Chebfun Version 2

Numerical computation with functions instead of numbers.

Description

The chebfun project is a collection of algorithms, and a 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. The mathematical basis of the system combines tools of Chebyshev expansions, fast Fourier transform, barycentric interpolation, Clenshaw-Curtis quadrature, and recursive zerofinding.

CHEBFUN VERSION 3 IS NOW AVAILABLE FROM http://www.maths.ox.ac.uk/chebfun/

Here are three quick examples:

What's the integral of exp(-sqrt(x)) from 0 to 10?
>> x = chebfun('x',[0 10]); sum(exp(-sqrt(x)))
ans = 1.647628069579947

What's the maximum of sin(x)+sin(x2) over the same interval?
>> max(sin(x)+sin(x.^2))
ans = 1.985446580874099

How many roots does the Bessel function J0(x) have between 0 and 1000?
>> length(roots(chebfun(@(x) besselj(0,x),[0 1000])))
ans = 318

Chebfun Version 2 is compatible with MATLAB 7.4 (2007a) 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).

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.maths.ox.ac.uk/chebfun

• Acknowledgements: This submission has inspired the following:; Gauss quadrature nodes and weights.
• MATLAB release: MATLAB 7.7 (R2008b)
• Published M Files: CHEBFUN GUIDE 1: INTRODUCTION, CHEBFUN GUIDE 2: INTEGRATION AND DIFFERENTIATION, CHEBFUN GUIDE 3: ROOTFINDING AND MINIMA AND MAXIMA, CHEBFUN GUIDE 4: CHEBFUNS AND APPROXIMATION THEORY, CHEBFUN GUIDE 5: COMPLEX CHEBFUNS, CHEBFUN GUIDE 6: QUASIMATRICES AND LEAST-SQUARES, CHEBFUN GUIDE 7: LINEAR DIFFERENTIAL EQUATIONS WITH CHEBOPS, CHEBFUN GUIDE 8: CHEBFUN PREFERENCES
• HTML Files: Chebfun Toolbox,; Chebfun Toolbox Software License

