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# Table of Contents
![GitHub](https://img.shields.io/github/license/jishnurajendran/Numerical-analysis?style=for-the-badge) ![GitHub forks](https://img.shields.io/github/forks/jishnurajendran/Numerical-analysis?style=for-the-badge) ![GitHub Repo stars](https://img.shields.io/github/stars/jishnurajendran/Numerical-analysis?style=for-the-badge) ![GitHub watchers](https://img.shields.io/github/watchers/jishnurajendran/Numerical-analysis?style=for-the-badge)

1. [Root Finding Methods](#org770afdb)
1. [Newton’s method](#org013f1b7)
2. [Fixed point method](#orgde6567b)
3. [Secant method](#org4ebbe87)
2. [Interpolation techniques](#org9e2a72e)
1. [Hermite Interpolation](#orgd63ca7f)
2. [Lagrange Interpolation](#org1e8da43)
3. [Newton’s Interpolation](#orgd4f58aa)
3. [Integration methods](#org8e7e5c8)
1. [Euler Method](#org351da1a)
2. [Newton–Cotes Method](#org75020aa)
3. [Predictor–Corrector Method](#orgcf8f14e)
4. [Trapizoidal method](#orgf561a2c)
1. [Root Finding Methods](#org97f8dc1)
1. [Newton’s method](#org4ec5a5a)
2. [Fixed point method](#orgd92eb51)
3. [Secant method](#org5e86b54)
2. [Interpolation techniques](#org7879a30)
1. [Hermite Interpolation](#org01982a3)
2. [Lagrange Interpolation](#org1020c9c)
3. [Newton’s Interpolation](#orgd08b2ee)
3. [Integration methods](#orgf7b000b)
1. [Euler Method](#orge64619c)
2. [Newton–Cotes Method](#orgb51f88e)
3. [Predictor–Corrector Method](#org2f8adfb)
4. [Trapizoidal method](#org4dbe660)

\![GitHub](<https://img.shields.io/github/license/jishnurajendran/Numerical-analysis?style=for-the-badge>) \![GitHub forks](<https://img.shields.io/github/forks/jishnurajendran/Numerical-analysis?style=for-the-badge>) \![GitHub Repo stars](<https://img.shields.io/github/stars/jishnurajendran/Numerical-analysis?style=for-the-badge>) \![GitHub watchers](<https://img.shields.io/github/watchers/jishnurajendran/Numerical-analysis?style=for-the-badge>)

:TOC: :include all


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# Root Finding Methods


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## [Newton&rsquo;s method](https://en.wikipedia.org/wiki/Newton%27s_method)

Newton&rsquo;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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## [Fixed point method](https://en.wikipedia.org/wiki/Fixed-point_iteration)

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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## [Secant method](https://en.wikipedia.org/wiki/Secant_method)

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&rsquo;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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# Interpolation techniques


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## Hermite Interpolation

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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## Lagrange Interpolation

Lagrange polynomials are used for polynomial interpolation. See [Wikipedia](https://en.wikipedia.org/wiki/Lagrange_polynomial)


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## Newton&rsquo;s Interpolation

Newton&rsquo;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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# Integration methods


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## Euler Method

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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## Newton–Cotes Method

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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## Predictor–Corrector Method

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:

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


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## Trapizoidal method

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#+TITLE: Numerical-analysis
#+AUTHOR: Jishnu Rajendran

![GitHub](https://img.shields.io/github/license/jishnurajendran/Numerical-analysis?style=for-the-badge) ![GitHub forks](https://img.shields.io/github/forks/jishnurajendran/Numerical-analysis?style=for-the-badge) ![GitHub Repo stars](https://img.shields.io/github/stars/jishnurajendran/Numerical-analysis?style=for-the-badge) ![GitHub watchers](https://img.shields.io/github/watchers/jishnurajendran/Numerical-analysis?style=for-the-badge)

:PROPERTIES:
:TOC: :include all
:END:

* Root Finding Methods
** [[https://en.wikipedia.org/wiki/Newton%27s_method][Newton's method]]
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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** Predictor–Corrector Method
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:
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.
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.
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.
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.
** Trapizoidal method
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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