
    • +

      SQR (x)
      Description Return x2. -

    • -
    • +
    • +

      CUB (x)
      Description Return x3.

    +class="rm-lmr-8">3.

    + + +

    3.1.3 Classifying a floating–point number
      -
    • +

      double pnl_nan ()
      Description Return NaN - - - -

    • -
    • +
    • +

      double pnl_posinf ()
      Description Return + infinity -

    • -
    • +
    • +

      double pnl_neginf ()
      Description Return - infinity -

    • -
    • +
    • +

      int pnl_isnan (double x)
      Description Return +1 if x=NaN -

    • -
    • +
    • +

      int pnl_isinf (double x)
      Description Return +1 if x=+Inf, -1 if x=-Inf and 0 otherwise. -

    • -
    • +
    • +

      int pnl_isfinite (double x)
      Description Return 1 if x!=+-Inf

    +class="newline" /> Description Return 1 if x!=+-Inf

    3.1.4 Rouding a floating–point number
      -
    • +

      int pnl_itrunc (double s)
      Description This function is similar to the trunc function (provided by the C library) but the result is typed as an integer instead of a double. Digits may be lost if s exceeds MAX_INT. -

    • -
    • +
    • +

      long int pnl_ltrunc (double s)
      Description This function is similar to the trunc function (provided by the C library) but the result is typed as a long integer instead of a double. -

    • -
    • +
    • +

      double pnl_trunc (double s)
      Description Return the nearest integer not greater than the absolute value of s. This function is part of C99 as trunc. -

    • -
    • +
    • +

      double pnl_round (double s)
      Description Return the integral value nearest to x rounding half-way cases away from zero, regardless of the current rounding direction. This function is part of C99 as round. - - - -

    • -
    • +
    • +

      int pnl_iround (double s)
      Description This function is similar to the round function (provided by the C library) but the result is typed as an integer instead of a double. Digits may be lost if s exceeds MAX_INT. -

    • -
    • +
    • +

      long int pnl_lround (double s)
      Description This function is similar to the round function (provided by the C library) - but the result is typed as a long integer instead of a double.

    + but the result is typed as a long integer instead of a double.

    3.1.5 Some standard mathematical functions
      -
    • +

      double pnl_fact (double x)
      Description See pnl_sf_fact -

    • -
    • pnl_sf_fact +

    • +
    • +

      double pnl_lgamma (double x)
      Description See pnl_sf_log_gamma -

    • -
    • pnl_sf_log_gamma + + + +

    • +
    • +

      double pnl_tgamma (double x)
      Description See pnl_sf_gamma -

    • -
    • pnl_sf_gamma +

    • +
    • +

      double pnl_acosh (double x)
      Description Compute acosh(x). -

    • -
    • +
    • +

      double pnl_asinh (double x)
      Description Compute asinh(x). -

    • -
    • +
    • +

      double pnl_atanh (double x)
      Description Compute atanh(x). -

    • -
    • +
    • +

      double pnl_log1p (double x)
      Description Compute log(1+x) accurately for small values of x -

    • -
    • +
    • +

      double pnl_expm1 (double x)
      Description Compute exp(x)-1 accurately for small values of x - - - -

    • -
    • +
    • +

      double pnl_cosm1 (double x)
      Description Compute cos(x)-1 accurately for small values of x -

    • -
    • +
    • +

      double pnl_pow_i (double x, int n)
      Description Compute x^n for an integer n.

    +class="ec-lmtt-10x-x-109">x^n for an integer n.

    3.1.6 Comparison of floating–point numbers
    + + +
      -
    • +

      int pnl_isequal_rel (double x, double y, double relerr)
      Description Compare two floating–point numbers up to a relative precision relerr -

    • -
    • +
    • +

      int pnl_isequal_abs (double x, double y, double abserr)
      Description Compare two floating–point numbers up to an absolute precision abserr -

    • -
    • +
    • +

      int pnl_isequal (double x, double y, double relerr)
      Description Equivalent to pnl_isequal_abs if |x| < 1 and to pnl_isequal_abs - otherwise.

    + otherwise.

    3.2 3.2.1

    The first native implementation of complex numbers in the C language appeared in C99, which is unfortunately not available on all platforms. For this reason, we provide here an implementation of complex numbers. - - -

    -
    -typedef struct { - 
        double r; /*!< real part */ - 
        double i; /*!< imaginary part */ - 
    } dcomplex; -
    -

    -

    + +

    typedef struct { 
    +    double r; /*!< real part */ 
    +    double i; /*!< imaginary part */ 
    +} dcomplex;
    +

    3.2.2 Constants

    +class="td11">
    + + +

    3.2.3 Functions

    Most algebraic operations on complex numbers are implemented using the following naming for the functions

      -
    • All these function names begin in +

      All these function names begin in C_op_, +

    • +
    • +

      The small letters a, b denote two complex numbers whereas d is a real number, -

    • -
    • The small letters a, b denote two complex numbers whereas d is a real number, -
    • -
    • The letter
    • +
    • +

      The letter i denotes the multiplication by the pure imagniary number i, -

    • -
    • The letter
    • +
    • +

      The letter c indicates that the next coming number is conjugated. -

    • -
    • The letters
    • +
    • +

      The letters p, m denote the two standard operations plus and minus respectively.

    +class="ec-lmri-10x-x-109">minus respectively.

    For example C_op_idamcb is id(   -)
+src=. So functions are :

    + a - b" class="left" align="middle" />.

    diff --git a/docs/manual-html/pnl-manualse4.html b/docs/manual-html/pnl-manualse4.html index a05c7435..18e5db22 100644 --- a/docs/manual-html/pnl-manualse4.html +++ b/docs/manual-html/pnl-manualse4.html @@ -1,13 +1,13 @@ - + Linear Algebra - - - + + + @@ -37,61 +37,437 @@
    4.1.1 - - - -

    -
    -typedef struct _PnlVect { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlVect pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; /*!< size of the vector */ - 
      int mem_size; /*!< size of the memory block allocated for array */ - 
      double *array; /*!< pointer to store the data */ - 
      int owner; /*!< 1 if the object owns its array member, 0 otherwise */ - 
    } PnlVect; - 

    typedef struct _PnlVectInt { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlVectInt pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; /*!< size of the vector */ - 
      int mem_size; /*!< size of the memory block allocated for array */ - 
      int *array; /*!< pointer to store the data */ - 
      int owner; /*!< 1 if the object owns its array member, 0 otherwise */ - 
    } PnlVectInt; - 

    typedef struct _PnlVectComplex { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlVectComplex pointer to be cast - 
       * to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; /*!< size of the vector */ - 
      int mem_size; /*!< size of the memory block allocated for array */ - 
      dcomplex *array; /*!< pointer to store the data */ - 
      int owner; /*!< 1 if the object owns its array member, 0 otherwise */ - 
    } PnlVectComplex; -
    -

    size is the size of the vector, array is a pointer containing the data and owner is an integer to know + id="dx6-29003">

    +
    typedef struct _PnlVect { 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlVect pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; /*!< size of the vector */ 
    +  int mem_size; /*!< size of the memory block allocated for array */ 
    +  double *array; /*!< pointer to store the data */ 
    +  int owner; /*!< 1 if the object owns its array member, 0 otherwise */ 
    +} PnlVect; 
    + 
    +typedef struct _PnlVectInt { 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlVectInt pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; /*!< size of the vector */ 
    +  int mem_size; /*!< size of the memory block allocated for array */ 
    +  int *array; /*!< pointer to store the data */ 
    +  int owner; /*!< 1 if the object owns its array member, 0 otherwise */ 
    +} PnlVectInt; 
    + 
    +typedef struct _PnlVectComplex { 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlVectComplex pointer to be cast 
    +   * to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; /*!< size of the vector */ 
    +  int mem_size; /*!< size of the memory block allocated for array */ 
    +  dcomplex *array; /*!< pointer to store the data */ 
    +  int owner; /*!< 1 if the object owns its array member, 0 otherwise */ 
    +} PnlVectComplex;
    +

    size is the size of the vector, array is a pointer containing the data and owner is an integer to know if the vector owns its array pointer (owner=1) or shares it with another structure (owner=0). mem_size is the number of elements the vector can hold at most.

    - - -

    4.1.2 Functions

    General functions These functions exist for all types of vector no matter what the basic type is. The following conventions are used to name functions operating on vectors. Here is the table of prefixes used for @@ -100,9 +476,13 @@

    4.1.2

    +

    + + +
    4.1.2
    dcomplexpnl_vect_complexdcomplex
    +class="td11">dcomplex

    In this paragraph, we present the functions operating on PnlVect which exist for all types. To deduce the prototypes of these functions for other basic types, one must replace 4.1.2 double according the above table.

    -

    Constructors and destructors There are no special functions to access the size of a vector, instead the field size should be accessed directly.

    -

    +

    Resizing vectors

      -
    • int  + + + +

      int pnl_vect_resize (PnlVect *v, int size)
      Description Resize a PnlVect . It copies as much of the old data to fit in the resized object. -

    • -
    • +
    • +

      int pnl_vect_resize_from_scalar (PnlVect *v, int size, double x)
      Description Resize a PnlVect . Copy as much of the old data as possible and fill the + new cells with x. +

    • +
    • +

      int pnl_vect_resize_from_ptr (PnlVect *v, int size, double *t)
      Description Resize a PnlVect and uses t to fill the vector. t must be of size size.

    -

    PnlVect and uses t to fill the vector. t must be of size size.

    +

    Accessing elements If it is supported by the compiler, the following functions are declared inline. To speed up these functions, you can define the macro PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an +class="ec-lmtt-10x-x-109">PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an explanation. -

    Accessing elements of a vector is faster using the following macros

    - - - +

    Accessing elements of a vector is faster using the following macros

      -
    •   +

       GET (PnlVect *v, int i)
      Description Return v[i] for reading, eg. x=GET(v,i) -

    • -
    • +
    • +

       GET_INT (PnlVectInt *v, int i)
      Description Same as GET but for an integer vector. -

    • -
    • GET but for an integer vector. +

    • +
    • +

       GET_COMPLEX (PnlVectComplex *v, int i)
      Description Same as GET but for a complex vector. -

    • -
    • GET but for a complex vector. +

    • +
    • +

       LET (PnlVect *v, int i)
      Description Return v[i] as a lvalue for writing, eg. LET(v,i)=x -

    • -
    • +
    • +

       LET_INT (PnlVectInt *v, int i)
      Description Same as LET but for an integer vector. -

    • -
    • LET but for an integer vector. + + + +

    • +
    • +

       LET_COMPLEX (PnlVectComplex *v, int i)
      Description Same as LET but for a complex vector.

    +href="#x6-34000doc">LET but for a complex vector.

      -
    • +

      void pnl_vect_set (PnlVect *v, int i, double x)
      Description Set v[i]=x. -

    • -
    • +
    • +

      double pnl_vect_get (const PnlVect *v, int i)
      Description Return the value of v[i]. -

    • -
    • +
    • +

      void pnl_vect_lget (PnlVect *v, int i)
      Description Return the address of v[i]. -

    • -
    • +
    • +

      void pnl_vect_set_all (PnlVect *v, double x)
      Description Set all elements to x. -

    • -
    • +
    • +

      void pnl_vect_set_zero (PnlVect *v)
      Description Set all elements to zero.

    - - - -

    Description Set all elements to zero.

    +

    Printing vector

    -

    Description Print a vector to a file in a format compatible with Nsp.

    +

    Applying external operation to vectors

      -
    • void  +

      void pnl_vect_minus (PnlVect *lhs)
      Description In-place unary minus -

    • -
    • +
    • +

      void pnl_vect_plus_scalar (PnlVect *lhs, double x)
      Description In-place vector scalar addition -

    • -
    • +
    • +

      void pnl_vect_minus_scalar (PnlVect *lhs, double x)
      Description In-place vector scalar substraction -

    • -
    • +
    • +

      void pnl_vect_mult_scalar (PnlVect 4.1.2 -

    • +
    • +

      void pnl_vect_div_scalar (PnlVect *lhs, double x)
      Description In-place vector scalar division

    -

    Description In-place vector scalar division

    +

    Element wise operations

    -

    +

    Scalar products and norms

    - - - -

    Comparison functions

    -

    Description Test if all the components of v are equal to x. Return TRUE or FALSE.

    +

    Ordering functions The following functions are not defined for PnlVectComplex because there is no total ordering on Complex numbers

    +

    ind contains on exit the indices i for which the function f returned 1. This function returns OK or FAIL when something went wrong (size mismatch between matrices, invalid string type).

    -

    Misc

      -
    • void  +

      void pnl_vect_swap_elements (PnlVect *v, int i, int j)
      Description Exchange v[i] and v[j]. -

    • -
    • +
    • +

      void pnl_vect_reverse (PnlVect *v)
      Description Perform a mirror operation on v. On output v[i] = v[n-1-i] for i=0,…,n-1 - where n is the length of the vector.

