Numerical Methods
I plan to put together short explanations and animations on various numerical methods and math-related topics here. It will take me a long time to put this together totally. So, please be patient.
This page was inspired by John H Mathew’s website (Mathew 2000-2019). In 2019, the page was taken down. I tried contacting Cal State Fullerton in 2019 but they did not respond. Fastforward to April-2026, I realized I could pull it back from the Internet Archive (The Wayback Machine) at least partially. Therefore, you will notice that this is inspired from his page.
Note that as of 2026-04-27, I have used AI to generate the new page from Mathew’s old page. I expect there to be a large amount of errors even if the animations look correct. I will be going through the pages and fixing errors as I find them and have time to do so. If you find any errors, please let me know.
The Solution of Nonlinear Equations f(x) = 0
- Fixed Point Iteration
- Bisection Method
- False Position or Regula Falsi Method
- Newton-Raphson Method
- Secant Method
- Muller’s Method
- Aitken’s Method & Steffensen’s Acceleration
- Accelerated & Modified Newton-Raphson
- Improved Newton Method
- Halley’s Method
- Horner’s Method
- Brent’s Method
- Graeffe’s Method
- Nonlinear Systems
- Broyden’s Method
- Damped Newton with Acceleration (Madsen-Reid Method)
The Solution of Linear Systems AX = B
- Triangular Systems and Back Substitution
- Gauss-Jordan Elimination and Pivoting
- Tri-Diagonal Matrices
- Inverse Matrix
- LU Factorization
- Cholesky, Doolittle and Crout Factorizations
- Jacobi and Gauss-Seidel Iteration
- Successive Over Relaxation - SOR
- Pivoting Methods
- Iterative Refinement
- Row Reduced Echelon Form
- Homogeneous Linear Systems
- Kirchoff’s Law
- Leontief Model
- Linear Programming-Simplex Method
Function Approximation
Information from a point expanded in a neighborhood
Information at several points. Interpolation and Polynomial Approximation
- Lagrange Polynomial Interpolation and Approximation
- Newton Interpolation Polynomial
- Hermite Polynomial Interpolation
- Cubic Splines
- B-Splines
- Bézier Curves
- Chebyshev Approximation Polynomial
- Rational Approximation
- Aitken’s and Neville’s Interpolation
- Legendre Polynomials
- The Tangent Parabola
- Catenary
Other Series Approximations
Curve Fitting
Numerical Differentiation
Numerical Integration
- Riemann Sums
- Midpoint Rule
- Newton-Cotes Integration
- Trapezoidal Rule for Numerical Integration
- Simpson’s Rule for Numerical Integration
- Simpson’s 3/8 Rule for Numerical Integration
- Boole’s Rule
- Romberg Integration
- Adaptive Simpson’s Rule
- Gauss-Legendre Quadrature
- Cubic Spline Quadrature
- Monte Carlo Pi
- Monte Carlo Integration
- 2D Trapezoidal and Simpson Rules
Solution of Differential Equations
- Euler’s Method for ODE’s
- Taylor Series Method for ODE’s
- Runge-Kutta Method
- Runge-Kutta-Fehlberg Method
- Adams-Bashforth-Moulton Method
- Milne-Simpson’s Method
- Predictor-Corrector Methods
- Shooting Methods for ODE’s
- Finite Difference Method for ODE’s
- Galerkin’s Method
- Painleve Property
- Lotka-Volterra Model
- Pendulum
- Projectile Motion
- Lorenz Attractor
- van der Pol System
- Harvesting Model
- Frobenius Series Solution
- Picard Iteration
- Spring-Mass Systems