Other Files

chebfun/@chebfun/abs.m,
chebfun/@chebfun/acos.m,
chebfun/@chebfun/acosh.m,
chebfun/@chebfun/acot.m,
chebfun/@chebfun/acoth.m,
chebfun/@chebfun/acsc.m,
chebfun/@chebfun/acsch.m,
chebfun/@chebfun/asec.m,
chebfun/@chebfun/asech.m,
chebfun/@chebfun/asin.m,
chebfun/@chebfun/asinh.m,
chebfun/@chebfun/atan.m,
chebfun/@chebfun/atanh.m,
chebfun/@chebfun/bvp4c.m,
chebfun/@chebfun/bvp5c.m,
chebfun/@chebfun/ceil.m,
chebfun/@chebfun/cf.m,
chebfun/@chebfun/chebfun.m,
chebfun/@chebfun/chebpade.m,
chebfun/@chebfun/chebpoly.m,
chebfun/@chebfun/chebpolyplot.m,
chebfun/@chebfun/comet.m,
chebfun/@chebfun/complex.m,
chebfun/@chebfun/cond.m,
chebfun/@chebfun/conj.m,
chebfun/@chebfun/conv.m,
chebfun/@chebfun/cos.m,
chebfun/@chebfun/cosh.m,
chebfun/@chebfun/cot.m,
chebfun/@chebfun/coth.m,
chebfun/@chebfun/csc.m,
chebfun/@chebfun/csch.m,
chebfun/@chebfun/ctranspose.m,
chebfun/@chebfun/cumprod.m,
chebfun/@chebfun/cumsum.m,
chebfun/@chebfun/define.m,
chebfun/@chebfun/diag.m,
chebfun/@chebfun/diff.m,
chebfun/@chebfun/display.m,
chebfun/@chebfun/domain.m,
chebfun/@chebfun/eq.m,
chebfun/@chebfun/erf.m,
chebfun/@chebfun/erfc.m,
chebfun/@chebfun/erfcx.m,
chebfun/@chebfun/erfinv.m,
chebfun/@chebfun/exp.m,
chebfun/@chebfun/feval.m,
chebfun/@chebfun/find.m,
chebfun/@chebfun/fix.m,
chebfun/@chebfun/fliplr.m,
chebfun/@chebfun/flipud.m,
chebfun/@chebfun/floor.m,
chebfun/@chebfun/get.m,
chebfun/@chebfun/gmres.m,
chebfun/@chebfun/horzcat.m,
chebfun/@chebfun/imag.m,
chebfun/@chebfun/interp1.m,
chebfun/@chebfun/inv.m,
chebfun/@chebfun/isempty.m,
chebfun/@chebfun/isequal.m,
chebfun/@chebfun/isreal.m,
chebfun/@chebfun/ldivide.m,
chebfun/@chebfun/legpoly.m,
chebfun/@chebfun/length.m,
chebfun/@chebfun/log.m,
chebfun/@chebfun/log10.m,
chebfun/@chebfun/log2.m,
chebfun/@chebfun/max.m,
chebfun/@chebfun/mean.m,
chebfun/@chebfun/min.m,
chebfun/@chebfun/minandmax.m,
chebfun/@chebfun/minus.m,
chebfun/@chebfun/mldivide.m,
chebfun/@chebfun/mrdivide.m,
chebfun/@chebfun/mtimes.m,
chebfun/@chebfun/ne.m,
chebfun/@chebfun/norm.m,
chebfun/@chebfun/not.m,
chebfun/@chebfun/null.m,
chebfun/@chebfun/ode113.m,
chebfun/@chebfun/ode15s.m,
chebfun/@chebfun/ode45.m,
chebfun/@chebfun/orth.m,
chebfun/@chebfun/pinv.m,
chebfun/@chebfun/plot.m,
chebfun/@chebfun/plot3.m,
chebfun/@chebfun/plus.m,
chebfun/@chebfun/poly.m,
chebfun/@chebfun/polyfit.m,
chebfun/@chebfun/power.m,
chebfun/@chebfun/private/auto.m,
chebfun/@chebfun/private/comp.m,
chebfun/@chebfun/private/compose.m,
chebfun/@chebfun/private/ctor_2.m,
chebfun/@chebfun/private/ctor_3.m,
chebfun/@chebfun/private/ctor_ini.m,
chebfun/@chebfun/private/detectedge.m,
chebfun/@chebfun/private/getfun.m,
chebfun/@chebfun/private/jumpvals.m,
chebfun/@chebfun/private/merge.m,
chebfun/@chebfun/private/odesol.m,
chebfun/@chebfun/private/overlap.m,
chebfun/@chebfun/private/rescale.m,
chebfun/@chebfun/private/scale.m,
chebfun/@chebfun/private/unwrap_arg.m,
chebfun/@chebfun/private/update_vscl.m,
chebfun/@chebfun/private/vectorcheck.m,
chebfun/@chebfun/private/vectorwrap.m,
chebfun/@chebfun/prod.m,
