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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 4th 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