# Miscellaneous Mathematical Utilities

This page contains links to several mathematical utilities. More will be added as I write them. The algorithms underlying these utilities come from the BLAS, EISPACK, and LINPACK collection of subprograms, written by some of the brightest mathematicians and computer scientists (I have cited sources when I found them). Those subprograms incorporate excellent basic algorithms and programming techniques to optimize the routines for speed and accuracy.

A note about machine precision.

**Because these utilities are written in Javascript, make sure Javascript is enabled in your Internet browser.**

## Root-finding Routines

Numerical Solver for Kepler's Equation for Elliptical Motion

Quadratic Equation

Cubic Equation

Quartic Equation

Polynomial Root-finder (Real Coefficients)

Root-Finder Utility #1: f(M) = **a**_{1}cos(M)sin^{2}(M) + **a**_{2}cos^{2}(M)sin^{2}(M) + **a**_{3}

Root-Finder Utility #2: f(x) = **a**_{1}/ √ (1 - x ^{2}) - (1/x^{2}) [( (1 + **a**_{2} x^{2})/(1 + **a**_{2}) )^{a3} - 1]

Multi-dimensional Root-Finder #1: Two Non-linear Equations in Two Unknowns, Mixed Polynomials

## Utilities for Finding Maxima/Minima of Functions

Maximum-Finder Utility #1: f(M) = **a**_{1}cos(M)sin^{2}(M) + **a**_{2}cos^{2}(M)sin^{2}(M) + **a**_{3}

Multi-dimensional Optimization Routine #1 (Four Dimensions): **F(w, x, y, x)**

Quadratic Programming Utility, Three Variables

## N Equations in N Unknowns

Three Equations in Three Unknowns

Four Equations in Four Unknowns

Five Equations in Five Unknowns

Simultaneous Equation Solver (up to 12 equations in 12 real unknowns)

A Downloadable Console Utility, N Equations in N Unknowns

## Eigenvalues and Eigenvectors

Eigenvalues of a 2 by 2 Matrix

Eigenvalues of a 3 by 3 Matrix

Eigenvalues of a 4 by 4 Matrix

Eigenvalues of a 5 by 5 Matrix

Eigenvalues of a 6 by 6 Matrix

Eigenvalues and Eigenvectors Calculator (matrices up to size 12 X 12, real coefficients)

A Downloadable Console Utility, N X N, Real, General Matrix

Eigenvalues Only of a 3 by 3 Real, Symmetric Matrix

## Numerical Integration

Numerical Integration Utility #1: f(x) = sin(x)

Numerical Integration Utility #2: f(x) = √ x sin(x)

## Data Fitting

Linear Least-Squares Data-Fitting Utility, Polynomial Fit, up to Fourth Degree

## Interpolation

Cubic Spline Interpolation Utility

## Examples of Applications of Some of These Utilities

Aerodynamics Example #1: Finding the Critical Mach Number of an Airfoil

Quadratic Programming Example #1: No Constraints

Two Papers on Optimization/Quadratic Programming

## Other Utilities

Matrix Multiplication Utility

Cholesky Decomposition

Discrete Fourier Transform (forward)

A Transformation Utility for Successive Rotations Around Axes

A 2D Positioning Utility (by angulation)

A 3D Positioning Utility (by angulation)

Dynamic Arrays in C++ (source code)

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