chebfun/@chebfun/qr.m,
chebfun/@chebfun/range.m,
chebfun/@chebfun/rank.m,
chebfun/@chebfun/rdivide.m,
chebfun/@chebfun/real.m,
chebfun/@chebfun/remez.m,
chebfun/@chebfun/repmat.m,
chebfun/@chebfun/restrict.m,
chebfun/@chebfun/roots.m,
chebfun/@chebfun/round.m,
chebfun/@chebfun/sec.m,
chebfun/@chebfun/sech.m,
chebfun/@chebfun/set.m,
chebfun/@chebfun/sign.m,
chebfun/@chebfun/sin.m,
chebfun/@chebfun/sinh.m,
chebfun/@chebfun/size.m,
chebfun/@chebfun/spy.m,
chebfun/@chebfun/sqrt.m,
chebfun/@chebfun/std.m,
chebfun/@chebfun/subsasgn.m,
chebfun/@chebfun/subspace.m,
chebfun/@chebfun/subsref.m,
chebfun/@chebfun/sum.m,
chebfun/@chebfun/svd.m,
chebfun/@chebfun/tan.m,
chebfun/@chebfun/tanh.m,
chebfun/@chebfun/times.m,
chebfun/@chebfun/transpose.m,
chebfun/@chebfun/uminus.m,
chebfun/@chebfun/uplus.m,
chebfun/@chebfun/vander.m,
chebfun/@chebfun/var.m,
chebfun/@chebfun/vertcat.m,
chebfun/@chebop/and.m,
chebfun/@chebop/blkdiag.m,
chebfun/@chebop/char.m,
chebfun/@chebop/chebop.m,
chebfun/@chebop/display.m,
chebfun/@chebop/domain.m,
chebfun/@chebop/eigs.m,
chebfun/@chebop/end.m,
chebfun/@chebop/exp.m,
chebfun/@chebop/expm.m,
chebfun/@chebop/feval.m,
chebfun/@chebop/gmres.m,
chebfun/@chebop/horzcat.m,
chebfun/@chebop/isempty.m,
chebfun/@chebop/minus.m,
chebfun/@chebop/mldivide.m,
chebfun/@chebop/mpower.m,
chebfun/@chebop/mrdivide.m,
chebfun/@chebop/mtimes.m,
chebfun/@chebop/plus.m,
chebfun/@chebop/private/bdyreplace.m,
chebfun/@chebop/private/blockeye.m,
chebfun/@chebop/private/chebopdefaults.m,
chebfun/@chebop/private/domaincheck.m,
chebfun/@chebop/private/filter.m,
chebfun/@chebop/private/getbc.m,
chebfun/@chebop/private/newIDnum.m,
chebfun/@chebop/private/setbc.m,
chebfun/@chebop/size.m,
chebfun/@chebop/subsasgn.m,
chebfun/@chebop/subsref.m,
chebfun/@chebop/uminus.m,
chebfun/@chebop/uplus.m,
chebfun/@chebop/vertcat.m,
chebfun/@domain/char.m,
chebfun/@domain/chebpts.m,
chebfun/@domain/cumsum.m,
chebfun/@domain/diff.m,
chebfun/@domain/display.m,
chebfun/@domain/domain.m,
chebfun/@domain/double.m,
chebfun/@domain/end.m,
chebfun/@domain/eq.m,
chebfun/@domain/eye.m,
chebfun/@domain/fred.m,
chebfun/@domain/interp1.m,
chebfun/@domain/isempty.m,
chebfun/@domain/length.m,
chebfun/@domain/linspace.m,
chebfun/@domain/minus.m,
chebfun/@domain/mtimes.m,
chebfun/@domain/ode113.m,
chebfun/@domain/ode15s.m,
chebfun/@domain/ode45.m,
chebfun/@domain/ones.m,
chebfun/@domain/plus.m,
chebfun/@domain/private/odesol.m,
chebfun/@domain/subsref.m,
chebfun/@domain/times.m,
chebfun/@domain/uminus.m,
chebfun/@domain/volt.m,
chebfun/@domain/zeros.m,
chebfun/@fun/chebpoly.m,
chebfun/@fun/compfun.m,
chebfun/@fun/conj.m,
chebfun/@fun/cumsum.m,
chebfun/@fun/diff.m,
chebfun/@fun/display.m,
chebfun/@fun/feval.m,
chebfun/@fun/fieldnames.m,
chebfun/@fun/flipud.m,
chebfun/@fun/fun.m,
chebfun/@fun/get.m,
chebfun/@fun/imag.m,
chebfun/@fun/isempty.m,
chebfun/@fun/isequal.m,
chebfun/@fun/length.m,
chebfun/@fun/max.m,
chebfun/@fun/min.m,
chebfun/@fun/minandmax.m,
chebfun/@fun/minus.m,
chebfun/@fun/mrdivide.m,
chebfun/@fun/mtimes.m,
chebfun/@fun/plus.m,
