Chebfun

Numerical computation with functions instead of numbers.

Description

Chebfun is a collection of algorithms and an open-source software system in object-oriented MATLAB which extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. It also implements continuous analogues of linear algebra notions like the QR decomposition and the SVD, and solves ordinary differential equations. The mathematical basis of the system combines tools of Chebyshev expansions, fast Fourier transform, barycentric interpolation, recursive zerofinding, and automatic differentiation.

Current release: V4.2.2194

Here are three quick examples:

What's the integral of sin(sin(x)) from 0 to 10?
>> x = chebfun('x',[0 10]); sum(sin(sin(x)))
ans = 1.629603118459496

What's the maximum of sin(x)+sin(x2) over the same interval?
>> max(sin(x)+sin(x.^2))
ans = 1.985446580874100

What's the solution to u"-xu=1 with zero boundary conditions on [-20,20]?
>> L = chebop(@(x,u)diff(u,2)-x.*u,[-20,20],'dirichlet'); plot(L\1)

For much more extensive list of more in-depth examples, see http://www.chebfun.org/examples/ and http://www.chebfun.org/examples/new/

The current release is compatible with MATLAB 7.6 (2008a) and more recent versions.
 
Installation instructions:

1. Unzip the contents of the zip file to a directory. We suggest the name "chebfun" for this directory. Make sure to maintain the existing subdirectory structure of the zip package. (Please note: If you install into the "toolbox" subdirectory of the MATLAB program hierarchy, you will need to click the button "Update toolbox path cache" from the File/Preferences... dialog in MATLAB.)

2. In MATLAB, add the chebfun directory to your path. This can be done by selecting "File/Set Path..." from the main or Command window menus. We recommend that you select the "Save" button on this dialog so that chebfuns are on the path automatically in future MATLAB sessions. (Alternatively, you can put an "addpath" command in your startup.m file, as described in the MATLAB documentation.)

3. A restart of MATLAB may be needed if you want to access the user guides via the Help browser.

4. To check whether the system appears to be working, type "chebtest" at the command window prompt. If a problem is reported, please contact [e-mail] with the details (including output of the "ver" command and the file chebtest_report.txt).

5. Point your web browser to the guide/html directory of the Chebfun installation in order to read the user guides.

Please see the file LICENSE.txt for licensing information. For full documentation, guides, and examples, visit http://www.chebfun.org

Acknowledgements

This file inspired Chebpack, Gauss Quadrature Nodes And Weights., and Dynamical System Viewer.

• Required Products: MATLAB
• MATLAB release: MATLAB 7.14 (R2012a)

[Nick Hale]

Hi Jeff

Sorry for the delay in replying. In your first problem, you haven't quite the boundary condition syntax correct. The following should work:

L = chebop(@(x,u) diff(u,4), [0,1]);
B = chebop(@(x,u) sin(x) + 0*u, [0,1]);
L.lbc = @(u) [diff(u,2), diff(u,3)];
L.rbc = @(u) [diff(u,2), diff(u,3)];

The y''''(x)=lambda*f(x)*y(x) problem should be possible also. Try something like the following, where sin is replaced by the m-file you wish to use to construct f:

f = chebfun(@sin, [0, 1]);
[V, D] = eigs(L, diag(f), 4);

Regards

Nick

P.S. Emailing [e-mail] or [e-mail] should result in speedier replies! :)

[Peter Green]

Another question: I want to solve the eigenvalues and eigenfunctions of the following equation

y''''(x)=lambda*f(x)*y(x)

Here f(x) is a function defined in a seperate .m file. Is it possible to use chebfun to solve the eigenvalues and eigenfunctions?

Thanks.
Jeff

[Peter Green]

An excellent software!!! Definitely 5 star.

A quick question. I am using the chebfun function to find the eigenvalues and eigenfunctions of the following boundary value problem

D^4 y=lambda*y, y''(0)=y'''(0)=y''(1)=y'''(1)=0,

where D is the differential operator, lambda is the eigenvalue and the solution y is the corresponding eigenfunction.

I used the following code in chebfun:

L = chebop(@(x,u) diff(u,4), [0,1]);
L.lbc = @(u) diff(u,2);
L.lbc =@(u) diff(u,3);
L.rbc = @(u) diff(u,2);
L.rbc =@(u) diff(u,3);
[V,D] = eigs(L,trunc);
disp(diag(D)), toc

HOWEVER, the below error was reported

Error using eig
Matrix must be square.

Error in linop/eigs>bc_eig (line 288)
[V,D] = eig(full(Amat),full(Pmat));

Error in linop/eigs (line 94)
[V1,D1] = bc_eig(A,B,33,33,0,map,breaks);

Error in chebop/eigs (line 57)
[varargout{1:nargout}] = eigs(L,varargin{:});

Looks like I am on the right track but some nonsquare matrix caused that error. Sincerely appreciate your comments and suggestions.

Thanks.
Jeff

[Raghuram]

 

[Shashank]

Perfect toolbox for somebody doing numerical analysis for the first time!

[Mark Shore]

There is just so much to learn from this topnotch collection of powerful routines.

[Mark Shore]

There is just so much to learn from this topnotch collection of powerful routines.

[Mark Shore]

 

[Chebfun Team]

Hi Jiro,

Thanks! We're currently in the process of moving to the new OOP system in our development release, which you can find on our website: http://www.maths.ox.ac.uk/chebfun/

[Jiro Doke]

I really like the comprehensive User's Guide that you include with this package.

Do you have a plan to implement this using the new MATLAB Class system?

[Erdal Bizkevelci]

 

[Siyi Deng]

this toolbox is awesome.

[Erdal Bizkevelci]

 

[John D'Errico]

Thanks Tobin - that was what i was looking for. Those files are very readable, clearly explaining what a chebfun is and how to use them. I'll now call this a complete package.

I've had the time to do a little more perusal of these tools now too. They work on my Mac, which is sometimes an issue.

[Toby Driscoll]

The guide directory included in the package has 8 published scripts that contain dozens of examples in tutorial style. The resulting HTML files are also installed in the Help Browser under the "Chebfun Toolbox" heading.

[John D'Errico]

I've not tried this out, nor tested it yet. That will take some time. But what I've seen suggests this will be interesting and potentially useful. A lot of effort has been invested here in these tools.

The only thing I'd have liked to see is a published set of demos, with more explanation around how to use it and what the demos are showing. There is a directory (chebtests) of some tests of the tools, but they have virtually no comments in them. Derek has a good suggestion, to read the paper by Trefethen. Here are a few links of interest:

http://www.comlab.ox.ac.uk/nick.trefethen/trefethen_functions.pdf

http://www.comlab.ox.ac.uk/nick.trefethen/ppt_revised.pdf

http://www.comlab.ox.ac.uk/chebfun/publications/driscoll_born_tref.pdf

The papers shown are quite readable. I'd highly recommend them.

My only source of dissatisfaction here is with a lack of examples. These files need extensive demos and DOCUMENTED examples that explain how to use them for the person who might be interested in using them. The fact is, it does not look like that would be difficult to do, and I'd welcome that addition.

My gut says these tools are worth a 5 star rating, IF you know how to use them and what they do. Without a good set of clearly documented examples, my gut wants to rate them as only 4 stars.

[Derek O'Connor]

An excellent system. Passed all tests with Matlab R2008a 64 bit with Windows Vista 64.

Read Trefethen's paper "Computing Numerically with Functions Instead of Numbers" which is a nice short introduction to Chebfun.

http://www.comlab.ox.ac.uk/nick.trefethen/home.html

Derek O'Connor