    -

    n is the length of the vector.

    +

    Complex vector functions

    -

    There exist functions to directly access the real or imaginary parts of an element of a complex + of v are stored into re and the imaginary parts into im.

    +

    There exist functions to directly access the real or imaginary parts of an element of a complex vector. These functions also have inlined versions that are used if the variable HAVE_INLINE was declared at compilation time.

      -
    • +

      double pnl_vect_complex_get_real (const PnlVectComplex *v, int i)
      Description Return the real part of v[i]. -

    • -
    • +
    • +

      double pnl_vect_complex_get_imag (const PnlVectComplex *v, int i)
      Description Return the imaginary part of v[i]. -

    • -
    • +
    • +

      double* pnl_vect_complex_lget_real (const PnlVectComplex *v, int i)
      Description Return the real part of v[i] as a lvalue. -

    • -
    • +
    • +

      double* pnl_vect_complex_lget_imag (const PnlVectComplex *v, int i)
      Description Return the imaginary part of v[i] as a lvalue. -

    • -
    • +
    • +

      void pnl_vect_complex_set_real (const PnlVectComplex *v, int i, double re)
      Description Set the real part of v[i] to re. -

    • -
    • +
    • +

      void pnl_vect_complex_set_imag (const PnlVectComplex *v, int i, double im)
      Description Set the imaginary part of v[i] to im.

    -

    Equivalently to these functions, there exist macros. When the compiler is able to handle inline +class="newline" /> Description Set the imaginary part of v[i] to im.

    +

    Equivalently to these functions, there exist macros. When the compiler is able to handle inline code, there is no gain in using macros instead of inlined functions at least in principle.

    + + +
      -
    • +

       GET_REAL (v, i)
      Description Return the real part of v[i]. - - - -

    • -
    • +
    • +

       GET_IMAG (v, i)
      Description Return the imaginary part of v[i]. -

    • -
    • +
    • +

       LET_REAL (v, i)
      Description Return the real part of v[i] as a lvalue. -

    • -
    • +
    • +

       LET_IMAG (v, i)
      Description Return the imaginary part of v[i] as a lvalue.

    -

    +class="newline" /> Description Return the imaginary part of v[i] as a lvalue.

    +

    4.2 Compact Vectors

    -

    +

    4.2.1 Short description
    - - - -
    -typedef struct PnlVectCompact { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlVectCompact pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; /* size of the vector */ - 
      double val; /* single value */ - 
      double *array; /* Pointer to double values */ - 
      char convert; /* ’a’, ’d’ : array, double */ - 
    } PnlVectCompact; -
    -

    -

    + +

    typedef struct PnlVectCompact { 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlVectCompact pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; /* size of the vector */ 
    +  double val; /* single value */ 
    +  double *array; /* Pointer to double values */ 
    +  char convert; /* a’, d : array, double */ 
    +} PnlVectCompact;
    +

    4.2.2 Functions
    -

    + to a non compact storage.

    + + + +

    4.3 Matrices

    -

    +

    4.3.1 Overview
    -

    The structures and functions related to matrices are declared in

    The structures and functions related to matrices are declared in pnl/pnl_matrix.h. @@ -1242,58 +2019,511 @@

    4.3.1 - - -

    -
    -typedef struct _PnlMat{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlMat pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int m; /*!< nb rows */ - 
      int n; /*!< nb columns */ - 
      int mn; /*!< product m*n */ - 
      int mem_size; /*!< size of the memory block allocated for array */ - 
      double *array; /*!< pointer to store the data row-wise */ - 
      int owner; /*!< 1 if the object owns its array member, 0 otherwise */ - 
    } PnlMat; - 

    typedef struct _PnlMatInt{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlMatInt pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int m; /*!< nb rows */ - 
      int n; /*!< nb columns */ - 
      int mn; /*!< product m*n */ - 
      int mem_size; /*!< size of the memory block allocated for array */ - 
      int *array; /*!< pointer to store the data row-wise */ - 
      int owner; /*!< 1 if the object owns its array member, 0 otherwise */ - 
    } PnlMatInt; - 

    typedef struct _PnlMatComplex{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlMatComplex pointer to be cast - 
       * to a PnlObject - 
       */ - 
      PnlObject object; - 
      int m; /*!< nb rows */ - 
      int n; /*!< nb columns */ - 
      int mn; /*!< product m*n */ - 
      int mem_size; /*!< size of the memory block allocated for array */ - 
      dcomplex *array; /*!< pointer to store the data row-wise */ - 
      int owner; /*!< 1 if the object owns its array member, 0 otherwise */ - 
    } PnlMatComplex; - - - -
    -

    m is the number of rows, n is the number of columns. array is a pointer containing the data of + +

    typedef struct _PnlMat{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlMat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< nb rows */ 
    +  int n; /*!< nb columns */ 
    +  int mn; /*!< product m*n */ 
    +  int mem_size; /*!< size of the memory block allocated for array */ 
    +  double *array; /*!< pointer to store the data row-wise */ 
    +  int owner; /*!< 1 if the object owns its array member, 0 otherwise */ 
    +} PnlMat; 
    + 
    +typedef struct _PnlMatInt{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlMatInt pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< nb rows */ 
    +  int n; /*!< nb columns */ 
    +  int mn; /*!< product m*n */ 
    +  int mem_size; /*!< size of the memory block allocated for array */ 
    +  int *array; /*!< pointer to store the data row-wise */ 
    +  int owner; /*!< 1 if the object owns its array member, 0 otherwise */ 
    +} PnlMatInt; 
    + 
    +typedef struct _PnlMatComplex{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlMatComplex pointer to be cast 
    +   * to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< nb rows */ 
    +  int n; /*!< nb columns */ 
    +  int mn; /*!< product m*n */ 
    +  int mem_size; /*!< size of the memory block allocated for array */ 
    +  dcomplex *array; /*!< pointer to store the data row-wise */ 
    +  int owner; /*!< 1 if the object owns its array member, 0 otherwise */ 
    +} PnlMatComplex;
    +

    m is the number of rows, n is the number of columns. array is a pointer containing the data of the matrix stored line wise, The element (i, j) of the matrix is 4.3.1 array pointer (owner=1) or shares it with another structure (owner=0). mem_size is the number of elements the matrix can hold at most. -

    The following operations are implemented on matrices and vectors. alpha +

    The following operations are implemented on matrices and vectors. alpha and beta are numbers, A and B are matrices and x and y are vectors.

    -

    +href="#x6-56000doc">pnl_mat_dger A := alpha x * y’ + A +

    4.3.2 Generic Functions
    -

    These functions exist for all types of matrices no matter what the basic type is. The following +

    These functions exist for all types of matrices no matter what the basic type is. The following conventions are used to name functions operating on matrices. Here is the table of prefixes used for the different basic types.

    -

    +

    +

    4.3.2
    dcomplexpnl_mat_complexdcomplex
    -

    In this paragraph we present the functions operating on dcomplex + + + +

    In this paragraph we present the functions operating on PnlMat which exist for all types. To deduce the prototypes of these functions for other basic types, one must replace pnl_mat and double according the above table.

    -

    Constructors and destructors There are no special functions to access the sizes of a matrix, instead the fields m, 4.3.2

    -

    M(i:i+m_block-1, j:j+n_block-1) = block.

    +

    Accessing elements. If it is supported by the compiler, the following functions are declared inline. To speed up these functions, you can define the macro PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an +class="ec-lmtt-10x-x-109">PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an explanation. -

    Accessing elements of a matrix is faster using the following macros

    - - - +

    Accessing elements of a matrix is faster using the following macros

      -
    •   +

       MGET (PnlMat *M, int i, int j)
      Description Return M[i,j] for reading, eg. x=MGET(M,i,j) -

    • -
    • +
    • +

       MGET_INT (PnlMatInt *M, int i, int j)
      Description Same as MGET but for an integer matrix. -

    • -
    • MGET but for an integer matrix. +

    • +
    • +

       MGET_COMPLEX (PnlMatComplex *M, int i, int j)
      Description Same as MGET but for a complex matrix. -

    • -
    • MGET but for a complex matrix. +

    • +
    • +

       MLET (PnlMat *M, int i, int j)
      Description Return M[i,j] as a lvalue for writing, eg. MLET(M,i,j)=x -

    • -
    • +
    • +

       MLET_INT (PnlMatInt *M, int i, int j)
      Description Same as MLET but for an integer matrix. -

    • -
    • MLET but for an integer matrix. +

    • +
    • +

       MLET_COMPLEX (PnlMatComplex *M, int i, int j)
      Description Same as MLET but for a complex matrix.

    +href="#x6-50000doc">MLET but for a complex matrix.

    -

    Printing Matrices

    + + + -

    Description Print a matrix to a file in a format compatible with Nsp.

    +

    Applying external operations

      -
    • void  +

      void pnl_mat_plus_scalar (PnlMat *lhs, double x)
      Description In-place matrix scalar addition -

    • -
    • +
    • +

      void pnl_mat_minus_scalar (PnlMat *lhs, double x)
      Description In-place matrix scalar substraction -

    • -
    • +
    • +

      void pnl_mat_mult_scalar (PnlMat *lhs, double x)
      Description In-place matrix scalar multiplication - - - -

    • -
    • +
    • +

      void pnl_mat_div_scalar (PnlMat *lhs, double x)
      Description In-place matrix scalar division +

    -

    Element wise operations

    -

    ik.

    +

    Comparison functions

    - - - -

    TRUE or FALSE.

    +

    Ordering operations

    +

    (indi, indj) contains on exit the indices (i,j) for which the function f returned 1. Note that if indj == NULL on entry, a linear indexing is used for matrices, which means that matrices are seen as large vectors built up be stacking rows. This function returns OK or FAIL if something went wrong (size mismatch between matrices, invalid string type).

    -

    Standard matrix operations

    -

    NULL.

    +

    Linear systems and matrix decompositions The following functions are designed to solve linear system of the from A x = b where A is a matrix and b is a vector except in the functions pnl_mat_syslin_mat, pnl_mat_lu_syslin_mat and +href="#x6-57000doc">pnl_mat_syslin_mat, pnl_mat_lu_syslin_mat and pnl_mat_chol_syslin_mat which expect the right hand side member to be a matrix too. Whenever +href="#x6-57000doc">pnl_mat_chol_syslin_mat which expect the right hand side member to be a matrix too. Whenever the vector b is not needed once the system is solved, you should consider using “inplace” functions. -

    All the functions described in this paragraph return OK if the computations have been carried out +

    All the functions described in this paragraph return OK if the computations have been carried out successfully and FAIL otherwise.

    -

    The following functions are designed to invert matrices. The authors provide these functions +

    The following functions are designed to invert matrices. The authors provide these functions although they cannot find good reasons to use them. Note that to solve a linear system, one must used the syslin functions and not invert the system matrix because it is much longer.

    - - - -

    + this function.

    +

    4.3.3 Functions specific to base type double
    -

    Linear systems and matrix decompositions The following functions are designed to solve linear system of the from A x = b where A is a matrix and b is a vector except in the functions pnl_mat_syslin_mat, pnl_mat_lu_syslin_mat and +href="#x6-57000doc">pnl_mat_syslin_mat, pnl_mat_lu_syslin_mat and pnl_mat_chol_syslin_mat which expect the right hand side member to be a matrix too. Whenever +href="#x6-57000doc">pnl_mat_chol_syslin_mat which expect the right hand side member to be a matrix too. Whenever the vector b is not needed once the system is solved, you should consider using “inplace” functions. -

    All the functions described in this paragraph return OK if the computations have been carried out +

    All the functions described in this paragraph return OK if the computations have been carried out successfully and FAIL otherwise.

    -

    + + + +

    4.3.4 Functions specific to base type dcomplex
      -
    • +

      PnlMatComplex * 4.3.4 PnlMat *R)
      Description Create a complex matrix using a real one. The complex parts of the - entries of the returned matrix are all set to zero.

    -

    + entries of the returned matrix are all set to zero.

    +

    4.3.5 Permutations
    - - - -
    -typedef PnlVectInt PnlPermutation; -
    -

    -

    The +

    typedef PnlVectInt PnlPermutation;
    +

    The PnlPermutation type is actually nothing else than a vector of integers, i.e. a PnlVectInt. It is used to store the partial pivoting with row interchanges transformation needed in the LU @@ -3162,41 +4721,49 @@

    4.3.5 p(i).

    -

    +class="newline" /> Description Print a permutation to the standard output.

