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mesti.m
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function [S, info] = mesti(syst, B, C, D, opts)
%MESTI Multi-source frequency-domain electromagnetic simulations.
% [field_profiles, info] = MESTI(syst, B) returns the spatial field profiles
% of Ez(x,y) for 2D transverse-magnetic (TM) fields satisfying
% [- (d/dx)^2 - (d/dy)^2 - (omega/c)^2*epsilon(x,y)] Ez(x,y) = source(x,y),
% or of Hz(x,y) for 2D transverse-electric (TE) fields satisfying
% [- (d/dx)*(1/epsilon(x,y))_yy*(d/dx) - (d/dy)*(1/epsilon(x,y))_xx*(d/dy)
% + (d/dy)*(1/epsilon(x,y))_xy*(d/dx) + (d/dx)*(1/epsilon(x,y))_yx*(d/dy)
% - (omega/c)^2] Hz(x,y) = source(x,y).
% The polarization (TM or TE), relative permittivity profile epsilon(x,y),
% frequency omega, and boundary conditions are specified by structure 'syst'.
% Each column of matrix 'B' specifies a distinct input source profile.
% Electric and magnetic current sources can both be specified (for either
% polarization) with appropriate conversion.
% The returned 'field_profiles' is a 3D array, with field_profiles(:,:,i)
% being the field profile of Ez or Hz given the i-th input source profile. The
% information of the computation is returned in structure 'info'.
%
% MESTI uses finite-difference discretization on the Yee lattice, after which
% the differential operator becomes an (nx*ny)-by-(nx*ny) sparse matrix A
% where [ny, nx] is the number of sites Ez or Hz is discretized onto, and
% field_profiles = reshape(inv(A)*B, ny, nx, []).
%
% [S, info] = MESTI(syst, B, C) returns S = C*inv(A)*B where the solution
% inv(A)*B is projected onto the output channels or locations of interest
% through matrix C; each row of matrix 'C' is a distinct output projection
% profile, discretized into a 1-by-(nx*ny) vector in the same order as matrix
% A. When the MUMPS function zmumps() is available, this is done by computing
% the Schur complement of an augmented matrix K = [A,B;C,0] through a partial
% factorization.
%
% [S, info] = MESTI(syst, B, C, D) returns S = C*inv(A)*B - D. This can be
% used for the computation of scattering matrices, where S is the scattering
% matrix, and matrix D can be derived analytically or computed as D =
% C*inv(A0)*B - S0 from a reference system A0 for which the scattering matrix
% S0 is known.
%
% [field_profiles, info] = MESTI(syst, B, [], [], opts),
% [S, info] = MESTI(syst, B, C, [], opts), and
% [S, info] = MESTI(syst, B, C, D, opts) allow detailed options to be
% specified with structure 'opts'.
%
% MESTI considers nonmagnetic materials with isotropic (i.e., scalar)
% permittivity. Even though epsilon(x,y) is originally a scalar in the
% continuous system, subpixel smoothing (which is needed to ensure smooth
% variation with respect to parameters and to reach second-order accuracy for
% TE polarization) creates anisotropy where epsilon(x,y) becomes a symmetric
% tensor. TM polarization uses the zz component of epsilon(x,y). TE
% polarization uses the xx, yy, and xy (equals yx) components of epsilon(x,y).
% The other components of epsilon(x,y) are zero.
%
% This file checks and parses the parameters, and it can build matrices B and
% C from its nonzero elements specified by the user (see details below). It
% calls function mesti_build_fdfd_matrix() to build matrix A and function
% mesti_matrix_solver() to compute C*inv(A)*B or inv(A)*B, where most of the
% computation is done.
%
% === Input Arguments ===
% syst (scalar structure; required):
% A structure that specifies the system, used to build the FDFD matrix A.
% It contains the following fields:
% syst.polarization (character vector; optional):
% Polarization. Possible choices are:
% 'TM' - Transverse-magnetic field (Hx, Hy, Ez)
% 'TE' - Transverse-electric field (Ex, Ey, Hz)
% TM field uses syst.epsilon, and TE field uses syst.inv_epsilon. If
% only one of syst.epsilon and syst.inv_epsilon is given,
% syst.polarization is optional and will be automatically picked based
% on which one is given. If syst.epsilon and syst.inv_epsilon are both
% given, then syst.polarization must be specified.
% syst.epsilon (numeric matrix, real or complex; required for TM):
% An ny_Ez-by-nx_Ez matrix discretizing the relative permittivity
% profile epsilon(x,y). Specifically, syst.epsilon(m,n) is the scalar
% epsilon(x,y) averaged over a square with area (syst.dx)^2 centered at
% the point (x_n, y_m) where Ez(x,y) is located on the Yee lattice. It
% is the zz component of the discretized epsilon(x,y) tensor from
% subpixel smoothing, used by TM fields. We choose
% (x_n, y_m) = (n-0.5, m-0.5)*syst.dx,
% with n = 1, ..., nx_Ez, m = 1, ..., ny_Ez.
% such that the lower corner of the first pixel syst.epsilon(m=1,n=1) is
% at (x,y) = (0,0).
% Note that y corresponds to the first index m, and x corresponds to
% the second index n.
% The positive imaginary part of syst.epsilon describes absorption,
% and the negative imaginary part describes linear gain.
% One can use syst.epsilon with ny_Ez = 1 and with a periodic or
% Bloch periodic boundary in y to simulate 1D systems where the relative
% permittivity profile is translationally invariant in y.
% syst.inv_epsilon (cell array; required for TE):
% The xx, yy, and xy components of the discretized inverse relative
% permittivity 1/epsilon(x,y) tensor from subpixel smoothing, used by TE
% fields. It has three elements
% inv_epsilon{1}: (1/epsilon(x,y))_xx, size [ny_Ez, nx_Hz]
% inv_epsilon{2}: (1/epsilon(x,y))_yy, size [ny_Hz, nx_Ez]
% inv_epsilon{3}: (1/epsilon(x,y))_xy, size [ny_Ez, nx_Ez]
% The third element, inv_epsilon{3}, is optional and is treated as zero
% when syst.inv_epsilon only has two elements. The yx component is not
% specified since we only consider symmetric 1/epsilon tensors where
% (1/epsilon)_yx = (1/epsilon)_xy.
% The different components are located at different points:
% - Hz(x,y) at (x_{n-0.5}, y_{m-0.5}); size [ny_Hz, nx_Hz].
% - (1/epsilon(x,y))_xx and Dx ~ dHz/dy at (x_{n+0.5}, y_m).
% - (1/epsilon(x,y))_yy and Dy ~ dHz/dx at (x_n, y_{m+0.5}).
% - (1/epsilon(x,y))_xy at (x_n, y_m), same as Ez.
% Here, (x_n, y_m) is the location of Ez and syst.epsilon above.
