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, implementation ...

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 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: The Chebyshe ...

, 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 family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...

and

Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...

with the speed of floating-point numerics. The Chebfun project is based in the Mathematical Institute at the

University of Oxford , mottoeng = The Lord is my light , established = , endowment = £6.1 billion (including colleges) (2019) , budget = £2.145 billion (2019–20) , chancellor ...

and was initiated in 2002 by

Lloyd N. Trefethen Lloyd Nicholas Trefethen (born 30 August 1955) is an American mathematician, professor of numerical analysis and head of the Numerical Analysis Group at the Mathematical Institute, University of Oxford. Education Trefethen was born 30 August 19 ...

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 the 4th of 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 This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity, dimensionless quantities and probability, probabilities. Each number is given a name in the Long and short scales, short scale ...

say. >> x = chebfun('x',

; 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, http://www.chebfun.org/ * Related projects and partial replacements in other languages

Mathematical software