    + + + +

    4.4 Tridiagonal Matrices

    -

    +

    4.4.1 Overview
    -

    The structures and functions related to tridiagonal matrices are declared in

    The structures and functions related to tridiagonal matrices are declared in pnl/pnl_tridiag_matrix.h. -

    We only store the three main diagonals as three vectors. +

    We only store the three main diagonals as three vectors. - - -

    -
    -typedef struct PnlTridiagMat{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlTridiagMat pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; /*!< number of rows, the matrix must be square */ - 
      double *D; /*!< diagonal elements */ - 
      double *DU; /*!< upper diagonal elements */ - 
      double *DL; /*!< lower diagonal elements */ - 
    } PnlTridiagMat; -
    -

    size is the size of the matrix, D is an array of size size containing the diagonal terms. DU, DL are + +

    typedef struct PnlTridiagMat{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlTridiagMat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; /*!< number of rows, the matrix must be square */ 
    +  double *D; /*!< diagonal elements */ 
    +  double *DU; /*!< upper diagonal elements */ 
    +  double *DL; /*!< lower diagonal elements */ 
    +} PnlTridiagMat;
    +

    size is the size of the matrix, D is an array of size size containing the diagonal terms. DU, DL are two arrays of size size-1 containing respectively the upper diagonal (Mi,i4.4.1 1,i). - - - + id="dx6-63013">

    -
    -typedef struct PnlTridiagMatLU{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlTridiagMatLU pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int size; /*!< number of rows, the matrix must be square */ - 
      double *D; /*!< diagonal elements */ - 
      double *DU; /*!< upper diagonal elements */ - 
      double *DU2; /*!< second upper diagonal elements */ - 
      double *DL; /*!< lower diagonal elements */ - 
      int *ipiv; /*!< Permutation: row i has been interchanged with row ipiv(i) */ - 
    }; -
    -

    This type is used to store the LU decomposition of a tridiagonal matrix. -

    + +

    typedef struct PnlTridiagMatLU{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlTridiagMatLU pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int size; /*!< number of rows, the matrix must be square */ 
    +  double *D; /*!< diagonal elements */ 
    +  double *DU; /*!< upper diagonal elements */ 
    +  double *DU2; /*!< second upper diagonal elements */ 
    +  double *DL; /*!< lower diagonal elements */ 
    +  int *ipiv; /*!< Permutation: row i has been interchanged with row ipiv(i) */ 
    +};
    +

    This type is used to store the LU decomposition of a tridiagonal matrix. +

    4.4.2 Functions
    -

    Constructors and destructors

    -

    PnlTridiagMat .

    + + + +

    Accessing elements. If it is supported by the compiler, the following functions are declared inline. To speed up these functions, you can use the macro constant PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an +class="ec-lmtt-10x-x-109">PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an explanation.

    - - - -

    }.

    +

    Printing Matrix

      -
    • void  +

      void pnl_tridiag_mat_fprint (FILE *fic, const PnlTridiagMat *M)
      Description Print a tri-diagonal matrix to a file. -

    • -
    • +
    • +

      void pnl_tridiag_mat_print (const PnlTridiagMat *M)
      Description Print a tridiagonal matrix to the standard output.

    -

    Description Print a tridiagonal matrix to the standard output.

    +

    Algebra operations

      -
    • void  +

      void pnl_tridiag_mat_plus_tridiag_mat (PnlTridiagMat *lhs, const @@ -3550,8 +5405,12 @@

      4.4.2 PnlTridiagMat *rhs)
      Description In-place matrix matrix addition -
    • -
    • +
    • +

      void pnl_tridiag_mat_minus_tridiag_mat (PnlTridiagMat 4.4.2 PnlTridiagMat *rhs)
      Description In-place matrix matrix substraction -

    • -
    • +
    • +

      void pnl_tridiag_mat_plus_scalar (PnlTridiagMat *lhs, double x)
      Description In-place matrix scalar addition -

    • -
    • +
    • +

      void pnl_tridiag_mat_minus_scalar (PnlTridiagMat *lhs, double x)
      Description In-place matrix scalar substraction -

    • -
    • +
    • +

      void pnl_tridiag_mat_mult_scalar (PnlTridiagMat *lhs, double x)
      Description In-place matrix scalar multiplication -

    • -
    • +
    • +

      void pnl_tridiag_mat_div_scalar (PnlTridiagMat *lhs, double x)
      Description In-place matrix scalar division

    -

    Description In-place matrix scalar division

    +

    Element-wise operations

    - - -
      -
    • void  +

      void pnl_tridiag_mat_mult_tridiag_mat_term (PnlTridiagMat *lhs, const @@ -3605,8 +5467,9 @@

      4.4.2 PnlTridiagMat *rhs)
      Description In-place matrix matrix term by term product -
    • -
    • +
    • +

      void pnl_tridiag_mat_div_tridiag_mat_term (PnlTridiagMat 4.4.2 PnlTridiagMat *rhs)
      Description In-place matrix matrix term by term division -

    • -
    • +
    • +

      void pnl_tridiag_mat_map_inplace (PnlTridiagMat *lhs, double(*f)(double))
      Description lhs = f(lhs). -

    • -
    • +
    • +

      void pnl_tridiag_mat_map_tridiag_mat_inplace (PnlTridiagMat 4.4.2 PnlTridiagMat *rhs, double(*f)(double, double))
      Description lhs = f(lhs, rhs).

    -

    Description lhs = f(lhs, rhs).

    +

    Standard matrix operations & Linear systems

    - - - -

    +href="#x6-70000doc">pnl_tridiag_mat_lu_compute.

    +

    4.5 Band Matrices

    -

    +

    4.5.1 Overview
    - - - -
    -typedef struct - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlBandMat pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int m; /*!< nb rows */ - 
      int n; /*!< nb columns */ - 
      int nu; /*!< nb of upperdiagonals */ - 
      int nl; /*!< nb of lowerdiagonals */ - 
      int m_band; /*!< nb rows of the band storage */ - 
      int n_band; /*!< nb columns of the band storage */ - 
      double *array;  /*!< a block to store the bands */ - 
    } PnlBandMat; -
    -

    -

    The structures and functions related to band matrices are declared in +

    typedef struct 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlBandMat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< nb rows */ 
    +  int n; /*!< nb columns */ 
    +  int nu; /*!< nb of upperdiagonals */ 
    +  int nl; /*!< nb of lowerdiagonals */ 
    +  int m_band; /*!< nb rows of the band storage */ 
    +  int n_band; /*!< nb columns of the band storage */ 
    +  double *array;  /*!< a block to store the bands */ 
    +} PnlBandMat;
    +

    The structures and functions related to band matrices are declared in pnl/pnl_band_matrix.h. -

    +

    4.5.2 Functions
    -

    Constructors and destructors

    -

    nl lower diagonals.

    +

    Accessing elements. If it is supported by the compiler, the following functions are declared inline. To speed up these functions, you can use the macro constant PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an +class="ec-lmtt-10x-x-109">PNL_RANGE_CHECK_OFF, see Section 1.3.2 for an explanation.

    - - - -

    Description Print a band matrix in a full format.

    +

    Element wise operations

    -

    Description lhs = f(lhs,rhs)

    + + + +

    Standard matrix operations & Linear system

    - - - -

    +href="#x6-77000doc">pnl_band_mat_lu and p the associated permutation.

    +

    4.6 Sparse Matrices

    -

    +

    4.6.1 Short description
    -

    The structures and functions related to matrices are declared in

    The structures and functions related to matrices are declared in pnl/pnl_sp_matrix.h. @@ -4210,80 +6281,717 @@

    4.6.1 - - -

    -
    -typedef struct _PnlSpMat - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows a PnlSpMat pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int m; /*!< number of rows */ - 
      int n; /*!< number of columns */ - 
      int nz; /*!< number of non-zero elements */ - 
      int *J; /*!< column indices, vector of size nzmax */ - 
      int *I; /*!< row offset integer vector, - 
                array[I[i]] is the first element of row i. - 
                Vector of size (m+1) */ - 
      double *array; /*!< pointer to store the data of size nzmax*/ - 
      int nzmax; /*!< size of the memory block allocated for array */ - 
    } PnlSpMat; - 

    typedef struct _PnlSpMatInt - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows a PnlSpMat pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int m; /*!< number of rows */ - 
      int n; /*!< number of columns */ - 
      int nz; /*!< number of non-zero elements */ - 
      int *J; /*!< column indices, vector of size nzmax */ - 
      int *I; /*!< row offset integer vector, - 
                array[I[i]] is the first element of row i. - 
                Vector of size (m+1) */ - 
      int *array; /*!< pointer to store the data of size nzmax */ - 
      int nzmax; /*!< size of the memory block allocated for array */ - 
    } PnlSpMatInt; - 

    typedef struct _PnlSpMatComplex - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows a PnlSpMat pointer to be cast to a PnlObject - 
       */ - - - - 
      PnlObject object; - 
      int m; /*!< number of rows */ - 
      int n; /*!< number of columns */ - 
      int nz; /*!< number of non-zero elements */ - 
      int *J; /*!< column indices, vector of size nzmax */ - 
      int *I; /*!< row offset integer vector, - 
                array[I[i]] is the first element of row i. - 
                Vector of size (m+1) */ - 
      dcomplex *array; /*!< pointer to store the data of size nzmax */ - 
      int nzmax; /*!< size of the memory block allocated for array */ - 
    } PnlSpMatComplex; -
    -

    -

    The non zero elements of row i are stored in array between the indices I[i] and I[i+1]-1. The array J + +

    typedef struct _PnlSpMat 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows a PnlSpMat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< number of rows */ 
    +  int n; /*!< number of columns */ 
    +  int nz; /*!< number of non-zero elements */ 
    +  int *J; /*!< column indices, vector of size nzmax */ 
    +  int *I; /*!< row offset integer vector, 
    +            array[I[i]] is the first element of row i. 
    +            Vector of size (m+1) */ 
    +  double *array; /*!< pointer to store the data of size nzmax*/ 
    +  int nzmax; /*!< size of the memory block allocated for array */ 
    +} PnlSpMat; 
    + 
    +typedef struct _PnlSpMatInt 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows a PnlSpMat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< number of rows */ 
    +  int n; /*!< number of columns */ 
    +  int nz; /*!< number of non-zero elements */ 
    +  int *J; /*!< column indices, vector of size nzmax */ 
    +  int *I; /*!< row offset integer vector, 
    +            array[I[i]] is the first element of row i. 
    +            Vector of size (m+1) */ 
    +  int *array; /*!< pointer to store the data of size nzmax */ 
    +  int nzmax; /*!< size of the memory block allocated for array */ 
    +} PnlSpMatInt; 
    + 
    +typedef struct _PnlSpMatComplex 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows a PnlSpMat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int m; /*!< number of rows */ 
    +  int n; /*!< number of columns */ 
    +  int nz; /*!< number of non-zero elements */ 
    +  int *J; /*!< column indices, vector of size nzmax */ 
    +  int *I; /*!< row offset integer vector, 
    +            array[I[i]] is the first element of row i. 
    +            Vector of size (m+1) */ 
    +  dcomplex *array; /*!< pointer to store the data of size nzmax */ 
    +  int nzmax; /*!< size of the memory block allocated for array */ 
    +} PnlSpMatComplex;
    +

    The non zero elements of row i are stored in array between the indices I[i] and I[i+1]-1. The array J contains the column indices of every element of array.
    -

    Sparse matrices are defined using the internal template approach and can be used for integer, float +

    Sparse matrices are defined using the internal template approach and can be used for integer, float or complex base data according to the following table

    -

    +

    +

    4.6.1
    dcomplex pnl_sp_mat_complexPnlSpMatComplex
    -

    +class="td11">PnlSpMatComplex +

    4.6.2 Functions
    -

    Constructors and destructors

    + + +
      -
    • +

      PnlSpMat * pnl_sp_mat_new ()
      Description Create an empty sparse matrix. -

    • -
    • +
    • +

      PnlSpMat * pnl_sp_mat_create (int m, int n, int nzmax)
      Description Create a sparse matrix with size m x n designed to hold at most nzmax non zero elements. - - - -

    • -
    • +
    • +

      void pnl_sp_mat_clone (PnlSpMat 4.6.2 *src)
      Description Clone src into dest, which is automatically resized. On output, dest and src are equal but independent. -

    • -
    • +
    • +

      PnlSpMat * 4.6.2 PnlSpMat *src)
      Description Create an independent copy of src. -

    • -
    • +
    • +

      void pnl_sp_mat_free (PnlSpMat **)
      Description Delete a sparse matrix. -

    • -
    • +
    • +

      int pnl_sp_mat_resize (PnlSpMat 4.6.2 PnlSpMat to become a m x n sparse matrices holding at most nzmax. Note that no old data are kept except if M->m is left unchanged and we only call this function to increase M->nzmax. Return OK or FAIL. -

    • -
    • +
    • +

      PnlMat * 4.6.2 *M)
      Description Create a dense
      PnlMat from a spare one. -

    • -
    • +
    • +

      PnlSpMat * 4.6.2 PnlMat *M)
      Description Create a sparse matrix from a dense one. -

    • -
    • void pnl_sp_mat_create_from_file (char *file)
      Description Read a sparse matrix from the file with name file. We use the Matrix Market - Exchange Format +

    • +
    • -
      - M  N  L                                   | <--- rows, columns, entries -  
      I1  J1  A(I1, J1)                         | <--+ -  
      I2  J2  A(I2, J2)                         |    | -  
      I3  J3  A(I3, J3)                         |    |-- L lines -  
          . . .                                 |    | -  
      IL JL  A(IL, JL)                          | <--+ -  
         -
      -

      The format

      void pnl_sp_mat_create_from_file (char *file)
      Description Read a sparse matrix from the file with name file. We use the Matrix Market + Exchange Format

      +
      M  N  L                                   | <--- rows, columns, entries 
      +I1  J1  A(I1, J1)                         | <--+ 
      +I2  J2  A(I2, J2)                         |    | 
      +I3  J3  A(I3, J3)                         |    |-- L lines 
      +    . . .                                 |    | 
      +IL JL  A(IL, JL)                          | <--+
      + +

      The format (I1, J1) A(I1,4.6.2 # or % is ignored up to the end of the line.