% inv_epsilon{1}, inv_epsilon{2}, and inv_epsilon{3} should each be
% (1/epsilon(x,y))_xx, (1/epsilon(x,y))_yy, and (1/epsilon(x,y))_xy from
% subpixel smoothing, averaged over a square with area (syst.dx)^2
% centered at the points where each of them is located. The
% 1/epsilon(x,y) tensor from subpixel smoothing is given by Eq. (1) of
% Farjadpour et al, Optics Letters 31, 2972 (2006). Where these points
% start and end depend on the boundary condition.
% For periodic, Bloch periodic, and PMCPEC boundary conditions in x,
% nx_Hz = nx_Ez, and all of the sites on x_{n+0.5} are half a pixel
% after the corresponding sites on x_n.
% For PEC boundary condition in x, nx_Hz = nx_Ez + 1, and the sites
% on x_{n+0.5} start from half a pixel before the first site of x_n and
% end on half a pixel after the last site of x_n.
% For PMC boundary condition in x, nx_Hz = nx_Ez - 1, and the sites
% on x_{n+0.5} start from half a pixel after the first site of x_n and
% end on half a pixel before the last site of x_n.
% For PECPMC boundary condition in x, nx_Hz = nx_Ez, and all of the
% sites on x_{n+0.5} are half a pixel before the corresponding sites on
% x_n.
% Similar applies to boundary conditions in y.
% syst.length_unit (anything; optional):
% Length unit, such as micron, nm, or some reference wavelength. This
% code only uses dimensionless quantities, so syst.length_unit is never
% used. This syst.length_unit is meant to help the user interpret the
% units of (x,y), dx, wavelength, kx_B, ky_B, etc.
% syst.wavelength (numeric scalar, real or complex; required):
% Vacuum wavelength 2*pi*c/omega, in units of syst.length_unit.
% syst.dx (positive scalar; required):
% Discretization grid size, in units of syst.length_unit.
% syst.PML (scalar structure or cell array; optional):
% Parameters of the perfectly matched layer (PML) used to simulate an
% open boundary. Note that PML is not a boundary condition; it is a
% layer placed within the simulation domain (just before the boundary)
% that attenuates outgoing waves with minimal reflection.
% In mesti(), the PML starts from the interior of the system
% specified by syst.epsilon or syst.inv_epsilon, and ends at the first
% or last pixel inside syst.epsilon or syst.inv_epsilon. (Note: this is
% different from the function mesti2s() that handles two-sided
% geometries, where the homogeneous spaces on the left and right are
% specified separately through syst.epsilon_L and syst.epsilon_R, and
% where PML is placed in such homogeneous space, outside of the
% syst.epsilon or syst.inv_epsilon there.)
% When only one set of PML parameters is used in the system (as is
% the most common), such parameters can be specified with a scalar
% structure syst.PML that contains the following fields:
% npixels (positive integer scalar; required): Number of PML pixels.
% Note this is within syst.epsilon or syst.inv_epsilon, not in
% addition to.
% direction (character vector; optional): Direction(s) where PML is
% placed. Available choices are (case-insensitive):
% 'all' - (default) PML in both x and y directions
% 'x' - PML in x direction
% 'y' - PML in y direction
% side (character vector; optional): Side(s) where PML is placed.
% Available choices are (case-insensitive):
% 'both' - (default) PML on both sides
% '-' - one-sided PML; end at the first pixel (n=1 or m=1)
% '+' - one-sided PML; end at the last pixel (n=nx or m=ny)
% power_sigma (non-negative scalar; optional): Power of the
% polynomial grading for the conductivity sigma; defaults to 3.
% sigma_max_over_omega (non-negative scalar; optional):
% Conductivity at the end of the PML; defaults to
% 0.8*(power_sigma+1)/((2*pi/wavelength)*dx*sqrt(epsilon_bg)).
% where epsilon_bg is the average relative permittivity along the
% last slice of the PML. This is used to attenuate propagating
% waves.
% power_kappa (non-negative scalar; optional): Power of the
% polynomial grading for the real-coordinate-stretching factor
% kappa; defaults to 3.
% kappa_max (real scalar no smaller than 1; optional):
% Real-coordinate-stretching factor at the end of the PML;
% defaults to 15. This is used to accelerate the attenuation of
% evanescent waves. kappa_max = 1 means no real-coordinate
% stretching.
% power_alpha (non-negative scalar; optional): Power of the
% polynomial grading for the CFS alpha factor; defaults to 1.
% alpha_max_over_omega (non-negative scalar; optional): Complex-
% frequency-shifting (CFS) factor at the beginning of the PML.
% This is typically used in time-domain simulations to suppress
% late-time (low-frequency) reflections. We don't use it by
% default (alpha_max_over_omega = 0) since we are in frequency
% domain.
% We use the following PML coordinate-stretching factor:
% s(p) = kappa(p) + sigma(p)./(alpha(p) - i*omega)
% with
% sigma(p)/omega = sigma_max_over_omega*(p.^power_sigma),
% kappa(p) = 1 + (kappa_max-1)*(p.^power_kappa),
% alpha(p)/omega = alpha_max_over_omega*((1-p).^power_alpha),
% where omega is frequency, and p goes linearly from 0 at the beginning
% of the PML to 1 at the end of the PML.
% By default, syst.PML = {}, which means no PML on any side. PML is
% only placed on the side(s) specified by syst.PML.
% When multiple sets of PML parameters are used in the system (e.g.,
% a thinner PML on one side, a thicker PML on another side), these
% parameters can be specified with a cell array
% syst.PML = {PML_1, PML_2, ...},
% with PML_1 and PML_2 each being a structure containing the above
% fields; they can specify different PML parameters on different sides.
% Each side cannot be specified more than once.
% With real-coordinate stretching, PML can attenuate evanescent waves
% more efficiently than free space, so there is no need to place free
% space in front of PML.
% The PML thickness should be chosen based on the acceptable level of
% reflectivity given the discretization resolution and the range of wave
% numbers (i.e., angles) involved; more PML pixels gives lower
% reflectivity. Typically 10-40 pixels are sufficient.
% syst.PML_type (character vector; optional):
% Type of PML. Available choices are (case-insensitive):
% 'UPML' - (default) uniaxial PML
% 'SC-PML' - stretched-coordinate PML
% The two are mathematically equivalent, but matrix A using UPML is
% symmetric (unless Bloch periodic boundary is used) while that using
% SC-PML has lower condition number.
