Numerical computing

This project begins with the idea to implement my own C++ numerical computing platform as an analogy to MATLAB.

This repository contains necessary packages for symbolic computation, matrix algebra, etc., and also offers data visualization – graphs.

Symbolic computation is used SymbolicC++ library, for data visualization and plotting, is used Gnuplot-iostream

Optimization application

    TwoDimensional::Function myFunction((x - 2)*(x + 5) + y*y);
    TwoDimensional::Point startPoint(1.5, 5);

    Optimization::NelderMead myMethod(myFunction, startPoint);
    myMethod.findMinimum();

Numerical computing

Open-source Interactive numerical computing environment

Toolbox specification

This repository contains the necessary packages for symbolic computation,
matrix algebra etc., also offer data visualization – graphs.

Symbolic computation is used SymbolicC++ library, for data visualization and
plotting is used Gnuplot-iostream.

Area of use

Matrices algebra

Symbolic differential computation

Graphical solutions and visualizations

Mathematical function properties analysis

Numerical computing library

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

Point

Object Points represent points 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 represents mathematical function by symbolic variables.

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

  1. Function value
    myFunction.value(Point);
    return function value of the 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 second’s derivatives – hessian.

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

Plot

Object Plot offers graphical visualization and data plotting.

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


My application

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

Numerical optimization

One-dimensional optimization problem

  • Davidon’s search
  • Golden selection search

Two-dimensional optimization problem

  • Nelder & Mead’s method
  • Newton & Raphson’s 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();