Chebfun is a freely available software system written in MATLAB

MATLAB

MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...

 for numerical computation with functions of a real variable. It is based on the idea of overloading MATLAB's commands for vectors and matrices to analogous commands for functions and operators. Thus, for example, whereas the SUM command in MATLAB adds up the elements of a vector, the SUM command in Chebfun evaluates a definite integral. Similarly the backslash command in MATLAB becomes a Chebfun command for solving differential equations.

The mathematical basis of Chebfun is numerical algorithms involving piecewise polynomial interpolants and Chebyshev polynomials

Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...

, and this is where the name "Cheb" comes from. The package aims to combine the feel of symbolic computing systems like Maple

Maple (software)

Maple is a general-purpose commercial computer algebra system. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada....

 and Mathematica

Mathematica

Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

 with the speed of floating-point numerics.

The Chebfun project is based in the Mathematical Institute at the University of Oxford

University of Oxford

The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...

 and was initiated in 2002 by Lloyd N. Trefethen and his student Zachary Battles. The most recent version, Version 4.0, was released in February 2011.

Features

• Approximation of functions
• Quadrature
• Rootfinding
• 1D global optimisation
• Ordinary differential equations
• Partial differential equations

Example usage

A user may begin by initialising the variable x, on the interval [0,10], say.


>> x = chebfun('x',[0,10]);


This variable can now be used to perform further computations, for example, computing and plotting roots of a function:


>> f = sin(x) + sin(x.^2); plot(f)
>> r = roots(f); hold on, plot(r,f(r),'.r'), hold off

The definite integral can be computed with:


>> sum(f)
ans
= 2.422742429006079