    -

    Accessing elements

      -
    • void  +

      void pnl_sp_mat_set (PnlSpMat *M, int i, int j, double x)
      Description Set M[i,j] = x. This function increases M->nzmax if necessary. -

    • -
    • +
    • +

      double pnl_sp_mat_get (const PnlSpMat *M, int i, int j)
      Description Return M[i,j]. If M has no entry with such an index, zero is returned.

    -

    Description Return M[i,j]. If M has no entry with such an index, zero is returned.

    +

    Applying external operations

    -

    Description In-place matrix scalar division

    +

    Standard matrix operations

    - - - -

    Description Return the Kroenecker product of A and B.

    +

    Comparison functions

    -

    + component–wise. Return TRUE or FALSE.

    +

    4.7 Hyper Matrices

    -

    + + + +

    4.7.1 Short description
    -

    The Hyper matrix types and related functions are defined in the header

    The Hyper matrix types and related functions are defined in the header pnl/pnl_matrix.h. @@ -4623,74 +7612,598 @@

    4.7.1 - - -

    -
    -typedef struct PnlHmat{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlHmat pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int ndim; /*!< nb dimensions */ - 
      int *dims; /*!< pointer to store the values of the ndim dimensions */ - 
      int mn; /*!< product dim_1 *...*dim_ndim */ - 
      int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1] - 
                    with pdims[ndim - 1] = 1 */ - 
      double *array; /*!< pointer to store */ - 
    } PnlHmat; - 

    typedef struct PnlHmatInt{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlHmatInt pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int ndim; /*!< nb dimensions */ - 
      int *dims; /*!< pointer to store the value of the ndim dimensions */ - 
      int mn; /*!< product dim_1 *...*dim_ndim */ - 
      int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1] - 
                    with pdims[ndim - 1] = 1 */ - 
      int *array; /*!< pointer to store */ - 
    } PnlHmatInt; - 

    typedef struct PnlHmatComplex{ - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlHmatComplex pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int ndim; /*!< nb dimensions */ - 
      int *dims; /*!< pointer to store the value of the ndim dimensions */ - 
      int mn; /*!< product dim_1 *...*dim_ndim */ - 
      int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1] - 
                    with pdims[ndim - 1] = 1 */ - 
      dcomplex *array; /*!< pointer to store */ - 
    } PnlHmatComplex; -
    - - - -

    ndim is the number of dimensions, dim is an array to store the size of each dimension and nm + +

    typedef struct PnlHmat{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlHmat pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int ndim; /*!< nb dimensions */ 
    +  int *dims; /*!< pointer to store the values of the ndim dimensions */ 
    +  int mn; /*!< product dim_1 *...*dim_ndim */ 
    +  int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1] 
    +                with pdims[ndim - 1] = 1 */ 
    +  double *array; /*!< pointer to store */ 
    +} PnlHmat; 
    + 
    +typedef struct PnlHmatInt{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlHmatInt pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int ndim; /*!< nb dimensions */ 
    +  int *dims; /*!< pointer to store the value of the ndim dimensions */ 
    +  int mn; /*!< product dim_1 *...*dim_ndim */ 
    +  int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1] 
    +                with pdims[ndim - 1] = 1 */ 
    +  int *array; /*!< pointer to store */ 
    +} PnlHmatInt; 
    + 
    +typedef struct PnlHmatComplex{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlHmatComplex pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int ndim; /*!< nb dimensions */ 
    +  int *dims; /*!< pointer to store the value of the ndim dimensions */ 
    +  int mn; /*!< product dim_1 *...*dim_ndim */ 
    +  int *pdims; /*!< array of size ndim, s.t. pdims[i] = dims[ndim-1] x ... dims[i+1] 
    +                with pdims[ndim - 1] = 1 */ 
    +  dcomplex *array; /*!< pointer to store */ 
    +} PnlHmatComplex;
    +

    ndim is the number of dimensions, dim is an array to store the size of each dimension and nm contains the product of the sizes of each dimension. array is an array of size mn containing the data. The integer array pdims is used to create the one–to–one map between the natural indexing and the linear indexing used in array. -

    +

    4.7.2 Functions
    -

    These functions exist for all types of hypermatrices no matter what the basic type is. The following +

    These functions exist for all types of hypermatrices no matter what the basic type is. The following conventions are used to name functions operating on hypermatrices. Here is the table of prefixes used for the different basic types.

    -

    +

    +

    4.7.2
    dcomplex pnl_hmat_complexPnlHmatComplex
    -

    In this paragraph, we present the functions operating on PnlHmatComplex +

    In this paragraph, we present the functions operating on PnlHmat which exist for all types. To deduce the prototypes of these functions for other basic types, one must replace pnl_hmat and double according the above table.

    -

    Constructors and destructors

    -

    PnlHmat .

    +

    Accessing elements

      -
    • void  +

      void pnl_hmat_set (PnlHmat *self, int *tab, double x)
      Description Set the element of index tab to x. -

    • -
    • +
    • +

      double pnl_hmat_get (const PnlHmat *self, int *tab)
      Description Return the value of the element of index tab -

    • -
    • +
    • +

      double* pnl_hmat_lget (4.7.2 *self, int *tab)
      Description Return the address of self[tab] for use as a lvalue. -

    • -
    • +
    • +

      PnlMat  pnl_mat_wrap_hmat (4.7.2 PnlMat not a pointer holding the data H(t,:,:). Note that t must be of size ndim-2 and that it cannot be checked within the function. The returned matrix shares its data with H, it is only a view not a true copy. -

    • -
    • +
    • +

      PnlVect  pnl_vect_wrap_hmat (4.7.2 PnlVect not a pointer holding the data H(t,:). Note that t must be of size ndim-1 and that it cannot be checked within the function. The returned vector shares its data with H, it is only a view not a true copy. +

    - - - -

    Printing hypermatrices

      -
    • void  +

      void pnl_hmat_print (const PnlHmat *H)
      Description Print an hypermatrix.

    -

    Description Print an hypermatrix.

    +

    Term by term operations

      -
    • void  +

      void pnl_hmat_plus_hmat (PnlHmat *lhs, const PnlHmat *rhs)
      Description Compute lhs += rhs. -

    • -
    • +
    • +

      void pnl_hmat_mult_scalar (PnlHmat *lhs, double x)
      Description Compute lhs *= x where x is a real number.

    -

    +class="newline" /> Description Compute lhs *= x where x is a real number.

    +

    4.8 Iterative Solvers

    -

    +

    4.8.1 Overview
    -

    The structures and functions related to solvers are declared in

    The structures and functions related to solvers are declared in pnl/pnl_linalgsolver.h. @@ -4918,111 +8455,651 @@

    4.8.1 - - -

    -
    -typedef struct _PnlIterationBase PnlIterationBase; - 
    typedef struct _PnlCgSolver PnlCgSolver; - 
    typedef struct _PnlBicgSolver PnlBicgSolver; - 
    typedef struct _PnlGmresSolver PnlGmresSolver; - 

    struct _PnlIterationBase - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlVectXXX pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int iteration; - 
      int max_iter; - 
      double normb; - 
      double tol_; - 
      double resid; - 
      int error; - 
      /* char *  err_msg; */ - 
    }; - 

    /* When you repeatedly use iterative solvers, do not malloc each time */ - 
    struct _PnlCgSolver - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlCgSolver  pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      PnlVect * r; - 
      PnlVect * z; - 
      PnlVect * p; - 
      PnlVect * q; - 
      double rho; - 
      double oldrho; - 
      double beta; - 
      double alpha; - 
      PnlIterationBase * iter; - 
    } ; - 

    struct _PnlBicgSolver - 
    { + +
    typedef struct _PnlIterationBase PnlIterationBase; 
    +typedef struct _PnlCgSolver PnlCgSolver; 
    +typedef struct _PnlBicgSolver PnlBicgSolver; 
    +typedef struct _PnlGmresSolver PnlGmresSolver; 
    + 
    +struct _PnlIterationBase 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlVectXXX pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int iteration; 
    +  int max_iter; 
    +  double normb; 
    +  double tol_; 
    +  double resid; 
    +  int error; 
    +  /* char *  err_msg; */ 
    +}; 
    + 
    +/* When you repeatedly use iterative solvers, do not malloc each time */ 
    +struct _PnlCgSolver 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlCgSolver  pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  PnlVect * r; 
    +  PnlVect * z; 
    +  PnlVect * p; 
    +  PnlVect * q; 
    +  double rho; 
    +  double oldrho; 
    +  double beta; 
    +  double alpha; 
    +  PnlIterationBase * iter; 
    +} ; 
    + 
    +struct _PnlBicgSolver 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlBicgSolver pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  double rho_1, rho_2, alpha, beta, omega; 
    +  PnlVect * p; 
    +  PnlVect * phat; 
    +  PnlVect * s; 
    +  PnlVect * shat; 
    +  PnlVect * t; 
    +  PnlVect * v; 
    +  PnlVect * r; 
    +  PnlVect *  rtilde; 
    +  PnlIterationBase * iter; 
    +} ; 
    + 
    +struct _PnlGmresSolver 
    +{ 
    +  /** 
    +   * Must be the first element in order for the object mechanism to work 
    +   * properly. This allows any PnlGmresSolver pointer to be cast to a PnlObject 
    +   */ 
    +  PnlObject object; 
    +  int restart; 
    +  double beta; 
    +  PnlVect * s; 
    +  PnlVect * cs; 
    +  PnlVect * sn; 
    +  PnlVect * w; 
    +  PnlVect * r; 
    +  PnlMat * H; 
    +  PnlVect * v[MAX_RESTART]; 
    +  PnlIterationBase *iter; 
    +  PnlIterationBase *iter_inner; 
    +} ;
    +

    A Left preconditioner solves the problem : +

    +
    +PM x = P b,
+
    +

    and whereas right preconditioner solves +


      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlBicgSolver pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      double rho_1, rho_2, alpha, beta, omega; - 
      PnlVect * p; - 
      PnlVect * phat; - 
      PnlVect * s; - 
      PnlVect * shat; - 
      PnlVect * t; - 
      PnlVect * v; - 
      PnlVect * r; - 
      PnlVect *  rtilde; - 
      PnlIterationBase * iter; - 
    } ; - 

    struct _PnlGmresSolver - 
    { - 
      /** - 
       * Must be the first element in order for the object mechanism to work - 
       * properly. This allows any PnlGmresSolver pointer to be cast to a PnlObject - 
       */ - 
      PnlObject object; - 
      int restart; - 
      double beta; - 
      PnlVect * s; - 
      PnlVect * cs; - 
      PnlVect * sn; - 
      PnlVect * w; - 
      PnlVect * r; - 
      PnlMat * H; - 
      PnlVect * v[MAX_RESTART]; - 
      PnlIterationBase *iter; - 
      PnlIterationBase *iter_inner; - 
    } ; -
    -

    -

    A Left preconditioner solves the problem : -

    -
    -PM x = P b,
    and -whereas right preconditioner solves -
    +
    M  Py = b,    P y = x.
-
    -

    More information is given in +

    More information is given in Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed. ed.). SIAM. ISBN 0898715342. OCLC 51266114. The reader will find in this book some discussion about right or/and left preconditioner and a description of the following algorithms. -

    These algorithms, we implemented with a left preconditioner. Right preconditioner can be easily +

    These algorithms, we implemented with a left preconditioner. Right preconditioner can be easily computed changing matrix vector multiplication operator from M x to 4.8.1 y = x at the end of algorithm. -

    +

    4.8.2 Functions
    -

    Three methods are implemented : Conjugate Gradient, BICGstab and GMRES with restart. For +

    Three methods are implemented : Conjugate Gradient, BICGstab and GMRES with restart. For each of them a structure is created to store temporary vectors used in the algorithm. In some cases, we have to apply iterative methods more than once : for example to solve at each time step a discrete form of an elliptic problem come from parabolic problem. In the cases, do not call the constructor and destructor at each time, but instead use the initialization and solve procedures. -

    Formally we have, - - - -

    -
    -Create iterative method - 
    For each time step - 
      Initialisation of iterative method - 
      Solve linear system link to elliptic problem - 
    end for - 
    free iterative method -
    -

    -

    In these functions, we don’t use any particular matrix structure. We give the matrix vector +

    Formally we have,

    +
    Create iterative method 
    +For each time step 
    +  Initialisation of iterative method 
    +  Solve linear system link to elliptic problem 
    +end for 
    +free iterative method
    +

    In these functions, we don’t use any particular matrix structure. We give the matrix vector multiplication as a parameter of the solver.

