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# Table of Contents

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)
1. [Root Finding Methods](#orgefe5c09)
1. [Newton’s method](#org7d679d2)
2. [Fixed point method](#orgeb54040)
3. [Secant method](#org82e886b)
2. [Interpolation techniques](#orgcd5aecd)
1. [Hermite Interpolation](#org564bc6d)
2. [Lagrange Interpolation](#org2720a8a)
3. [Newton’s Interpolation](#orgc7ba82a)
3. [Integration methods](#org09c0dc7)
1. [Euler Method](#orgb8d1fca)
2. [Newton–Cotes Method](#org10d601a)
3. [Predictor–Corrector Method](#org555de04)
4. [Trapizoidal method](#orgcf2b400)

![img](c:/Users/rjish/notes-orgmode/.attach/45/5c46bb-952b-4978-b48e-554565046442/_20240120_041518num-ana.png)

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: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 $$ x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}} $$
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

Expand All @@ -104,7 +105,7 @@ Predictor–Corrector methods belong to a class of algorithms designed to integr
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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:PROPERTIES:
:ID: 455c46bb-952b-4978-b48e-554565046442
:END:
#+TITLE: Numerical-analysis
#+AUTHOR: Jishnu Rajendran

[[attachment:_20240120_041518num-ana.png]]


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: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 $$ x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}} $$
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
$$ x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}} $$

** [[https://en.wikipedia.org/wiki/Fixed-point_iteration][Fixed point method]]
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
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