% syst.xBC (character vector; optional):
% Boundary condition (BC) at the two ends in x direction, effectively
% specifying Ez(m,n) or Hz(m,n) at n=0 and n=nx_Ez+1 or nx_Hz+1, one
% pixel beyond the computation domain. Available choices are:
% 'Bloch' - Ez(m,n+nx_Ez) = Ez(m,n)*exp(1i*syst.kx_B*nx_Ez*syst.dx)
% Hz(m,n+nx_Hz) = Hz(m,n)*exp(1i*syst.kx_B*nx_Hz*syst.dx)
% 'periodic' - equivalent to 'Bloch' with syst.kx_B = 0
% 'PEC' - Ez(m,0) = Ez(m,nx_Ez+1) = 0
% Hz(m,0) = Hz(m,1); Hz(m,nx_Hz+1) = Hz(m,nx_Hz)
% 'PMC' - Ez(m,0) = Ez(m,1); Ez(m,nx_Ez+1) = Ez(m,nx_Ez)
% Hz(m,0) = Hz(m,nx_Hz+1) = 0
% 'PECPMC' - Ez(m,0) = 0; Ez(m,nx_Ez+1) = Ez(m,nx_Ez)
% Hz(m,0) = Hz(m,1); Hz(m,nx_Hz+1) = 0
% 'PMCPEC' - Ez(m,0) = Ez(m,1); Ez(m,nx_Ez+1) = 0
% Hz(m,0) = 0; Hz(m,nx_Hz+1) = Hz(m,nx_Hz)
% where PEC stands for perfect electric conductor (for which Ez = 0 and
% Ey ~ dHz/dx = 0 at the boundary) and PMC stands for perfect magnetic
% conductor (for which Hz = 0 and Hy ~ dEz/dx = 0 at the boundary).
% By default,
% syst.xBC = 'Bloch' if syst.kx_B is given; otherwise,
% syst.xBC = 'PEC' if syst.polarization = 'TM',
% syst.xBC = 'PMC' if syst.polarization = 'TE'.
% The choice of syst.xBC has little effect on the numerical accuracy
% when PML is used.
% syst.kx_B (real scalar; optional):
% Bloch wave number in x direction, in units of 1/syst.length_unit.
% syst.kx_B is only used when syst.xBC = 'Bloch'. It is allowed to
% specify a complex-valued syst.kx_B, but a warning will be displayed.
% syst.yBC (character vector; optional):
% Boundary condition in y direction, analogous to syst.xBC.
% syst.ky_B (real scalar; optional):
% Bloch wave number in y direction, analogous to syst.kx_B.
% syst.self_energy (sparse matrix; optional):
% Self-energy matrix, used as A = A - syst.self_energy to achieve exact
% radiation boundary condition. In mesti2s() when syst.xBC = 'outgoing',
% the self-energy matrix will be built and passed to mesti(). On the
% sides where self-energy is used, Dirichlet boundary condition (PEC for
% TM, PMC for TE) should be use with no PML.
% B (numeric matrix or structure array; required):
% Matrix specifying the input source profiles B in the C*inv(A)*B - D or
% C*inv(A)*B or inv(A)*B returned. When the input argument B is a matrix,
% it is directly used, and size(B,1) must equal ny_Ez*nx_Ez for TM,
% ny_Hz*Hx_Hz for TE; each column of B specifies a source profile, placed
% on the grid points of Ez or Hz.
% Note that matrix A is (syst.dx)^2 times the differential operator and
% is unitless, so each column of B is (syst.dx)^2 times the source(x,y) on
% the right-hand side of the differential equation and has the same unit as
% Ez or Hz.
% Instead of specifying matrix B directly, one can specify only its
% nonzero parts, from which mesti() will build the sparse matrix B. To do
% so, B in the input argument should be set as a structure array; here we
% refer to such structure array as B_struct to distinguish it from the
% resulting matrix B. If for every column of matrix B, all of its nonzero
% elements are spatially located within a rectangle (e.g., line sources or
% block sources), one can use the following fields:
% B_struct.pos (four-element integer vector): B_struct.pos =
% [m1, n1, h, w] specifies the location and the size of the
% rectangle. Here, (m1, n1) is the index of the (y,x) coordinate of
% the smaller-y, smaller-x corner of the rectangle, at the location
% of f(m1, n1) where f = Ez or Hz; (h, w) is the height and width of
% the rectangle, such that (m2, n2) = (m1+h-1, n1+w-1) is the index
% of the higher-index corner of the rectangle.
% B_struct.data (2D or 3D numeric array): nonzero elements of matrix B
% within the rectangle specified by B_struct.pos.
% When it is a 3D array, B_struct.data(m',n',a) is the a-th input
% source at the location of f(m=m1+m'-1, n=n1+n'-1), which becomes
% B(m+(n-1)*ny, a). In other words, B_struct.data(:,:,a) gives the
% sources at the rectangle f(m1+(0:(h-1)), n1+(0:(w-1))). So,
% size(B_struct.data, [1,2]) must equal [h, w], and
% size(B_struct.data, 3) is the number of inputs.
% Alternatively, B_struct.data can be a 2D array that is
% equivalent to reshape(data_in_3D_array, h*w, []), in which case
% size(B_struct.data, 2) is the number of inputs; in this case,
% B_struct.data can be a sparse matrix, and its sparsity will be
% preserved when building matrix B.
% If different inputs are located within different rectangles (e.g.,
% inputs from line sources on the left and separate inputs from line
% sources on the right), B_struct can be a structure array with multiple
% elements [e.g., B_struct(1).pos and B_struct(1).data specify line sources
% on the left; B_struct(2).pos and B_struct(2).data specify line sources on
% the right]; these inputs are treated separately, and the total number of
% inputs is size(B_struct(1).data, 3) + size(B_struct(2).data, 3) + ... +
% size(B_struct(end).data, 3).
% If the nonzero elements of matrix B do not have rectangular shapes in
% space [e.g., for total-field/scattered-field (TF/SF) simulations], one
% can use a structure array with the following fields:
% B_struct.ind (integer vector): linear indices of the spatial
% locations of the nonzero elements of matrix B, such that
% f(B_struct.ind) are the points where the source is placed. Such
% linear indices can be constructed from sub2ind().
% B_struct.data (2D numeric matrix): nonzero elements of matrix B at
% the locations specified by B_struct.ind. Specifically,
% B_struct.data(i,a) is the a-th input source at the location of
% f(B_struct.ind(i)), which becomes B(B_struct.ind(i), a). So,
% size(B_struct.data, 1) must equal numel(B_struct.ind), and
% size(B_struct.data, 2) is the number of inputs.
% Similarly, one can use B_struct(1).ind, B_struct(2).ind etc together
% with B_struct(1).data, B_struct(2).data etc to specify inputs at
% different sets of locations. Every element of the structure array must
% have the same fields [e.g., one cannot specify B_struct(1).pos and
% B_struct(2).ind], so the more general B_struct.ind syntax should be used
% when some of the inputs are rectangular and some are not.
% C (numeric matrix or structure array or 'transpose(B)' or []; optional):
% Matrix specifying the output projections in the C*inv(A)*B - D or
% C*inv(A)*B returned. When the input argument C is a matrix, it is
% directly used, and size(C,2) must equal ny_Ez*nx_Ez for TM, ny_Hz*nx_Hz
% for TE; each row of C specifies a projection profile, placed on the grid
% points of Ez or Hz.