    -

    Conjugate Gradient method Only available for symmetric and positive matrices.

    - - - -

    +

    BICG stab

    -

    +

    GMRES with restart See Saad, Yousef (2003) for a discussion about the restart parameter. For GMRES we need to @@ -5213,19 +9331,22 @@

    4.8.2 = restart and restarts the algorithm with the previously computed solution as initial guess. -

    Note that if restart equals

    Note that if restart equals m, we have a classical GMRES algorithm.

    -

    In the next paragraph, we write all the solvers for +

    In the next paragraph, we write all the solvers for PnlMat . This will be done as follows: construct -an application matrix vector. - - - -

    -
    -static void pnl_mat_mult_vect_applied(const void *mat, const PnlVect *vec, - 
                                          const double a , const double b, - 
                                          PnlVect *lhs) - 
    {pnl_mat_lAxpby(a, (PnlMat*)mat, vec, b, lhs);} -
    -

    and give it as the parameter of the iterative method - - - -

    -
    -int pnl_mat_cg_solver_solve(const PnlMat * Matrix, const PnlMat * PC, - 
                                PnlVect * x, const PnlVect *b, PnlCgSolver * Solver) - 
    { return pnl_cg_solver_solve(pnl_mat_mult_vect_applied, - 
                                 Matrix, pnl_mat_mult_vect_applied, - 
                                 PC, x, b, Solver);} -
    -

    -

    In practice, we cannot define all iterative methods for all structures. With this implementation, the +an application matrix vector.

    +
    static void pnl_mat_mult_vect_applied(const void *mat, const PnlVect *vec, 
    +                                      const double a , const double b, 
    +                                      PnlVect *lhs) 
    +{pnl_mat_lAxpby(a, (PnlMat*)mat, vec, b, lhs);}
    +

    and give it as the parameter of the iterative method

    +
    int pnl_mat_cg_solver_solve(const PnlMat * Matrix, const PnlMat * PC, 
    +                            PnlVect * x, const PnlVect *b, PnlCgSolver * Solver) 
    +{ return pnl_cg_solver_solve(pnl_mat_mult_vect_applied, 
    +                             Matrix, pnl_mat_mult_vect_applied, 
    +                             PC, x, b, Solver);}
    +

    In practice, we cannot define all iterative methods for all structures. With this implementation, the user can easily :

      -
    • implement right precondioner, -
    • -
    • implement method with sparse matrix and diagonal preconditioner, or special +
    • +

      implement right precondioner, +

    • +
    • +

      implement method with sparse matrix and diagonal preconditioner, or special combination of this form

    -

    +

    Iterative algorithms for PnlMat

      -
    • int  +

      int pnl_mat_cg_solver_solve (const PnlMat *M, const 4.8.2 PnlCgSolver *Solver)
      Description Solve the linear system M x = b with preconditionner PC. -

    • -
    • +
    • +

      int pnl_mat_bicg_solver_solve (const PnlMat 4.8.2 PnlBicgSolver *Solver)
      Description Solve the linear system M x = b with preconditionner PC. -

    • -
    • +
    • + + + +

      int pnl_mat_gmres_solver_solve (const PnlMat 4.8.2 *b, PnlGmresSolver *Solver)
      Description Solve the linear system M x = b with preconditionner PC.

    +class="newline" /> Description Solve the linear system M x = b with preconditionner PC.

    diff --git a/docs/manual-html/pnl-manualse5.html b/docs/manual-html/pnl-manualse5.html index a4920425..5ed522b6 100644 --- a/docs/manual-html/pnl-manualse5.html +++ b/docs/manual-html/pnl-manualse5.html @@ -1,13 +1,13 @@ - + Cumulative distribution Functions - - - + + + @@ -29,19 +29,23 @@

    5
    -
    +
                   ∫ x
+src=
    +" class="math-display" /> +

    +

    +
  • +

    bound is undefined if STATUS is 0. Bound exceeded by parameter number I if STATUS + is negative. Lower search bound if STATUS is 1. Upper search bound if STATUS is 2.

  • +class="newline" /> Description Inverse of the cumulative normal distribution function.

    diff --git a/docs/manual-html/pnl-manualse6.html b/docs/manual-html/pnl-manualse6.html index 0cb9e1a8..3fcf04e7 100644 --- a/docs/manual-html/pnl-manualse6.html +++ b/docs/manual-html/pnl-manualse6.html @@ -1,13 +1,13 @@ - + Random Number Generators - - - + + + @@ -41,7 +41,7 @@

    6 6 NIEDERREITER
    PNL_RNG_NIEDERREITERquasi
    +class="td11">
    Table 2: Indices of the random generators
    @@ -146,72 +146,289 @@

    6.1 - - - -

    -
    -typedef struct _PnlRng PnlRng; - 
    struct _PnlRng - 
    { - 
      PnlObject object; - 
      int type; /*!< generator type * - 
      void (*Compute)(PnlRng *g, double *sample); /*!< the function to compute the - 
                                                    next number in the sequence */ - 
      int rand_or_quasi; /*!< can be PNL_MC or PNL_QMC */ - 
      int dimension; /*!< dimension of the space in which we draw the samples */ - 
      int counter; /*!< counter = number of samples already drawn */ - 
      int has_gauss; /*!< Is a gaussian deviate available? */ - 
      double gauss; /*!< If has_gauss==1, gauss a gaussian sample */ - 
      int size_state; /*!< size in bytes of the state variable */ - 
      void *state; /*!< state of the random generator */ - 
    }; -
    -

    + +
    typedef struct _PnlRng PnlRng; 
    +struct _PnlRng 
    +{ 
    +  PnlObject object; 
    +  int type; /*!< generator type * 
    +  void (*Compute)(PnlRng *g, double *sample); /*!< the function to compute the 
    +                                                next number in the sequence */ 
    +  int rand_or_quasi; /*!< can be PNL_MC or PNL_QMC */ 
    +  int dimension; /*!< dimension of the space in which we draw the samples */ 
    +  int counter; /*!< counter = number of samples already drawn */ 
    +  int has_gauss; /*!< Is a gaussian deviate available? */ 
    +  double gauss; /*!< If has_gauss==1, gauss a gaussian sample */ 
    +  int size_state; /*!< size in bytes of the state variable */ 
    +  void *state; /*!< state of the random generator */ 
    +};
    +href="#x8-102016r6.1">pnl_rng_sseed count times.

    Some auxiliary functions internally used (to be used with caution)

      -
    • +

      PnlRng * pnl_rng_new ()
      Description Create an empty PnlRng . -

    • -
    • void 
    • +
    • +

      void pnl_rng_init (PnlRng *rng, int type)
      Description Initialize an empty PnlRng as returned by pnl_rng_new to become +href="#x8-102016r6.1">pnl_rng_new to become a generator of type type which can be any of the values PNL_RNG_XXX - listed in Table 2 which correspond to pseudo random number generators. Calling pnl_rng_create is equivalent to calling first pnl_rng_new and then pnl_rng_init. -

    • -
    • pnl_rng_create is equivalent to calling first pnl_rng_new and then pnl_rng_init. + + + +

    • +
    • +

      PnlRng * pnl_rng_get_from_id (int id)
      Description Return the global generator described by its macro name. The variable - id can be any of the values PNL_RNG_XXX listed in Table 2.

    + id can be any of the values PNL_RNG_XXX listed in Table 2.

    The following functions return one sample from the specified distribution.

      -
    • int  +

      int pnl_rng_bernoulli (double p, PnlRng *rng)
      6.1 ,1} with parameter p. - - - -

    • -
    • long 
    • +
    • +

      long pnl_rng_poisson (double lambda, PnlRng *rng)
      Description Generate a sample from the Poisson law with parameter lambda. -

    • -
    • double 
    • +
    • +

      double pnl_rng_exp (double lambda, PnlRng *rng)
      Description Generate a sample from the Exponential law with parameter lambda. -

    • -
    • double 
    • +
    • +

      double pnl_rng_dblexp (double lambda_p, double lambda_m, double p, PnlRng *rng)
      Description Generate a sample from the asymmetric exponential distribution with density +

      -
      - pλ e- λpy 1     + (1- p)λ  e-λm |y|1
-  p       {y>0}          m         {y<0}
-
      -

      - where +pλpe-λpy1{y>0} + (1 - p)λm e-λm|y|1{y<0}
+ + +

      where λp > 6.1 [0,1]. + + +

    • -
    • double  +

      double pnl_rng_uni (PnlRng *rng)
      6.1 ]0,1]. -

    • -
    • double 
    • +
    • +

      double pnl_rng_uni_ab (double a, double b, PnlRng *rng)
      6.1 [a,b]. -

    • -
    • double 
    • +
    • +

      double pnl_rng_normal (PnlRng *rng)
      Description Generate a sample from the standard normal distribution. -

    • -
    • double 
    • +
    • +

      double pnl_rng_lognormal (double m, double sigma2, PnlRng *rng)
      Description Generate a sample from the log-normal distribution. The underlying normal distribution has mean m and variance sigma2. -

    • -
    • double 
    • +
    • +

      double pnl_rng_invgauss (double mu, double lambda, PnlRng *rng)
      Description Generate a sample from the inverse Gaussian distribution with mean mu and shape parameter lambda. - - - -

    • -
    • long 
    • +
    • +

      long pnl_rng_poisson1 (double lambda, double t,PnlRng *rng)
      Description Generate a sample from a Poisson process with intensity lambda at time t. -

    • -
    • double 
    • +
    • +

      double pnl_rng_gamma (double a, double b, PnlRng *rng)
      6.1 Γ(a,b) distribution. -

    • -
    • double 
    • +
    • +

      double pnl_rng_chi2 (double df, PnlRng *rng)
      6.1 (df) distribution. -

    • -
    • double 
    • +
    • +

      double pnl_rng_ncchi2 (double df, double xnonc, PnlRng *rng)
      6.1 χ2 distribution with df degrees of freedom and non central parameter xnonc. -

    • -
    • double 
    • +
    • +

      double pnl_rng_bessel (double v, double a,PnlRng *rng)
      Description Generate a sample from the Bessel distribution with parameters v > -1 and a > 0. -

    • -
    • double 
    • +
    • +

      double pnl_rng_gauss (int d, int create_or_retrieve, int index, PnlRng *rng)
      Description The second argument can be either CREATE (to actually draw the sample) or RETRIEVE (to retrieve that element of index index). With CREATE, it draws d random normal variables and stores them for future usage. They can be withdrawn using RETRIEVE - with the index of the number to be retrieved.

    + with the index of the number to be retrieved.

    The following functions take an already existing PnlVect *as first argument and fill each entry of @@ -479,8 +719,9 @@

    6.1 quasi random number generator.

    The following functions take an already existing PnlMat 6.1 dimension.

    -

    Some examples - - - -

    -
    -#include <stdlib.h> - 
    #include "pnl/pnl_random.h" - 

    int main () - 
    { - 
      int i, M; - 
      PnlRng *rng = pnl_rng_create(PNL_RNG_MERSENNE); - 
      PnlVect *v = pnl_vect_new(); - 
      M = 10000; - 

      /* rng must be initialized. When sseed=0, a default - 
         value depending on the generator is used */ - 
         pnl_rng_sseed(rng, 0); - 

      for (i=0 ; i<M ; i++) - 
      { - 
        /* Simulates a normal random vector in R^{10} */ - 
        pnl_vect_rng_normal(v, 10, rng); - 
        /* Do something with v */ - 
      } - 

      pnl_vect_free(&v); - 
      pnl_rng_free(&rng); /* Frees the generator */ - 
      exit(0); - 
    } -
    -

    - - - -

    -
    -#include <stdlib.h> - 
    #include <time.h> - 
    #include "pnl/pnl_random.h" - 

    int main () - 
    { - 
      int i, M; - 
      double E; - 
      PnlRng *rng = pnl_rng_create(PNL_RNG_MERSENNE); - 
      M = 10000; - 

      /* rng must be initialized. */ - 
      pnl_rng_sseed(rng, time (NULL)); - 

      for (i=0 ; i<M ; i++) - 
      { - 
        /* Simulates an exponential random variable */ - 
        E = pnl_rng_exp(1, rng); - 
        /* Do something with E */ - 
      } - 

      pnl_rng_free(&rng); /* Frees the generator */ - 
      exit(0); - 
    } -
    -

    -

    + p(j).

