From e2608a91292e573fe5ebbd46d2fe63051d919d00 Mon Sep 17 00:00:00 2001 From: Jishnu Date: Sat, 20 Jan 2024 05:26:24 +0530 Subject: [PATCH] 12 --- README.org | 15 +++-- index.html | 159 +++++++++++++++++++++++++++++++++++++++++++ notebook.css | 187 +++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 356 insertions(+), 5 deletions(-) create mode 100644 index.html create mode 100644 notebook.css diff --git a/README.org b/README.org index a4165f6..b07d46f 100644 --- a/README.org +++ b/README.org @@ -1,12 +1,17 @@ :PROPERTIES: :ID: 455c46bb-952b-4978-b48e-554565046442 :END: -#+TITLE: Numerical-analysis +#+TITLE: Numerical Analysis #+AUTHOR: Jishnu Rajendran - -[[attachment:num-ana.png]] - -[[attachment:num-ana.png]] +#+HTML_HEAD: +#+OPTIONS: toc:nil +#+OPTIONS: title:nil +#+OPTIONS: html-style:nil +#+OPTIONS: html-scripts:nil +#+OPTIONS: html-postamble:nil +#+OPTIONS: broken-links:mark + +[[file: num-ana.png]] * Root Finding Methods ** [[https://en.wikipedia.org/wiki/Newton%27s_method][Newton's method]] diff --git a/index.html b/index.html new file mode 100644 index 0000000..6684d53 --- /dev/null +++ b/index.html @@ -0,0 +1,159 @@ + + + + + + + +Numerical Analysis + + + + + + + + +
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1. Root Finding Methods

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1.1. Newton’s method

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+Newton’s method (also known as the Newton–Raphson method) is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. The process is repeated as +\[ x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}} \] +

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1.2. Fixed point method

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+Fixed-point iteration is a method of computing fixed points of iterated functions. More specifically, given a function f defined on the real numbers with real values and given a point x0 in the domain of f, the fixed point iteration is +\[ x_{n+1}=f(x_{n}),\,n=0,1,2,\dots\] +

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1.3. Secant method

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+Secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton’s method. +\[ x_{n}=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}. \] +

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2. Interpolation techniques

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2.1. Hermite Interpolation

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+Hermite Interpolation is a method of interpolating data points as a polynomial function. The generated Hermite interpolating polynomial is closely related to the Newton polynomial, in that both are derived from the calculation of divided differences. +

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2.2. Lagrange Interpolation

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+Lagrange polynomials are used for polynomial interpolation. See Wikipedia +

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2.3. Newton’s Interpolation

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+Newton’s divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Divided differences is a recursive division process. The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form. +

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3. Integration methods

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3.1. Euler Method

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+Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. +\[ y_{n+1} = y_{n} + h f(t_{n} , y_{n}) \] +

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3.2. Newton–Cotes Method

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+Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulae for numerical integration (also called quadrature) based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes. +

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3.3. Predictor–Corrector Method

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+Predictor–Corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: +

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  1. The initial, “prediction” step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate (“anticipate”) this function’s value at a subsequent, new point.
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  2. The next, “corrector” step refines the initial approximation by using the predicted value of the function and another method to interpolate that unknown function’s value at the same subsequent point.
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3.4. Trapizoidal method

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+Trapezoidal rule is a technique for approximating the definite integral. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. +\[ \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}\] +

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