dhruvj22 / numerical-methods

Numerical Methods

Numerical Methods

This repository consists of various Numerical Methods algorithms that I wrote as part of a graduate level numerical methods course that I took at Purdue Unviersity, through my Astrodynamics research and out of my curiosity to investigate algorithms that can solve non-trivial mathematical problems.

Setup

The algorithms are implemented in Jupyter lab by leveraging numpy array. There are 6 files focused on each problem and various method for each problem are implemented. A discussion of the algorithms through examples is included to compare performance and assumptions.

List of methods implemented

1. Root-finding algorithms

   • Bisection method
   • Newton's method (Newton-Raphson method)
   • Secant method
   • Fixed-point iteration method

2. Linear Equation solvers

   • Gaussian Elimination with No Pivoting(opt = 1) and Scaled Partial Pivoting (opt = 2)
   • Conjugate Gradient Method (Minimization Method)
   • Gauss-Sidel Method (Splitting Method)
   • Jacobi Method (Splitting Method)
   • SOR Method (Splitting Method)

3. Numerical interpolation and Curve Fitting

   • Lagrange's form of the interpolation polynomial
   • Newton's form of the interpolation polynomial

4. Numerical Integration

   • Midpoint Rule
   • Trapezoidal Rule
   • Simpson's Rule
   • Forward Euler's Method
   • Backward Euler's Method
   • Runge-Kutta 4^th order
   • Composite Trapezoidal Rule
   • 2-Point Gaussian Quadrature Rule
   • 3-Point Gaussian Quadrature Rule

5. Numerical Differentiation

   • Forward Differentiation
   • Central Differentiation
   • Complex Step Differentiation

6. Differential Algebraic Equation / Partial Differenital Equation solvers

   • 1D Heat Equation
   • Advect Eqation
   • Wave Equation
   • Laplace Equation (Contour Plot!)

7. Numerical Continuation

   • Natural Paramter Continuation
   • Pseudo-arc length continuation

Checkout my contribution to Poliastro or part of my research code to see the implementation of Numerical Continuation schemes

Reference

• "A Friendly Introduction to Numerical Analysis" by Brian Bradle, 2006.
• "The Complex-Step Derivative Approximation" by Martins, Strudza and Alonso