    +

    Some examples

    +
    #include <stdlib.h> 
    +#include "pnl/pnl_random.h" 
    + 
    +int main () 
    +{ 
    +  int i, M; 
    +  PnlRng *rng = pnl_rng_create(PNL_RNG_MERSENNE); 
    +  PnlVect *v = pnl_vect_new(); 
    +  M = 10000; 
    + 
    +  /* rng must be initialized. When sseed=0, a default 
    +     value depending on the generator is used */ 
    +     pnl_rng_sseed(rng, 0); 
    + 
    +  for (i=0 ; i<M ; i++) 
    +  { 
    +    /* Simulates a normal random vector in R^{10} */ 
    +    pnl_vect_rng_normal(v, 10, rng); 
    +    /* Do something with v */ 
    +  } 
    + 
    +  pnl_vect_free(&v); 
    +  pnl_rng_free(&rng); /* Frees the generator */ 
    +  exit(0); 
    +}
    + +
    #include <stdlib.h> 
    +#include <time.h> 
    +#include "pnl/pnl_random.h" 
    + 
    +int main () 
    +{ 
    +  int i, M; 
    +  double E; 
    +  PnlRng *rng = pnl_rng_create(PNL_RNG_MERSENNE); 
    +  M = 10000; 
    + 
    +  /* rng must be initialized. */ 
    +  pnl_rng_sseed(rng, time (NULL)); 
    + 
    +  for (i=0 ; i<M ; i++) 
    +  { 
    +    /* Simulates an exponential random variable */ 
    +    E = pnl_rng_exp(1, rng); 
    +    /* Do something with E */ 
    +  } 
    + 
    +  pnl_rng_free(&rng); /* Frees the generator */ 
    +  exit(0); 
    +}
    +

    6.2 The 6.2 interface to random number generation although we strongly encourage users to use the new rng interface (see section 6.1). +class="ec-lmri-10x-x-109"> 6.1).

    Every generator is identified by an integer valued macro. One must NOT refer to a generator using directly the value of the macro PNL_RNG_XXX because there is no warranty that the order used @@ -772,35 +1370,40 @@

    6.2 -
  • +

    int pnl_rand_init (int type_generator, int simulation_dim, long samples)
    Description It resets the sample counter to 0 and checks that the generator described by type_generator can actually generate samples in dimension simulation_dim and - fixes the seed.

  • + fixes the seed.

      -
    • +

      int pnl_rand_or_quasi (int type_generator)
      Description Return the type the generator of index type_generator, PNL_MC or PNL_QMC -

    • -
    • +
    • +

      void pnl_rand_sseed ((int type_generator, unsigned long int seed))
      Description It sets the seed of the generator type_generator with seed. -

    • -
    • +
    • +

      const char * pnl_rand_name (int type_generator)
      Description Return the name of the generator of index type_generator

    +class="newline" /> Description Return the name of the generator of index type_generator

    Once a generator is chosen, there are several functions available in the library to draw samples according to a given law.

    The following functions return one sample from a specified law.

      -
    • +

      int pnl_rand_bernoulli (double p, int type_generator)
      Description Generate a sample from the Bernouilli law on 6.2 ,1} with parameter p. -

    • -
    • +
    • +

      long pnl_rand_poisson (double lambda, int type_generator)
      Description Generate a sample from the Poisson law with parameter lambda. -

    • -
    • +
    • +

      double pnl_rand_exp (double lambda, int type_generator)
      Description Generate a sample from the Exponential law with parameter lambda. -

    • -
    • +
    • +

      double pnl_rand_uni (int type_generator)
      Description Generate a sample from the Uniform law on [0,1]. -

    • -
    • +
    • +

      double pnl_rand_uni_ab (double a, double b, int type_generator)
      Description Generate a sample from the Uniform law on [a,b]. -

    • -
    • +
    • +

      double pnl_rand_normal (int type_generator)
      Description Generate a sample from the standard normal distribution. -

    • -
    • +
    • +

      long pnl_rand_poisson1 (double lambda, double t, int type_generator)
      Description Generate a sample from a Poisson process with intensity lambda at time t. - - - -

    • -
    • +
    • +

      double pnl_rand_gamma (double a, double b, int type_generator)
      Description Generate a sample from the Γ(a,b) distribution. -

    • -
    • +
    • +

      double pnl_rand_chi2 (double n, int type_generator)
      Description Generate a sample from the centered 6.2 (n) distribution. -

    • -
    • +
    • +

      double pnl_rand_bessel (double v, double a, int generator)
      Description Generate a sample from the Bessel distribution with parameters v > -1 - and a > 0.

    + and a > 0.

    The following functions take an already existing PnlVect * as its first argument and fill each entry @@ -886,7 +1498,8 @@

    6.2 Quasi random number generator.

      -
    • +

      void pnl_vect_rand_uni (PnlVect 6.2 [a,b]. -

    • -
    • +
    • + + + +

      void pnl_vect_rand_normal (PnlVect *G, int samples, int generator)
      Description G is a vector of independent and identically distributed samples from the standard normal distribution. -

    • -
    • +
    • +

      void pnl_vect_rand_uni_d (PnlVect 6.2 [a,b]d. -

    • -
    • +
    • +

      void pnl_vect_rand_normal_d (PnlVect *G, int d, int generator)
      Description G is a sample from the d-dimensional standard normal distribution. +

    The following functions take an already existing PnlMat 6.2 dimension.

    + distribution.

    Because of the use of Quasi random number generators, you may need to draw a set of samples at once because they represent one sample from a multi-dimensional distribution. The following function enables to draw one sample from the dimension-dimensional standard normal distribution and store it so that you can access the elements individually afterwards.

    + + +
      -
    • +

      double pnl_rand_gauss (int d, int create_or_retrieve, int index, int type_generator)
      Description The second argument can be either CREATE (to actually draw the sample) or RETRIEVE (to retrieve that element of index index). With CREATE, it draws d random normal variables and stores them for future usage. They can be - withdrawn using RETRIEVE with the index of the number to be retrieved.

    + withdrawn using RETRIEVE with the index of the number to be retrieved.

    diff --git a/docs/manual-html/pnl-manualse7.html b/docs/manual-html/pnl-manualse7.html index 5c64951b..13a31aa2 100644 --- a/docs/manual-html/pnl-manualse7.html +++ b/docs/manual-html/pnl-manualse7.html @@ -1,13 +1,13 @@ - + Function bases and regression - - - + + + @@ -29,89 +29,623 @@

    7.1 pnl/pnl_basis.h. - - -

    -
    -struct PnlBasis_t { - 
      PnlObject     object; - 
      /** The basis type */ - 
      int           id; - 
      /** The string to label the basis */ - 
      const char   *label; - 
      /** The number of variates */ - 
      int           nb_variates; - 
      /** The total number of elements in the basis */ - 
      int           nb_func; - 
      /** The tensor matrix */ - 
      PnlMatInt    *T; - 
      /** The sparse Tensor matrix */ - 
      PnlSpMatInt  *SpT; - 
      /** The number of functions in the tensor #T */ - 
      int           len_T; - 
      /** The i-th element of the one dimensional basis. */ - 
      double      (*f)(double    x, int i); - 
      /** The first derivative of i-th element of the one dimensional basis */ - 
      double      (*Df)(double   x, int i); - 
      /** The second derivative of the i-th element of the one dimensional basis */ - 
      double      (*D2f)(double  x, int i); - 
      /** TRUE if the basis is reduced */ - 
      int           isreduced; - 
      /** The center of the domain */ - 
      double       *center; - 
      /** The inverse of the scaling factor to map the domain to [-1, 1]^nb_variates */ - 
      double       *scale; - 
      /** An array of additional functions */ - 
      PnlRnFuncR  *func_list; - 
      /** The number of functions in #func_list */ - 
      int          len_func_list; - 
    }; -
    -

    -

    A PnlBasis is a family of multivariate functions with real values. Tow different kinds of functions -can be stored in these families: tensor functions — originally, this was the only possibility, and + +

    struct _PnlBasis 
    +{ 
    +  PnlObject     object; 
    +  /** The basis type */ 
    +  int           id; 
    +  /** The string to label the basis */ 
    +  const char   *label; 
    +  /** The number of variates */ 
    +  int           nb_variates; 
    +  /** The total number of elements in the basis */ 
    +  int           nb_func; 
    +  /** The tensor matrix */ 
    +  PnlMatInt    *T; 
    +  /** The sparse Tensor matrix */ 
    +  PnlSpMatInt  *SpT; 
    +  /** The number of functions in the tensor #T */ 
    +  int           len_T; 
    +  /** Compute the i-th element of the one dimensional basis. 
    +  As a convention, (*f)(x, 0) MUST be equal to 1 */ 
    +  double      (*f)(double x, int i, int dim, void *params); 
    +  /** Compute the first derivative of i-th element of the one dimensional basis */ 
    +  double      (*Df)(double x, int i, int dim, void *params); 
    +  /** Compute the second derivative of the i-th element of the one dimensional basis */ 
    +  double      (*D2f)(double x, int i, int dim, void *params); 
    +  /** PNL_TRUE if the basis is reduced */ 
    +  int           isreduced; 
    +  /** The center of the domain */ 
    +  double       *center; 
    +  /** The inverse of the scaling factor to map the domain to [-1, 1]^nb_variates */ 
    +  double       *scale; 
    +  /** An array of additional functions */ 
    +  PnlRnFuncR   *func_list; 
    +  /** The number of functions in #func_list */ 
    +  int           len_func_list; 
    +  /** Extra parameters to pass to basis functions */ 
    +  void         *params; 
    +  /** Size of params in bytes to be passed to malloc */ 
    +  size_t        params_size; 
    +};
    +

    A PnlBasis is a family of multivariate functions with real values. Two different kinds of functions +can be stored in these families: tensor functions — originally, this was the only possibility — and standard multivariate function typed as PnlRnFuncR. +href="#x15-131031r13.1">PnlRnFuncR. +

    -

    Tensor functions. - - - -Tensors functions are built as a tensor product of one dimensional elements. Hence, we only need a +

    7.1.1 Tensor functions
    +

    Tensor functions are built as a tensor product of one dimensional elements. Hence, we only need a tensor matrix T to describe a multi-dimensional basis in terms of the one dimensional one. These tensors functions can be easily evaluated and differentiated twice, see pnl_basis_eval, +href="#x9-109019r7.2">pnl_basis_eval, pnl_basis_eval_vect, pnl_basis_eval_D, pnl_basis_eval_D_vect, pnl_basis_eval_D2, +href="#x9-109019r7.2">pnl_basis_eval_vect, pnl_basis_eval_D, pnl_basis_eval_D_vect, pnl_basis_eval_D2, pnl_basis_eval_D2_vect, pnl_basis_eval_derivs, pnl_basis_eval_derivs_vect. -

    The two tensors T and SpT do actually store the same information — T(i,j) is the degree w.r.t the -j-th variable in the i-th function. Originally, we were only using the dense representation T, which is -far more convenient to use when building the basis but it slows down the evaluation of -the basis by a great deal. To overcome this lack of efficiency, a sparse storage has been -added. +href="#x9-109019r7.2">pnl_basis_eval_D2_vect, pnl_basis_eval_derivs, pnl_basis_eval_derivs_vect. +

    Three bases are already registered as listed in Table 3. A new tensor basis can be registered using +the function pnl_basis_type_register.

    -


    7.1
    PNL_BASIS_CANONICAL

    for the Canonical polynomials
    PNL_BASIS_HERMITE

    for the Hermite polynomials
    PNL_BASIS_TCHEBYCHEV

    for the Tchebychev polynomials
    +class="td11">
    +class="content">Names of the bases.

    -

    The Hermite polynomials are defined by +

    The Hermite polynomials are defined by

    -
    +
                n  x2-dn- - x2-
 Hn(x) = (- 1) e2 dxn e  2 .
-
    -

    -If + +

    If G is a real valued standard normal random variable, E[7.1 )] = n!1{n=m}. -
    -

    -

    Standard multivariate functions. -These functions are supposed to be PnlRnFuncR. -

    To make this toolbox more complete, it is now possible to add some extra functions, which are not -tensor product. They are stored using an independent mechanism in func_list. These additional -functions are only taken into account by the methods pnl_basis_i, pnl_basis_i_vect, -pnl_basis_eval and pnl_basis_eval_vect. Note in particular that it is not possible to -differentiate these functions. To add an extra function to an existing PnlBasis, call the function -pnl_basis_add_function. -

    +src="pnl-manual23x.png" alt="{n=m }" class="left" align="middle" />. +

    The two tensors T and SpT do actually store the same information — T(i,j) is the degree w.r.t the +j-th variable in the i-th function. Originally, we were only using the dense representation T, which is +far more convenient to use when building the basis but it slows down the evaluation of the basis by +a great deal. To overcome this lack of efficiency, a sparse storage was added. Such a basis can be +created using one of the following functions.

