NumLib

Numerical Library: Demonstration of various numerical methods/recipes.

Notes

Currently, this package is being written in modern Fortran (90 and newer). There may be extensions to Python, C/C++, and/or MATLAB as time permits.

Additionally, "package" is a bit of a misnomer right now, considering there is no compilation into a library at the moment.

Install

The package can be compiled by navigating to the NumLib/bin/ directory and either using the CMake file or the compile.sh file (which also executes a test program).

It is recommended having a relatively newer compiler to capture more modern Fortran language elements (such as F2003 and F2008).

• Compilation has been tested on GNU 9.1.0 and 9.2.0 version gfortran compilers.

Methods

Numerical methods supported:

• Linear Algebra

  • Inner and Outer Products (workarounds for intrinsics issues)
  • Gaussian Elimination (Basic, Row Echelon From, and Reduced REF)
  • Gauss-Jordan Inverse using RREF([A|I]) = [I|B] --> B = inv(A)
  • Thomas solver for tridiagonal matrix systems
  • Back substitution for solving Ax=b with Gaussia Elimination

• Numerical differentiation

  • Forward differencing (1st/2nd order stencils up to 4th derivatives)
  • Backward differencing (1st/2nd order stencils up to 4th derivatives)
  • Central differencing (2nd/4th order stencils up to 4th derivatives)
  • Richardson Extrapolation

• Numerical Integration

  • Ordinary Differential Equations
    • Explicit Runge Kutta Methods
      • Forward Euler, Midpoint, Heun's
      • Ralston's, SSPRK3, Generic RK4

• Unconstrained Optimization

  • Line Search w/ Backtracking for minimization
    • Steepest Descent, Inexact Newton, Broyden-Fletcher-Goldfarb-Shanno
  • Univariate Rootfinding
    • Newton's, Halley's, Steffenson's
    • Muller's, Ridder's, Secant, Bisection
    • False Position, ITP, Brent-Dekker
    • Inverse Quadratic Interpolation

Numerical methods in development:

• Unconstrained Optimization
  • Controlled Random Search for minimization (stochastic method)
• Linear Algebra
  • Forward Substitution
  • LU, LUP, LUPQ, LDU, and Cholesky-variant decompositions