% Scattering matrix computations often have C = transpose(B); if that
% is the case, the user can set C = 'transpose(B)' as a character vector,
% and it will be replaced by transpose(B) in the code. Doing so has an
% advantage: if matrix A is symmetric (which is the case with UPML without
% Bloch periodic boundary), C = 'transpose(B)', and opts.method = 'APF',
% the matrix K = [A,B;C,0] will be treated as symmetric when computing its
% Schur complement to lower computing time and memory usage.
% For field-profile computations, the user can simply omit C from the
% input arguments, as in mesti(syst, B), if there is no need to change the
% default opts. If opts is needed, the user can use
% mesti(syst, B, [], [], opts), namely setting C = [] and D = [].
% Similar to B, here one can specify only the nonzero parts of the
% output matrix C, from which mesti() will build the sparse matrix C. The
% syntax is the same as for B, summarized below. If for every row of matrix
% C, all of its nonzero elements are spatially located withing a rectangle
% (e.g., projection of fields on a line), one can set the input argument C
% to be a structure array (referred to as C_struct below) with the
% following fields:
% C_struct.pos (four-element integer vector): C_struct.pos =
% [m1, n1, h, w] specifies the location and the size of the
% rectangle. Here, (m1, n1) is the index of the (y,x) coordinate of
% the smaller-y, smaller-x corner of the rectangle, at the location
% of f(m1, n1) where f = Ez or Hz; (h, w) is the height and width of
% the rectangle, such that (m2, n2) = (m1+h-1, n1+w-1) is the index
% of the higher-index corner of the rectangle.
% C_struct.data (2D or 3D numeric array): nonzero elements of matrix C
% within the rectangle specified by C_struct.pos.
% When it is a 3D array, C_struct.data(m',n',b) is the b-th output
% projection at the location of f(m=m1+m'-1, n=n1+n'-1), which
% becomes C(b, m+(n-1)*ny). In other words, C_struct.data(:,:,b)
% gives the projection at the rectangle f(m1+(0:(h-1)),n1+(0:(w-1))).
% So, size(C_struct.data, [1,2]) must equal [h, w], and
% size(C_struct.data, 3) is the number of outputs.
% Alternatively, C_struct.data can be a 2D array that is
% equivalent to reshape(data_in_3D_array, h*w, []), in which case
% size(C_struct.data, 2) is the number of outputs; in this case,
% C_struct.data can be a sparse matrix, and its sparsity will be
% preserved when building matrix C.
% If the nonzero elements of matrix C do not have rectangular shapes in
% space [e.g., for near-field-to-far-field transformations], one can set C
% to a structure array with the following fields:
% C_struct.ind (integer vector): linear indices of the spatial
% locations of the nonzero elements of matrix C, such that
% f(C_struct.ind) are the points where the projection is placed. Such
% linear indices can be constructed from sub2ind().
% C_struct.data (2D numeric matrix): nonzero elements of matrix C at
% the locations specified by C_struct.ind. Specifically,
% C_struct.data(i,b) is the b-th projection at the location of
% f(C_struct.ind(i)), which becomes C(b, C_struct.ind(i)). So,
% size(C_struct.data, 1) must equal numel(C_struct.ind), and
% size(C_struct.data, 2) is the number of outputs.
% Like in B_struct, one can use structure arrays with multiple elements
% to specify outputs at different spatial locations.
% D (numeric matrix or []; optional):
% Matrix D in the C*inv(A)*B - D returned, which specifies the baseline
% contribution; size(D,1) must equal size(C,1), and size(D,2) must equal
% size(B,2).
% When D = [], it will not be subtracted from C*inv(A)*B. For field-
% profile computations where C = [], the user must also set D = [].
% opts (scalar structure; optional, defaults to an empty struct):
% A structure that specifies the options of computation; defaults to an
% empty structure. It can contain the following fields (all optional):
% opts.verbal (logical scalar; optional, defaults to true):
% Whether to print system information and timing to the standard output.
% opts.prefactor (numeric scalar, real or complex; optional):
% When opts.prefactor is given, mesti() will return
% opts.prefactor*C*inv(A)*B - D or opts.prefactor*C*inv(A)*B or
% opts.prefactor*inv(A)*B. Such prefactor makes it easier to use C =
% transpose(B) to take advantage of reciprocity. Defaults to 1.
% opts.exclude_PML_in_field_profiles (logical scalar; optional, defaults to false):
% When opts.exclude_PML_in_field_profiles = true, the PML pixels
% (specified by syst.PML.npixels) are excluded from the returned
% field_profiles on each side where PML is used; otherwise the full
% field profiles are returned. Only used for field-profile computations
% (i.e., when the output projection matrix C is not given).
% opts.solver (character vector; optional):
% The solver used for sparse matrix factorization. Available choices are
% (case-insensitive):
% 'MUMPS' - (default when MUMPS is available) Use MUMPS. Its MATLAB
% interface zmumps.m must be in MATLAB's search path.
% 'MATLAB' - (default when MUMPS is not available) Use the built-in
% lu() function in MATLAB, which uses UMFPACK with AMD
% ordering.
% MUMPS is faster and uses less memory than lu(), and is required for
% the APF method.
% opts.method (character vector; optional):
% The solution method. Available choices are (case-insensitive):
% 'APF' - Augmented partial factorization. C*inv(A)*B is obtained
% through the Schur complement of an augmented matrix
% K = [A,B;C,0] using a partial factorization. Must have
% opts.solver = 'MUMPS'. This is the most efficient method,
% but it cannot be used for computing the full field profile
% X=inv(A)*B or with iterative refinement.
% 'FG' - Factorize and group. Factorize A=L*U, and obtain C*inv(A)*B
% through C*inv(U)*inv(L)*B with optimized grouping. Must
% have opts.solver = 'MATLAB'. This is slightly better than
% 'FS' when MUMPS is not available, but it cannot be used for
% computing the full field profile X=inv(A)*B.
% 'FS' - Factorize and solve. Factorize A=L*U, solve for X=inv(A)*B
% with forward and backward substitutions, and project with
% C as C*inv(A)*B = C*X. Here, opts.solver can be either
% 'MUMPS' or 'MATLAB', and it can be used for computing
% the full field profile X=inv(A)*B or with iterative
% refinement.
% 'C*inv(U)*inv(L)*B' - Same as 'FG'.
% 'factorize_and_solve' - Same as 'FS'.
% By default, if C is given and opts.iterative_refinement = false, then
% 'APF' is used when opts.solver = 'MUMPS', and 'C*inv(U)*inv(L)*B' is
% used when opts.solver = 'MATLAB'. Otherwise, 'factorize_and_solve' is
% used.
% opts.clear_BC (logical scalar; optional, defaults to false):
% When opts.clear_BC = true, variables 'B' and 'C' will be cleared in
% the caller's workspace to reduce peak memory usage. Can be used when B
% and/or C take up significant memory and are not needed after calling
% mesti().