    -

    7.2 Functions

      -
    • int pnl_basis_type_register (const char *name, double (*f)(double, int), double - (*Df)(double, int), double (*D2f)(double, int))
      Description Register a new basis type and return the index to be passed to pnl_basis_create . - The variable name is a unique string identifier of the family. The variables f, Df, D2f are the - one dimensional basis functions, its first and second order derivatives. Each of these functions - must return a double and take two arguments : the first one is the point at which evaluating - the basis functions, the second one is the index of function. Here is a toy example to show - how the canonical basis is registered (this family is actually already available with - the id PNL_BASIS_CANONICAL, so the following example may look a little - - - - fake) - - - -
      -   double f(double x, int n) { return pnl_pow_i(x, n); } -  
        double Df(double x, int n) { return n * pnl_pow_i(x, n-1); } -  
        double f(double x, int n) { return n * (n-1) * pnl_pow_i(x, n-2); } -  
      -  
        int id = pnl_basis_register ("Canonic", f, Df, D2f); -  
        /* -  
         * B is the Canonical basis of polynomials with degree less or equal than 2 in -  
         * dimension 5. -  
         */ -  
        PnlBasis *B = pnl_basis_create_from_degree (id, 2, 5); -  
         -
      -

      -

    • -
    • PnlBasis * pnl_basis_new ()
      Description Create an empty PnlBasis . -
    • -
    • void pnl_basis_print (const PnlBasis *B)
      Description Print the characteristics of a basis. -
    • -
    • +

      PnlBasis * pnl_basis_create (int index, int nb_func, int nb_variates)
      Description Create a PnlBasis for the family defined by index (see Table 3 and +href="#x9-1050007.1">PnlBasis for the family defined by index (see Table ?? and pnl_basis_type_register) with nb_variates variates. The basis will contain nb_func. -

    • -
    • PnlBasis * pnl_basis_create_from_degree (int index, int degree, int nb_variates)
      pnl_basis_type_register) with nb_variates variates. The basis will contain nb_func. +

    • +
    • +

      PnlBasis * pnl_basis_create_from_degree (int index, int degree, int + nb_variates)
      Description Create a PnlBasis for the family defined by index (see Table 3 and +href="#x9-1050007.1">PnlBasis for the family defined by index (see Table ?? and pnl_basis_type_register) with total degree less or equal than degree and nb_variates +href="#x9-1090007.2">pnl_basis_type_register) with total degree less or equal than degree and nb_variates variates. The total degree is the sum of the partial degrees.
      For instance, calling pnl_basis_create_from_degree (index,pnl_basis_create_from_degree(index, 2, 4) is equivalent to calling - pnl_basis_create_from_tensor (index, 4) is equivalent to + calling pnl_basis_create_from_tensor(index, T) where T is given by -

      - (            )
-  0  0  0  0
-|| 1  0  0  0 ||
-|| 0  1  0  0 ||
-|| 0  0  1  0 ||
-|| 0  0  0  1 ||
-||            ||
-|| 1  1  0  0 ||
-|| 1  0  1  0 ||
-|| 1  0  0  1 ||
-|| 0  1  1  0 ||
-|| 0  1  0  1 ||
-|| 0  0  1  1 ||
-|| 2  0  0  0 ||
-|| 0  2  0  0 ||
-||            ||
-( 0  0  2  0 )
-  0  0  0  2
-
      -

      -

    • -
    • PnlBasis * pnl_basis_create_from_prod_degree (int index, int degree, int +

      + + + +
      +(             )
+   0  0  0  0
+||  1  0  0  0 ||
+||  0  1  0  0 ||
+||             ||
+||  0  0  1  0 ||
+||  0  0  0  1 ||
+||  1  1  0  0 ||
+||  1  0  1  0 ||
+||  1  0  0  1 ||
+||  0  1  1  0 ||
+||  0  1  0  1 ||
+||  0  0  1  1 ||
+||             ||
+||  2  0  0  0 ||
+||  0  2  0  0 ||
+||  0  0  2  0 ||
+(  0  0  0  2 )
+
      +
    • +
    • +

      PnlBasis * pnl_basis_create_from_prod_degree (int index, int degree, int nb_variates)
      Description Create a PnlBasis for the family defined by index (see Table 3 and +href="#x9-1050007.1">PnlBasis for the family defined by index (see Table ?? and pnl_basis_type_register) with total degree less or equal than degree and nb_variates +href="#x9-1090007.2">pnl_basis_type_register) with total degree less or equal than degree and nb_variates variates. The total degree is the product of MAX(1, d_i) where the d_i are the partial degrees. -

    • -
    • +
    • +

      PnlBasis * pnl_basis_create_from_hyperbolic_degree (int index, double + degree, double q, int n)
      Description Create a sparse basis of polynomial with n variates. We give the example + of the Canonical basis. A canonical polynomial with n variates writes X1α1X2α2Xnαn. + To be a member of the basis, it must satisfy  ∑n     q
+(  i=1 αi )1∕q degree. This kind of basis + based on an hyperbolic set of indices gives priority to polynomials associated to low + order interaction. +

    • +
    • +

      PnlBasis * pnl_basis_create_from_tensor (int index, PnlMatInt *T)
      Description Create a PnlBasis for the polynomial family defined by index (see Table 3) - using the basis described by the tensor matrix T. The number of lines of T is the number of - functions of the basis whereas the numbers of columns of T is the number of variates of the - functions. Note that T is not copied inside this function but only its address is stored, so - Description Create a PnlBasis for the polynomial family defined by index (see + Table ??) using the basis described by the tensor matrix T. The number of lines of + T is the number of functions of the basis whereas the numbers of columns of T is the + number of variates of the functions. Note that T is not copied inside this function but + + + + only its address is stored, so never free T. It will be freed when calling pnl_basis_free on the returned object. - i
      Here is an example of a tensor matrix. Assume you are working with three variate functions, - the basis pnl_basis_free + on the returned object. i
      Here is an example of a tensor matrix. Assume you are working with three variate + functions, the basis { 1, x,7.2  x^2, xy, yz, z^3} is decomposed in the one dimensional - canonical basis using the following tensor matrix -

      - (          )
-   0  0  0
-||  1  0  0 ||
-||  0  1  0 ||
-||  0  0  1 ||
-||  2  0  0 ||
-||          ||
-||  1  1  0 ||
-(  0  1  1 )
-   0  0  3
-
      -

      +class="ec-lmtt-10x-x-109"> z^3} is decomposed in the one + dimensional canonical basis using the following tensor matrix +

      +
      +(         )
+|  0 0  0 |
+||  1 0  0 ||
+||  0 1  0 ||
+||  0 0  1 ||
+||  2 0  0 ||
+||  1 1  0 ||
+|(  0 1  1 |)
+   0 0  3
+
      +
    • +
    • +

      void  pnl_basis_set_from_tensor (PnlBasis *b, int index, const PnlMatInt *T)
      Description Set an alredy existing basis b to a polynomial family defined by index + (see Table ??) using the basis described by the tensor matrix T. The number of lines + of T is the number of functions of the basis whereas the numbers of columns of T is the + number of variates of the functions.
      Same function as pnl_basis_create_from_tensor except that it operates on an already + existing basis. +

      +
    +
    7.1.2 Local basis functions
    +

    A local basis is a family of indicator functions of a Cartesian partition of [1,1]d. Let +(ni)i∈{1,,d} be the number of interval along each direction. An element of the partition write +Ak = + i=1d[-1 + ki∕ni,-1 + (ki + 1)∕ni] where k is the multi-index defined by ki ∈{0,,ni - 1} +for all i ∈{1,,d}. These functions are orthogonal; this property is used by pnl_basis_fit_ls. Note +that they are not differentiable. +

    We provide the helper function pnl_basis_local_get_index to compute the linear representation of +the multi-index k. +

    + + + +
      +
    • +

      PnlBasis * pnl_basis_local_create (int *n_intervals, int space_dim)
      Description Create a local basis with n_intervals[i - 1] intervals along dimension i + for all i ∈{1,,space_dim} +

    • +
    • +

      PnlBasis * pnl_basis_local_create_regular (int n_intervals, int space_dim)
      Description Equivalent to calling pnl_basis_local_create with an array of size + space_dim filled with n_intervals.

    • -
    • void  +

      int pnl_basis_local_get_index (const PnlBasis *basis, const double *x)
      Description Return the linear index of the cell containing x. It is an integer between + 0 and ( + i=1dni) - 1.

    +

    If the domain you want to consider is not [-1,1]d, use the functions pnl_basis_set_domain or +pnl_basis_set_reduced to map your product space into [-1,1]d. +

    +

    +
    7.1.3 Standard multivariate functions
    +

    These functions are supposed to be PnlRnFuncR. +

    To make this toolbox more complete, it is now possible to add some extra functions, which +are not tensor product. They are stored using an independent mechanism in func_list. +These additional functions are only taken into account by the methods pnl_basis_i, +pnl_basis_i_vect, pnl_basis_eval and pnl_basis_eval_vect. Note in particular that it +is not possible to differentiate these functions and that they are not sensitive to the +isreduced attribute. To add an extra function to an existing PnlBasis, call the function +

    +
      +
    • +

      void pnl_basis_add_function (PnlBasis *b, PnlRnFuncR PnlRnFuncR *f)
      Description Add the function f to the already existing basis b. -

    • -
    • void  Description Add the function f to the already existing basis b.

    +

    +

    +

    7.2 Functions

    + + + +
      +
    • +

      PnlBasis * pnl_basis_new ()
      Description Create an empty PnlBasis . +

    • +
    • +

      void pnl_basis_clone (PnlBasis *dest, const 7.2 *src)
      Description Clone src into dest. The basis dest must already exist before calling this function. On exit, dest and src are identical and independent. -

    • -
    • +
    • +

      PnlBasis * pnl_basis_copy (const PnlBasis *B)
      Description Create a copy of B. -

    • -
    • void  pnl_basis_set_from_tensor (PnlBasis *b, int index, const PnlMatInt - *T)
      Description Set an alredy existing basis b to a polynomial family defined by index (see - Table 3) using the basis described by the tensor matrix T. The number of lines of T is the - number of functions of the basis whereas the numbers of columns of T is the number of - variates of the functions.
      Same function as pnl_basis_create_from_tensor except that it operates on an already - existing basis. -
    • -
    • PnlBasis * pnl_basis_create_from_hyperbolic_degree (int index, double degree, - double q, int n)
      Description Create a sparse basis of polynomial with n variates. We give the example of the - Canonical basis. A canonical polynomial with n variates writes X1α1X2α2Xnαn. To be a - member of the basis, it must satisfy (∑n   α q)
-   i=1 i1∕q degree. This kind of basis based on - an hyperbolic set of indices gives priority to polynomials associated to low order - interaction. -
    • -
    • void 
    • +
    • +

      void pnl_basis_free (PnlBasis **basis)
      Description Free a PnlBasis created by pnl_basis_create. Beware that basis is the address - of a pnl_basis_create. Beware that basis is the + address of a PnlBasis *. - - - -

    • -
    • void 
    • +
    • +

      void pnl_basis_del_elt (PnlBasis *B, const PnlVectInt *d)
      Description Remove the function defined by the tensor product d from an existing basis - B. -

    • -
    • void  Description Remove the function defined by the tensor product d from an existing + basis B. +

    • +
    • +

      void pnl_basis_del_elt_i (PnlBasis *B, int i)
      Description Remove the i-th element of basis B. -

    • -
    • void 
    • +
    • +

      void pnl_basis_add_elt (PnlBasis *B, const PnlVectInt *d)
      Description Add the function defined by the tensor d to the Basis B. +

    • +
    • +

      int pnl_basis_type_register (const char *name, double (*f)(double, int, int, + void), double (*Df)(double, int, int, void), double (*D2f)(double, int, int, void), int + is_orthogonal)
      Description Register a new basis type and return the index to be passed to pnl_basis_create. + The variable name is a unique string identifier of the family. The variables f, Df, D2f are the + one dimensional basis functions, its first and second order derivatives. Each of these functions + must return a double and take two arguments : the first one is the point at which evaluating + + + + the basis functions, the second one is the index of function. The varis_orthogonal if the + elements of the basis are orthogonal for the L2 scalar product. Here is a toy example to show + how the canonical basis is registered (this family is actually already available with the + id PNL_BASIS_CANONICAL, so the following example may look a little fake) +

      +
        double f(double x, int n, int dim, void *params) { return pnl_pow_i(x, n); } 
      +  double Df(double x, int n, int dim, void *params) { return n * pnl_pow_i(x, n-1); } 
      +  double f(double x, int n, int dim, void *params) { return n * (n-1) * pnl_pow_i(x, n-2); } 
      + 
      +  int id = pnl_basis_register ("Canonic", f, Df, D2f, PNL_FALSE); 
      +  /* 
      +   * B is the Canonical basis of polynomials with degree less or equal than 2 in dimension 5. 
      +   */ 
      +  PnlBasis *B = pnl_basis_create_from_degree (id, 2, 5);
      + +
    • +
    • +

      void pnl_basis_print (const PnlBasis *B)
      Description Print the characteristics of a basis. +

    -

    Functional regression based on a least square approach often leads to ill conditioned linear systems. +

    Functional regression based on a least square approach often leads to ill conditioned linear systems. One way of improving the stability of the system is to use centered and renormalised polynomials so that the original domain of interest

    -

    +src="pnl-manual27x.png" alt=" ( ) +x ∈ D ↦- → xi --(bi +-ai)∕2 + (bi - ai)∕2 i=1,⋅⋅⋅,d +" class="math-display" /> +(1) +

    +
    +          ( xi - centeri)
+x ∈ D ↦-→    --scale----
+                   i    i=1,⋅⋅⋅,d
+
    + +

    +

    Note that this renormlization does not apply to the extra functions by pnl_basis_add_function but +only to the functions defined by the tensor T. +

    -

    The following functions are provided for compatibility purposes but are marked as deprecated. Use +

    The following functions are provided for compatibility purposes but are marked as deprecated. Use the functions with the _vect extension.

