no message

Author: jishnurajendran

README.md

@@ -0,0 +1,114 @@
+```
+title: Numerical Analysis
+author: Jishnu
+```
+
+# Table of Contents
+
+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)
+
+
+
+<a id="org770afdb"></a>
+
+# Root Finding Methods
+
+
+<a id="org013f1b7"></a>
+
+## [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 $$ x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}} $$
+
+
+<a id="orgde6567b"></a>
+
+## [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
+$$ x_{n+1}=f(x_{n}),\,n=0,1,2,\dots$$
+
+
+<a id="org4ebbe87"></a>
+
+## [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.
+$$ x_{n}=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}. $$
+
+
+<a id="org9e2a72e"></a>
+
+# Interpolation techniques
+
+
+<a id="orgd63ca7f"></a>
+
+## 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.
+
+
+<a id="org1e8da43"></a>
+
+## Lagrange Interpolation
+
+Lagrange polynomials are used for polynomial interpolation. See [Wikipedia](https://en.wikipedia.org/wiki/Lagrange_polynomial)
+
+
+<a id="orgd4f58aa"></a>
+
+## 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.
+
+
+<a id="org8e7e5c8"></a>
+
+# Integration methods
+
+
+<a id="org351da1a"></a>
+
+## 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.
+$$ y_{n+1} = y_{n} + h f(t_{n} , y_{n}) $$
+
+
+<a id="org75020aa"></a>
+
+## 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.
+
+
+<a id="orgcf8f14e"></a>
+
+## 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.
+
+
+<a id="orgf561a2c"></a>
+
+## 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.
+$$ \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}$$
+

README.org

@@ -25,7 +25,7 @@ Newton's divided differences is an algorithm, historically used for computing ta
 * Integration methods
 ** 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.
-$$ y_{n+1} = y_{n} + h f(t_n , y_n) $$
+$$ y_{n+1} = y_{n} + h f(t_{n} , y_{n}) $$
 
 ** 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.