% opts.clear_syst (logical scalar; optional, defaults to false):
% When opts.clear_syst = true, variable 'syst' will be cleared in the
% caller's workspace to reduce peak memory usage. This can be used when
% syst.epsilon/syst.inv_epsilon and/or syst.self_energy take up
% significant memory and are not needed after calling mesti().
% opts.clear_memory (logical scalar; optional, defaults to true):
% Whether or not to clear variables inside mesti() to reduce peak memory
% usage.
% opts.verbal_solver (logical scalar; optional, defaults to false):
% Whether to have the solver print detailed information to the standard
% output. Note the behavior of output from MUMPS depends on compiler.
% opts.use_METIS (logical scalar; optional, defaults to false):
% Whether to use METIS (instead of the default AMD) to compute the
% ordering in MUMPS. Using METIS can sometimes reduce memory usage
% and/or factorization and solve time, but it typically takes longer at
% the analysis (i.e., ordering) stage.
% opts.nrhs (positive integer scalar; optional):
% The number of right-hand sides (number of columns of the input matrix
% B) to consider simultaneously, used only when opts.method =
% 'factorize_and_solve' and C is given. Defaults to 1 if
% opts.iterative_refinement = true, 10 if opts.solver = 'MUMPS' with
% opts.iterative_refinement = false, 4 otherwise.
% opts.store_ordering (logical scalar; optional, defaults to false):
% Whether to store the ordering sequence (permutation) for matrix A or
% matrix K; only possible when opts.solver = 'MUMPS'. If
% opts.store_ordering = true, the ordering will be returned in
% info.ordering.
% opts.ordering (positive integer vector; optional):
% A user-specified ordering sequence for matrix A or matrix K, used only
% when opts.solver = 'MUMPS'. Using the ordering from a previous
% computation can speed up (but does not eliminate) the analysis stage.
% The matrix size must be the same, and the sparsity structure should be
% similar among the previous and the current computation.
% opts.analysis_only (logical scalar; optional, defaults to false):
% When opts.analysis_only = true, the factorization and solution steps
% will be skipped, and S = [] will be returned. The user can use
% opts.analysis_only = true with opts.store_ordering = true to return
% the ordering for A or K; only possible when opts.solver = 'MUMPS'.
% opts.nthreads_OMP (positive integer scalar; optional):
% Number of OpenMP threads used in MUMPS; overwrites the OMP_NUM_THREADS
% environment variable.
% opts.parallel_dependency_graph (logical scalar; optional):
% If MUMPS is multithread, whether to use parallel dependency graph in MUMPS.
% This typically improve the time performance, but marginally increase
% the memory usage.
% opts.iterative_refinement (logical scalar; optional, defaults to false):
% Whether to use iterative refinement in MUMPS to lower round-off
% errors. Iterative refinement can only be used when opts.solver =
% 'MUMPS' and opts.method = 'factorize_and_solve' and C is given, in
% case opts.nrhs must equal 1. When iterative refinement is used, the
% relevant information will be returned in info.itr_ref_nsteps,
% info.itr_ref_omega_1, and info.itr_ref_omega_2.
%
% === Output Arguments ===
% field_profiles (3D array):
% For field-profile computations (i.e., when the output projection matrix C
% is not given), the returned field_profiles are the spatial field profiles
% of Ez (for TM polarization) or Hz (for TE polarization) resulting from
% the input sources specified by B.
% When opts.exclude_PML_in_field_profiles = false, field_profiles =
% reshape(inv(A)*B, ny, nx, M) where [ny, nx] = [ny_Ez, nx_Ez] for TM,
% [ny_Hz, nx_Hz] for TE, and M = size(B, 2).
% When opts.exclude_PML_in_field_profiles = true, the PML pixels
% (specified by syst.PML.npixels) are excluded from field_profiles on each
% side where PML is used.
% S (full numeric matrix):
% The generalized scattering matrix S = C*inv(A)*B or S = C*inv(A)*B - D.
% info (scalar structure):
% A structure that contains the following fields:
% info.opts (scalar structure):
% The final 'opts' used, excluding the user-specified matrix ordering.
% info.timing (scalar structure):
% A structure containing timing of the various stages, in seconds, in
% fields 'total', 'init', 'build', 'analyze', 'factorize', 'solve'.
% info.nnz (scalar structure):
% A structure containing the number of nonzero elements for the various
% matrices, in fields 'A', 'B', 'C', 'S', 'X'.
% info.xPML (two-element cell array; optional);
% PML parameters on the two sides in x direction, if used.
% info.yPML (two-element cell array; optional);
% PML parameters on the two sides in y direction, if used.
% info.ordering_method (character vector; optional):
% Ordering method used in MUMPS.
% info.ordering (positive integer vector; optional):
% Ordering sequence returned by MUMPS when opts.store_ordering = true.
% info.itr_ref_nsteps (integer vector; optional):
% Number of steps of iterative refinement for each input, if
% opts.iterative_refinement = true; 0 means no iterative refinement.
% info.itr_ref_omega_1 (real vector; optional):
% Scaled residual omega_1 at the end of iterative refinement for each
% input; see MUMPS user guide section 3.3.2 for definition.
% info.itr_ref_omega_2 (real vector; optional):
% Scaled residual omega_2 at the end of iterative refinement for each
% input; see MUMPS user guide section 3.3.2 for definition.
%
% See also: mesti_build_fdfd_matrix, mesti_matrix_solver, mesti2s
%% Part 1.1: Check validity of syst, assign default values to its fields, and parse BC and PML specifications
t0 = clock;
if nargin < 2; error('Not enough input arguments.'); end
if ~(isstruct(syst) && isscalar(syst)); error('Input argument syst must be a scalar structure.'); end
if ~isfield(syst, 'wavelength'); error('Input argument syst must have field ''wavelength''.'); end
if ~isfield(syst, 'dx'); error('Input argument syst must have field ''dx''.'); end
if ~(isnumeric(syst.wavelength) && isscalar(syst.wavelength)); error('syst.wavelength must be a numeric scalar.'); end
if ~(isreal(syst.dx) && isscalar(syst.dx) && syst.dx > 0); error('syst.dx must be a positive scalar.'); end
% Pick the polarization to use; assign use_TM
if isfield(syst, 'polarization')
if strcmpi(syst.polarization, 'TM')
use_TM = true;
elseif strcmpi(syst.polarization, 'TE')
use_TM = false;
else
error('syst.polarization, if given, must be ''TM'' or ''TE''.');
end
else
% When syst.polarization is not given, we automatically pick based on whether syst.epsilon or syst.inv_epsilon is given.