    +href="#x9-109019r7.2">pnl_basis_eval_derivs_vect but takes a C array as + the point of evaluation.

    diff --git a/docs/manual-html/pnl-manualse8.html b/docs/manual-html/pnl-manualse8.html index 2d0e5f81..5c39ee3d 100644 --- a/docs/manual-html/pnl-manualse8.html +++ b/docs/manual-html/pnl-manualse8.html @@ -1,13 +1,13 @@ - + Numerical integration - - - + + + @@ -20,11 +20,11 @@ href="#tailpnl-manualse8.html">tail] [up]

    8 Numerical integration

    + id="x10-1100008">Numerical integration

    8.1 Overview

    + id="x10-1110008.1">Overview

    To use these functionalities, you should include pnl/pnl_integration.h.

    Numerical integration methods are designed to numerically evaluate the integral over a finite or @@ -33,44 +33,153 @@

    8.1 2). - - - -

    -
    -typedef struct { - 
      double (*function) (double x, void *params); - 
      void *params; - 
    } PnlFunc; - 

    typedef struct { - 
      double (*function) (double x, double y, void *params); - 
      void *params; - 
    } PnlFunc2D; -
    -

    -

    We provide the following two macros to evaluate a PnlFunc or PnlFunc2D at a given -point - - -

    -
    -#define PNL_EVAL_FUNC(F, x) (*((F)->function))(x, (F)->params) - 
    #define PNL_EVAL_FUNC2D(F, x, y) (*((F)->function))(x, y, (F)->params) -
    -

    -

    + +

    typedef struct { 
    +  double (*function) (double x, void *params); 
    +  void *params; 
    +} PnlFunc; 
    + 
    +typedef struct { 
    +  double (*function) (double x, double y, void *params); 
    +  void *params; 
    +} PnlFunc2D;
    +

    We provide the following two macros to evaluate a PnlFunc or PnlFunc2D at a given point +

    +
    #define PNL_EVAL_FUNC(F, x) (*((F)->function))(x, (F)->params) 
    +#define PNL_EVAL_FUNC2D(F, x, y) (*((F)->function))(x, y, (F)->params)
    +

    8.2 Functions

    + id="x10-1120008.2">Functions + functions F with integrable singularities on the interval [x0,x1].

    diff --git a/docs/manual-html/pnl-manualse9.html b/docs/manual-html/pnl-manualse9.html index 4deebba0..11652c98 100644 --- a/docs/manual-html/pnl-manualse9.html +++ b/docs/manual-html/pnl-manualse9.html @@ -1,13 +1,13 @@ - + Fast Fourier Transform - - - + + + @@ -20,75 +20,78 @@ href="#tailpnl-manualse9.html">tail
    ] [up]

    9 Fast Fourier Transform

    + id="x11-1130009">Fast Fourier Transform

    9.1 Overview

    -

    This toolbox uses C version of the Fortran FFTPack library available on Overview +

    This toolbox uses C version of the Fortran FFTPack library available on +http://www.netlib.org/fftpack. -
    http://www.netlib.org/fftpack.

    The forward Fourier transform of a vector c is defined by

    -
    +
         N∑ -1
-zj =     cke-ijk2π∕N ,  j = 0,⋅⋅⋅,N - 1
+src=
    -

    -The inverse Fourier transform enables to recover + +

    The inverse Fourier transform enables to recover c from z and is defined by

    -
    +
           N -1
-c =  1-∑   c eijk2π∕N,  j = 0,⋅⋅⋅,N - 1
- k   N k=0  k
-
    +src="pnl-manual31x.png" alt=" N∑- 1 +ck = -1 ckeijk2π∕N, j = 0,⋅⋅⋅ ,N - 1 + N k=0 +" class="math-display" /> + -

    -Note that the inverse Fourier transform is scaled by

    Note that the inverse Fourier transform is scaled by N1, such that the inverse Fourier transform applies to the Fourier transform just yields the original vector.

    The coefficients of the Fourier transform of a real function satisfy the following relation

    -
    +class="equation">
    +
    z  = z---,
- k    N-k
-
    (2)
    -

    -where +(2) +

    where N is the number of discretization points.

    A few remarks on the FFT of real functions and its inverse transform:

      -
    • We only need half of the coefficients. -
    • -
    • When a value is known to be real, its imaginary part is not stored. So the imaginary +
    • +

      We only need half of the coefficients. +

    • +
    • +

      When a value is known to be real, its imaginary part is not stored. So the imaginary part of the zero-frequency component is never stored as it is known to be zero. -

    • -
    • For a sequence of even length the imaginary part of the frequency
    • +
    • +

      For a sequence of even length the imaginary part of the frequency n∕2 is not stored - either, since the symmetry (2) implies that this is purely real too.

    + either, since the symmetry (2) implies that this is purely real too.

    FFTPack storage + The functions use the fftpack storage convention for half-complex sequences. In this convention, the half-complex transform of a real sequence is stored with frequencies in increasing order, starting from zero, with the real and imaginary parts of each frequency in neighboring @@ -102,24 +105,24 @@

    9.1 V ) and the values (PnlVectComplex C) that would be returned if the same real input sequence were passed to pnl_fft as a complex sequence (with +href="#x11-1170009.2.1">pnl_fft as a complex sequence (with imaginary parts set to 0),

    -
    -
    +
    +
    C(0) = V(0) + i0,
-
 C(1) = V(1) + iV (2),
 C(2) = V(3) + iV (4), -----
 C(3) = V(3) - iV (4) = C-(2),
 C(4) = V(1) + iV (2) = C (1)
-
    (3)
    +" class="math-display" /> +
    (3)

    The elements of index greater than N∕9.1
    -
    +class="equation">
    +
    C(0) = V(0) + i0,
 C(1) = V(1) + iV (2),
-C(2) = V(3) + iV (4),
-                  -----
+
+C(2) = V(3) + iV (4),---
 C(3) = V(5) - i0 = C (0),---
 C(4) = V(3) - iV (4) = C-(2),
 C(5) = V(1) + iV (2) = C (1)
-
    (4)
    +" class="math-display" />
    +(4)

    9.2 Functions

    + id="x11-1160009.2">Functions

    To use the following functions, you should include pnl/pnl_fft.h. @@ -160,76 +164,86 @@

    9.2

    9.2.1 Direct call functions
    + id="x11-1170009.2.1">Direct call functions

    All FFT functions need some extra memory to perform their computations. This is automatically handled by all the functions but you can these repeatedly, for instance inside a Monte Carlo loop, you should allocate a workspace once and for all and use the same at every iteration. In this case, -use the functions defined in Section 9.2.2. +use the functions defined in Section 9.2.2.

      -
    • int  +

      int pnl_fft_inplace (PnlVectComplex *data)
      Description Compute the FFT of data in place. The original content of data is lost. -

    • -
    • int 
    • +
    • +

      int pnl_ifft_inplace (PnlVectComplex *data)
      Description Compute the inverse FFT of data in place. The original content of data is lost. -

    • -
    • int 
    • +
    • +

      int pnl_fft (const PnlVectComplex *in, PnlVectComplex *out)
      Description Compute the FFT of in and stores it into out. -

    • -
    • int 
    • +
    • +

      int pnl_ifft (const PnlVectComplex *in, PnlVectComplex *out)
      Description Compute the inverse FFT of in and stores it into out. -

    • -
    • int 
    • +
    • +

      int pnl_fft2 (double *re, double *im, int n)
      Description Compute the FFT of the vector of length n whose real (resp. imaginary) parts are given by the arrays re (resp. im). The real and imaginary parts of the FFT are respectively stored in re and im on output. -

    • -
    • int 
    • +
    • +

      int pnl_ifft2 (double *re, double *im, int n)
      Description Compute the inverse FFT of the vector of length n whose real (resp. imaginary) parts are given by the arrays re (resp. im). The real and imaginary parts of the inverse FFT are respectively stored in re and im on output. -

    • -
    • int 
    • +
    • + + + +

      int pnl_real_fft (const PnlVect *in, PnlVectComplex *out)
      Description Compute the FFT of the real valued sequence in and stores it into out. -

    • -
    • int 
    • +
    • +

      int pnl_real_ifft (const PnlVectComplex *in, 9.2.1 *out)
      Description Compute the inverse FFT of in and stores it into out. The vector in is supposed to be the FFT of a real valued vector. -

    • -
    • int 
    • +
    • +

      int pnl_real_fft_inplace (double *data, int n)
      Description Compute the FFT of the real valued vector data of length n. The result - is stored in data using the FFTPack storage described above, see 9.1. -

    • -
    • int data using the FFTPack storage described above, see 9.1.0.0. +

    • +
    • +

      int pnl_real_ifft_inplace (double *data, int n)
      Description Compute the inverse FFT of the vector data of length n. data is supposed to be the FFT coefficients a real valued sequence stored using the FFTPack storage. On output, data contains the inverse FFT. -

    • -
    • int 
    • +
    • +

      int pnl_real_fft2 (double *re, double *im, int n)
      Description Compute the FFT of the real vector re of length n. im is only used on output to store the imaginary part the FFT. The real part is stored into re -

    • -
    • int 
    • +
    • +

      int pnl_real_ifft2 (double *re, double *im, int n)
      Description Compute the inverse FFT of the vector re + i * im of length n, which is supposed to be the FFT of a real valued sequence. On exit, im is unused. -

    • -
    • int 
    • +
    • +

      int pnl_fft2d_inplace (PnlMatComplex *data)
      Description Compute the 2D FFT of data. This function applies a 1D FFT to each row of the matrix and then a 1D FFT to each column of the modified matrix. -

    • -
    • int 
    • +
    • +

      int pnl_ifft2d_inplace (PnlMatComplex *data)
      Description Compute the inverse 2D FFT of data. This function is the inverse of the function pnl_fft2d_inplace. -

    • -
    • int pnl_fft2d_inplace. + + + +

    • +
    • +

      int pnl_fft2d (const PnlMatComplex *in, PnlMatComplex *out)
      Description Compute the 2D FFT of in and stores it into out. -

    • -
    • int 
    • +
    • +

      int pnl_ifft2d (const PnlMatComplex *in, PnlMatComplex *out)
      Description Compute the inverse 2D FFT of in and stores it into out. -

    • -
    • int 
    • +
    • +

      int pnl_real_fft2d (const PnlMat *in, PnlMatComplex *out)
      Description Compute the 2D FFT of the real matrix in and stores it into out. -

    • -
    • int 
    • +
    • +

      int pnl_real_ifft2d (const PnlMatComplex *in, 9.2.1 *out)
      Description Compute the inverse 2D FFT of the complex matrix in which is known to be the forward 2D FFT a real matrix. The result id stored it into out. Note that this - function modifies the input matrix in.

    - - - + function modifies the input matrix in.

    9.2.2 Function with workspace
    + id="x11-1180009.2.2">Function with workspace + supposed to be the FFT of a real valued vector.

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pnl_basis_create_from_tensor - pnl_basis_local_create - pnl_basis_local_create_regular pnl_basis_del_elt pnl_basis_del_elt_i pnl_basis_eval @@ -141,7 +139,6 @@ EXPORTS pnl_basis_eval_D_vect pnl_basis_eval_derivs pnl_basis_eval_derivs_vect - pnl_basis_eval_local pnl_basis_eval_vect pnl_basis_fit_ls pnl_basis_free @@ -153,6 +150,8 @@ EXPORTS pnl_basis_i_vect pnl_basis_ik pnl_basis_ik_vect + pnl_basis_local_create + pnl_basis_local_create_regular pnl_basis_local_get_index pnl_basis_new pnl_basis_print

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