Numerical computing

Open source Interactive numerical computing environment

Toolbox specification

This repository contains necessary packages to symbolic computation, matrix algebra etc., also offer data visualisation - graphs.

For symbolic computation is used SymbolicC++ library, for data visualisation and plotting is used Gnuplot-iostream.

Area of use

Matrices algebra

Symbolic differential computation

Graphical solutions and visualisations

Mathematical function properties analysis

Numerical computing library

My C++ implementation of utilities based on symbolic computation.

Point

Object Point represent point in n-dimensional space.

    OneDimensional::Point myPoint(2);
    TwoDimensional::Point myPoint(1.5, 5);
    ThreeDimensional::Point myPoint(1.5, 2, 5);

1. Point value
   myPoint[i]; return point value in i dimension. By call myPoint.matrix; are possible matrices operations, return 1 x n matrix.

Function

Object Function represent mathematical function by symbolic variables.

    TwoDimensional::Function myFunction((x - 2)*(x + 5) + 3*y);

1. Function value
   myFunction.value(Point); return function value of function at given Point.

2. Function gradient myFunction.gradient(); myFunction.gradient(Point); return matrix n x 1 of function's first derivatives - gradient.

3. Function hessian myFunction.hessian(); myFunction.hessian(Point); return matrix n x n of function's seconds derivatives - hessian.

4. Function character of stationary/inflection point myFunction.character(Point); return StationaryPoin::Minimum = 0 if Point is a locally minimum of function, return StationaryPoin::Maximum = 1 if Point is a locally maximum, or StationaryPoin::Saddle = 2 if Point is saddle point of function.

Plot

Object Plot offer graphical visualisation and data plotting.

Plot::function(myFunction.getSymbolic());

My application

I have develop this platform to implement Numerical Optimization comparative and gradient methods used for finding locally extremes of functions.

Numerical optimization

One dimensional optimization problem

• Davidon's search
• Golden selection search

Two dimensional optimization problem

• Nelder & Mead method
• Newton & Raphson method
• Levenberg & Marquardt method

Usage example

    TwoDimensional::Function myFunction((x - 2)*(x + 5) + y*y);
    TwoDimensional::Point startPoint(1.5, 5);
    
    Optimization::NelderMead myMethod(myFunction, startPoint);
    myMethod.findMinimum();