if isfield(syst, 'epsilon') && ~isfield(syst, 'inv_epsilon')
use_TM = true;
elseif ~isfield(syst, 'epsilon') && isfield(syst, 'inv_epsilon')
use_TM = false;
elseif isfield(syst, 'epsilon') && isfield(syst, 'inv_epsilon')
error('syst.polarization must be given when syst.epsilon and syst.inv_epsilon both exist.');
else % neither syst.epsilon nor syst.inv_epsilon exists
error('Input argument syst must have field ''epsilon'' or ''inv_epsilon''.');
end
end
% Check syst.epsilon (for TM) and syst.inv_epsilon (for TE)
if use_TM
if ~isfield(syst, 'epsilon')
error('syst.epsilon must be given when syst.polarization = ''TM''.');
elseif ~(isnumeric(syst.epsilon) && ismatrix(syst.epsilon))
error('syst.epsilon must be a numeric matrix, if given.');
end
syst.polarization = 'TM';
str_pol = 'Ez'; % for printing system info
else
if ~isfield(syst, 'inv_epsilon')
error('syst.inv_epsilon must be given when syst.polarization = ''TE''.');
elseif ~iscell(syst.inv_epsilon) || (numel(syst.inv_epsilon) ~= 2 && numel(syst.inv_epsilon) ~= 3)
error('syst.inv_epsilon must be a two-element or three-element cell array, if given.');
elseif ~(ismatrix(syst.inv_epsilon{1}) && isnumeric(syst.inv_epsilon{1}))
error('syst.inv_epsilon{1} must be a numeric matrix.');
elseif ~(ismatrix(syst.inv_epsilon{2}) && isnumeric(syst.inv_epsilon{2}))
error('syst.inv_epsilon{2} must be a numeric matrix.');
elseif numel(syst.inv_epsilon) == 3 && ~(ismatrix(syst.inv_epsilon{3}) && isnumeric(syst.inv_epsilon{3}))
error('syst.inv_epsilon{3} must be a numeric matrix, if given.');
end
syst.polarization = 'TE';
str_pol = 'Hz'; % for printing system info
end
% Check that the user did not accidentally use options only in mesti2s()
if isfield(syst, 'epsilon_L') && ~isempty(syst.epsilon_L)
warning('syst.epsilon_L is not used in mesti(); will be ignored.');
end
if isfield(syst, 'epsilon_R') && ~isempty(syst.epsilon_R)
warning('syst.epsilon_R is not used in mesti(); will be ignored.');
end
% Number of sites in y and x
if use_TM
% [ny, nx] = [ny_Ez, nx_Ez] for TM
[ny, nx] = size(syst.epsilon);
else
% [ny, nx] = [ny_Hz, nx_Hz] for TE
nx = size(syst.inv_epsilon{1}, 2); % inv_epsilon_xx
ny = size(syst.inv_epsilon{2}, 1); % inv_epsilon_yy
end
nxy = nx*ny;
% Check boundary condition in x
if isfield(syst, 'kx_B') && ~isempty(syst.kx_B)
if ~(isnumeric(syst.kx_B) && isscalar(syst.kx_B))
error('syst.kx_B must be a numeric scalar, if given.');
elseif (isfield(syst, 'xBC') && ~isempty(syst.xBC)) && (iscell(syst.xBC) || ~strcmpi(syst.xBC, 'Bloch'))
error('When syst.kx_B is given, syst.xBC must be ''Bloch'' if specified.');
end
syst.xBC = 'Bloch';
% mesti_build_fdfd_matrix() uses (kx_B,ky_B)*periodicity as the input arguments xBC and yBC for Bloch BC
xBC = (syst.kx_B)*(nx*syst.dx); % dimensionless
else
% Defaults to Dirichlet boundary condition unless syst.kx_B is given
if ~isfield(syst, 'xBC') || isempty(syst.xBC)
if use_TM
syst.xBC = 'PEC';
else
syst.xBC = 'PMC';
end
elseif ~((ischar(syst.xBC) && isrow(syst.xBC)) || (isstring(syst.xBC) && isscalar(syst.xBC)))
error('syst.xBC must be a character vector or string, if given.');
elseif ~ismember(lower(syst.xBC), lower({'Bloch', 'periodic', 'PEC', 'PMC', 'PECPMC', 'PMCPEC'}))
error('syst.xBC = ''%s'' is not a supported option; type ''help mesti'' for supported options.', syst.xBC);
elseif strcmpi(syst.xBC, 'Bloch')
error('syst.xBC = ''Bloch'' but syst.kx_B is not given.');
end
xBC = syst.xBC;
end
if isfield(syst, 'kx') && ~isempty(syst.kx)
warning('syst.kx will not be used; use syst.kx_B for Bloch wave number in x.');
end
% Check boundary condition in y
if isfield(syst, 'ky_B') && ~isempty(syst.ky_B)
if ~(isnumeric(syst.ky_B) && isscalar(syst.ky_B))
error('syst.ky_B must be a numeric scalar, if given.');
elseif (isfield(syst, 'yBC') && ~isempty(syst.yBC)) && (iscell(syst.yBC) || ~strcmpi(syst.yBC, 'Bloch'))
error('When syst.ky_B is given, syst.yBC must be ''Bloch'' if specified.');
end
syst.yBC = 'Bloch';
% mesti_build_fdfd_matrix() uses (kx_B,ky_B)*periodicity as the input arguments xBC and yBC for Bloch BC
yBC = (syst.ky_B)*(ny*syst.dx); % dimensionless
else
% Defaults to Dirichlet boundary condition unless syst.ky_B is given
if ~isfield(syst, 'yBC') || isempty(syst.yBC)
if use_TM
syst.yBC = 'PEC';
else
syst.yBC = 'PMC';
end
elseif ~((ischar(syst.yBC) && isrow(syst.yBC)) || (isstring(syst.yBC) && isscalar(syst.yBC)))
error('syst.yBC must be a character vector or string, if given.');
elseif ~ismember(lower(syst.yBC), lower({'Bloch', 'periodic', 'PEC', 'PMC', 'PECPMC', 'PMCPEC'}))
error('syst.yBC = ''%s'' is not a supported option; type ''help mesti'' for supported options.', syst.yBC);
elseif strcmpi(syst.yBC, 'Bloch')
error('syst.yBC = ''Bloch'' but syst.ky_B is not given.');
end
yBC = syst.yBC;
end
if isfield(syst, 'ky') && ~isempty(syst.ky)
warning('syst.ky will not be used; use syst.ky_B for Bloch wave number in y.');
end
% Defaults to no PML anywhere
if ~isfield(syst, 'PML') || isempty(syst.PML)
syst.PML = {};
elseif ~((isstruct(syst.PML) && isscalar(syst.PML)) || iscell(syst.PML))
error('syst.PML must be a scalar structure or a cell array, if given.');
elseif isstruct(syst.PML)
% convert to a single-element cell array if only one set of PML spec is given
syst.PML = {syst.PML};
end
% Parse the user-specified PML parameters to PML on the four sides
% PML_list = {xPML_low, xPML_high, yPML_low, yPML_high}
PML_list = {[], [], [], []};
str_sides = {'-x', '+x', '-y', '+y'};
