Chebfun is a
free/open-source software system written in
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
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
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
...
, and this is where the name "Cheb" comes from. The package aims to combine the feel of symbolic computing systems like
Maple
''Acer'' is a genus of trees and shrubs commonly known as maples. The genus is placed in the soapberry family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated si ...
and
Mathematica
Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
with the speed of floating-point numerics.
The Chebfun project is based in the Mathematical Institute at the
University of Oxford
The University of Oxford is a collegiate university, collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the List of oldest un ...
and was initiated in 2002 by
Lloyd N. Trefethen and his student Zachary Battles.
The most recent version, Version 5.7.0, was released on June 2, 2017.
Chebfun2, a software system that extends Chebfun to two dimensions, was made publicly available on 4 March 2013. Following Chebfun2, Spherefun (extension to the unit sphere) and Chebfun3 (extension to three dimensions) were made publicly available in May and July 2016.
Features
* Approximation of functions in 1D, including functions with jumps
* Approximation of smooth bivariate functions (Chebfun2)
* Approximation of smooth trivariate functions (Chebfun3)
* Approximation of smooth functions on the unit sphere (Spherefun)
* Quadrature
* Rootfinding
* 1D global optimisation
* Bivariate and trivariate rootfinding
* Ordinary differential equations
* Partial differential equations
* Vector calculus
Example usage
A user may begin by initialising the variable x, on the interval
,10 say.
>> x = chebfun('x', ,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
References
External links
* {{Official website, http://www.chebfun.org/
* Related projects and partial replacements in other languages
Mathematical software