use_PML = false;
for ii = 1:numel(syst.PML)
use_PML = true;
PML_ii = syst.PML{ii};
if ~(isstruct(PML_ii) && isscalar(PML_ii))
error('syst.PML{%d} must be a scalar structure.', ii)
end
% Number of PML pixels must be given
% Other fields are optional and will be checked in mesti_build_fdfd_matrix()
if ~isfield(PML_ii, 'npixels') || isempty(PML_ii.npixels)
error('syst.PML{%d} must contain field ''npixels''.', ii);
end
% If PML is specified, we put it on both x and y directions by default
if ~isfield(PML_ii, 'direction') || isempty(PML_ii.direction)
PML_ii.direction = 'all';
elseif ~((ischar(PML_ii.direction) && isrow(PML_ii.direction)) || (isstring(PML_ii.direction) && isscalar(PML_ii.direction)))
error('syst.PML{%d}.direction must be a character vector or string, if given.', ii);
elseif ~ismember(lower(PML_ii.direction), {'all', 'x', 'y'})
error('syst.PML{%d}.direction = ''%s'' is not a supported option; use ''all'', ''x'', or ''y''.', ii, PML_ii.direction);
end
% If PML is specified, we put it on both sides by default
if ~isfield(PML_ii, 'side') || isempty(PML_ii.side)
PML_ii.side = 'both';
elseif ~((ischar(PML_ii.side) && isrow(PML_ii.side)) || (isstring(PML_ii.side) && isscalar(PML_ii.side)))
error('syst.PML{%d}.side must be a character vector or string, if given.', ii);
elseif ~ismember(lower(PML_ii.side), {'both', '-', '+'})
error('syst.PML{%d}.side = ''%s'' is not a supported option; use ''both'', ''-'', or ''+''.', ii, PML_ii.side);
end
% Convert {PML_ii.direction and PML_ii.side} to a list of the PML locations
% 1=xPML_low, 2=xPML_high, 3=yPML_low, 4=yPML_high
if strcmpi(PML_ii.direction, 'all') % x & y
if strcmpi(PML_ii.side, 'both')
ind_ii = [1,2,3,4];
elseif strcmpi(PML_ii.side, '-')
ind_ii = [1,3];
else % PML_ii.side = or '+'
ind_ii = [2,4];
end
elseif strcmpi(PML_ii.direction, 'x')
if strcmpi(PML_ii.side, 'both')
ind_ii = [1,2];
elseif strcmpi(PML_ii.side, '-')
ind_ii = 1;
else % PML_ii.side = '+'
ind_ii = 2;
end
else % PML_ii.direction = 'y'
if strcmpi(PML_ii.side, 'both')
ind_ii = [3,4];
elseif strcmpi(PML_ii.side, '-')
ind_ii = 3;
else % PML_ii.side = '+'
ind_ii = 4;
end
end
% These two fields are no longer needed
PML_ii = rmfield(PML_ii, {'direction', 'side'});
% Specify PML at those locations
for jj = 1:numel(ind_ii)
ind_side = ind_ii(jj);
% Check that PML has not been specified at that location yet
if ~isempty(PML_list{ind_side})
error('PML on %s side is specified more than once in syst.PML.', str_sides{ind_side});
end
PML_list{ind_side} = PML_ii;
end
end
% Convert to two separate cell arrays for mesti_build_fdfd_matrix()
xPML = PML_list(1:2); % {xPML_low, xPML_high}
yPML = PML_list(3:4); % {yPML_low, yPML_high}
% Use UPML by default as it produces a symmetric matrix A (unless Bloch periodic boundary is used)
if ~isfield(syst, 'PML_type') || isempty(syst.PML_type)
syst.PML_type = 'UPML';
elseif ~((ischar(syst.PML_type) && isrow(syst.PML_type)) || (isstring(syst.PML_type) && isscalar(syst.PML_type)))
error('syst.PML_type must be a character vector or string, if given.');
elseif ~ismember(lower(syst.PML_type), {'upml', 'sc-pml', 'scpml'})
error('syst.PML_type = ''%s'' is not a supported option; use ''UPML'' or ''SC-PML''.', syst.PML_type);
end
if strcmpi(syst.PML_type, 'UPML')
use_UPML = true;
else
use_UPML = false;
end
%% Part 1.2: Check validity of the other input arguments and assign default values
if ~((ismatrix(B) && isnumeric(B)) || (isstruct(B) && ~isempty(B)))
error('Input argument B must be a numeric matrix or a non-empty structure array.');
end
% C is an optional argument
if nargin < 3
C = [];
end
% D is an optional argument
if nargin < 4
D = [];
end
if ~((ismatrix(D) && isnumeric(D)) || isempty(D))
error('Input argument D must be a numeric matrix or [], if given.');
end
% opts is an optional argument
if nargin < 5 || isempty(opts)
opts = struct();
end
if ~(isstruct(opts) && isscalar(opts))
error('Input argument opts must be a scalar structure or [], if given.');
end
% opts.return_field_profile is only used internally (but will be returned within info.opts)
if isempty(C) && ~isstruct(C)
opts.return_field_profile = true;
elseif (ismatrix(C) && isnumeric(C)) || (isstruct(C) && ~isempty(C))
opts.return_field_profile = false;
use_transpose_B = false;
elseif isequal(C, 'transpose(B)')
opts.return_field_profile = false;
use_transpose_B = true;
else
error('Input argument C must be a numeric matrix or a non-empty structure array or ''transpose(B)'' or [], if given.');
end
% Check that the user did not accidentally use options only in mesti2s()
if isfield(opts, 'symmetrize_K') && ~isempty(opts.symmetrize_K)
error('opts.symmetrize_K is not used in mesti(); to symmetrize matrix K = [A,B;C,0], set C = ''transpose(B)'', make sure matrix A is symmetric (syst.PML_type = ''UPML'' and no Bloch periodic boundary), set opts.solver = ''MUMPS'', and set opts.method = ''APF''.');
end
% Turn on verbal output by default
if ~isfield(opts, 'verbal') || isempty(opts.verbal)
opts.verbal = true;
elseif ~(islogical(opts.verbal) && isscalar(opts.verbal))
error('opts.verbal must be a logical scalar, if given.');
end
% Defaults the prefactor to 1
if ~isfield(opts, 'prefactor') || isempty(opts.prefactor)
opts.prefactor = 1;
elseif ~(isnumeric(opts.prefactor) && isscalar(opts.prefactor))
error('opts.prefactor must be a numeric scalar, if given.');
end
% By default, we don't exclude the PML pixels from the returned field_profiles.
if opts.return_field_profile
if ~isfield(opts, 'exclude_PML_in_field_profiles') || isempty(opts.exclude_PML_in_field_profiles)
opts.exclude_PML_in_field_profiles = false;
elseif ~(islogical(opts.exclude_PML_in_field_profiles) && isscalar(opts.exclude_PML_in_field_profiles))
error('opts.exclude_PML_in_field_profiles must be a logical scalar, if given.');
end
else
if isfield(opts, 'exclude_PML_in_field_profiles') && ~isempty(opts.exclude_PML_in_field_profiles)
warning('opts.exclude_PML_in_field_profiles is not used when output projection C is given; will be ignored.');
opts = rmfield(opts, 'exclude_PML_in_field_profiles');
end
end
% By default, we don't clear syst in the caller's workspace
if ~isfield(opts, 'clear_syst') || isempty(opts.clear_syst)
opts.clear_syst = false;
elseif ~(islogical(opts.clear_syst) && isscalar(opts.clear_syst))
error('opts.clear_syst must be a logical scalar, if given.');
end
% By default, we don't clear B and C in the caller's workspace
if ~isfield(opts, 'clear_BC') || isempty(opts.clear_BC)
opts.clear_BC = false;
elseif ~(islogical(opts.clear_BC) && isscalar(opts.clear_BC))
error('opts.clear_BC must be a logical scalar, if given.');
end
% By default, we will clear internal variables to save memory; this is only used in mesti_matrix_solver()
% Note that mesti_matrix_solver() defaults opts.clear_memory to false because some users that use mesti_matrix_solver() directly may want to keep the input arguments A,B,C after calling it. But the opts.clear_memory in mesti() here only deals with the variables internal to mesti() and mesti_matrix_solver(); it doesn't deal with the input arguments provided by the user (which are specified by opts.clear_syst and opts.clear_BC), so it is safe to default opts.clear_memory to true here.
if ~isfield(opts, 'clear_memory') || isempty(opts.clear_memory)
opts.clear_memory = true;
elseif ~(islogical(opts.clear_memory) && isscalar(opts.clear_memory))
error('opts.clear_memory must be a logical scalar, if given.');
end
% The following fields of opts will be checked/initialized in mesti_matrix_solver():
% opts.solver
% opts.method
% opts.verbal_solver
% opts.use_METIS
% opts.nrhs
% opts.store_ordering
% opts.ordering
% opts.analysis_only
% opts.nthreads_OMP
% opts.parallel_dependency_graph
% opts.iterative_refinement
if opts.verbal
% print basic system info if the calling function is not mesti2s()
st = dbstack;
if numel(st) > 1 && strcmp(st(2).name,'mesti2s')
called_from_mesti2s = true;
fprintf(' ... ');
else
called_from_mesti2s = false;
fprintf('System size: ny = %d, nx = %d; %s polarization\n', ny, nx, str_pol);
if use_PML
fprintf('%s on ', syst.PML_type);
for ind_side = 1:4
if ~isempty(PML_list{ind_side})
fprintf('%s ', str_sides{ind_side});
end
end
fprintf('sides; ');
else
fprintf('no PML; ');
end
fprintf('xBC = %s', syst.xBC);
if strcmpi(syst.xBC, 'Bloch'); fprintf(' (kx_B = %.4f)', syst.kx_B); end
fprintf('; yBC = %s', syst.yBC);
if strcmpi(syst.yBC, 'Bloch'); fprintf(' (ky_B = %.4f)', syst.ky_B); end
if isfield(syst, 'self_energy') && ~isempty(syst.self_energy); fprintf('; with self-energy'); end
fprintf('\nBuilding B,C... ');
end
end
t1 = clock; timing_init = etime(t1,t0); % Initialization time
%% Part 2.1: Build matrices B and C
% Build the input matrix B from its nonzero elements specified by user
if isstruct(B)
B_struct = B;
if ~isfield(B_struct, 'data')
error('Input argument B must have field ''data'' when B is a structure array.');
elseif ~isfield(B_struct, 'pos') && ~isfield(B_struct, 'ind')
error('Input argument B must have field ''pos'' or ''ind'' when B is a structure array.');
elseif isfield(B_struct, 'pos') && isfield(B_struct, 'ind')
error('Input argument B cannot have both field ''pos'' and field ''ind'' when B is a structure array.');
end
if isfield(B_struct, 'pos')
for ii = 1:numel(B_struct)
pos = B_struct(ii).pos;
if ~(isreal(pos) && isvector(pos) && numel(pos)==4 && isequal(pos, round(pos)) && min(pos)>0)
error('B(%d).pos must be a positive integer vector with 4 elements.', ii);
end
end
end
% We first pick the most efficient way to build matrix B.
% If all of the following are satisfied: (1) numel(B_struct) is small, (2) B_struct.pos is used, and (3) the rectangle specified by B_struct.pos is a single vertical slice or if it spans the full height of ny, then we will stack reshaped B_struct(ii).data with zeros. This avoids the overhead of building B with index-value pairs.
% If any of the above is not satisfied, we will build B with index-value pairs.
use_iv_pairs = false;
if numel(B_struct) > 10
use_iv_pairs = true;
elseif ~isfield(B_struct, 'pos')
use_iv_pairs = true;
else
for ii = 1:numel(B_struct)
pos = B_struct(ii).pos;
if ~(pos(4) == 1 || pos(3) == ny)
use_iv_pairs = true;
end
end
end
if use_iv_pairs
% Construct matrix B from the complete set of index-value pairs
N_tot = 0; % total number of nonzero elements in B
for ii = 1:numel(B_struct)
N_tot = N_tot + numel(B_struct(ii).data);
end
ind_list = zeros(N_tot, 0);
a_list = zeros(N_tot, 0);
v_list = zeros(N_tot, 0);
N = 0;
M = 0;
else
% Build matrix B incrementally
B = sparse(nxy, 0);
end
% Loop over different positions of the input source
for ii = 1:numel(B_struct)
data = B_struct(ii).data;
if isfield(B_struct, 'pos')
% B_struct(ii).pos specifies a rectangle inside [ny, nx]; (m1,n1) and (m2,n2) are its two diagonal corners
pos = B_struct(ii).pos;
m1 = pos(1); % first index in y
n1 = pos(2); % first index in x
m2 = m1 + pos(3) - 1; % last index in y
n2 = n1 + pos(4) - 1; % last index in x
nxy_data = pos(3)*pos(4); % number of elements in this rectangle
if m1 > ny
error('B(%d).pos(1) = %d exceeds ny = %d.', ii, m1, ny);
elseif n1 > nx
error('B(%d).pos(2) = %d exceeds nx = %d.', ii, n1, nx);
elseif m2 > ny
error('B(%d).pos(1) + B(%d).pos(3) - 1 = %d exceeds ny = %d.', ii, ii, m2, ny);
elseif n2 > nx
error('B(%d).pos(2) + B(%d).pos(4) - 1 = %d exceeds nx = %d.', ii, ii, n2, nx);
elseif ~(ndims(data) <= 3 && isnumeric(data))
error('B(%d).data must be a 2D or 3D numeric array when B.pos is given.', ii);
elseif isequal(size(data, [1,2]), [pos(3), pos(4)]) % this way to use size() is supported starting MATLAB R2019b
M_ii = size(data, 3); % number of inputs
elseif size(data, 1) == nxy_data
M_ii = size(data, 2); % number of inputs