chebfun/@fun/poly.m,
chebfun/@fun/private/ctor_1.m,
chebfun/@fun/private/ctor_2.m,
chebfun/@fun/private/ctor_3.m,
chebfun/@fun/private/ctor_ini.m,
chebfun/@fun/private/growfun.m,
chebfun/@fun/prolong.m,
chebfun/@fun/range.m,
chebfun/@fun/real.m,
chebfun/@fun/restrict.m,
chebfun/@fun/roots.m,
chebfun/@fun/set.m,
chebfun/@fun/simplify.m,
chebfun/@fun/struct.m,
chebfun/@fun/subsasgn.m,
chebfun/@fun/subsref.m,
chebfun/@fun/sum.m,
chebfun/@fun/times.m,
chebfun/@fun/uminus.m,
chebfun/@fun/uplus.m,
chebfun/@oparray/char.m,
chebfun/@oparray/display.m,
chebfun/@oparray/feval.m,
chebfun/@oparray/horzcat.m,
chebfun/@oparray/isempty.m,
chebfun/@oparray/minus.m,
chebfun/@oparray/mpower.m,
chebfun/@oparray/mtimes.m,
chebfun/@oparray/oparray.m,
chebfun/@oparray/plus.m,
chebfun/@oparray/size.m,
chebfun/@oparray/subsref.m,
chebfun/@oparray/uminus.m,
chebfun/@oparray/uplus.m,
chebfun/@oparray/vertcat.m,
chebfun/@varmat/char.m,
chebfun/@varmat/ctranspose.m,
chebfun/@varmat/display.m,
chebfun/@varmat/end.m,
chebfun/@varmat/feval.m,
chebfun/@varmat/horzcat.m,
chebfun/@varmat/isempty.m,
chebfun/@varmat/ldivide.m,
chebfun/@varmat/minus.m,
chebfun/@varmat/mldivide.m,
chebfun/@varmat/mpower.m,
chebfun/@varmat/mrdivide.m,
chebfun/@varmat/mtimes.m,
chebfun/@varmat/plus.m,
chebfun/@varmat/power.m,
chebfun/@varmat/private/op_scalar_expand.m,
chebfun/@varmat/rdivide.m,
chebfun/@varmat/subsref.m,
chebfun/@varmat/times.m,
chebfun/@varmat/transpose.m,
chebfun/@varmat/uminus.m,
chebfun/@varmat/uplus.m,
chebfun/@varmat/varmat.m,
chebfun/@varmat/vertcat.m,
chebfun/ATAPformats.m,
chebfun/bary.m,
chebfun/bary_weights.m,
chebfun/chebfunpref.m,
chebfun/cheblogo.m,
chebfun/cheboppref.m,
chebfun/chebpoly.m,
chebfun/chebpolyval.m,
chebfun/chebpts.m,
chebfun/chebtest.m,
chebfun/chebtests/adaptapply.m,
chebfun/chebtests/airy_extrema.m,
chebfun/chebtests/aliasing.m,
chebfun/chebtests/applycumsumop.m,
chebfun/chebtests/applydiffop.m,
chebfun/chebtests/bcchange.m,
chebfun/chebtests/besseljextrema.m,
chebfun/chebtests/besseljroots.m,
chebfun/chebtests/breakpoints.m,
chebfun/chebtests/bvp1.m,
chebfun/chebtests/bvp2.m,
chebfun/chebtests/bvptest.m,
chebfun/chebtests/callpref.m,
chebfun/chebtests/ceiltest.m,
chebfun/chebtests/cftest.m,
chebfun/chebtests/chebdomain.m,
chebfun/chebtests/chebpadetest.m,
chebfun/chebtests/chebpolytest.m,
chebfun/chebtests/circplate.m,
chebfun/chebtests/circplate.mat,
chebfun/chebtests/complexrotation.m,
chebfun/chebtests/composetest.m,
chebfun/chebtests/convspline.m,
chebfun/chebtests/cumsumcos100x.m,
chebfun/chebtests/eigsho.m,
chebfun/chebtests/exact_endpoints.m,
chebfun/chebtests/falknershan.m,
chebfun/chebtests/grammatrix.m,
chebfun/chebtests/guide2tests.m,
chebfun/chebtests/interp1test.m,
chebfun/chebtests/intops.m,
chebfun/chebtests/invtest.m,
chebfun/chebtests/ivp1.m,
chebfun/chebtests/ivp_ty_test.m,
chebfun/chebtests/ivptest.m,
chebfun/chebtests/ivptestcomplex.m,
chebfun/chebtests/legptstest.m,
chebfun/chebtests/mathieu.m,
chebfun/chebtests/matrixnorms.m,
chebfun/chebtests/max_and_imps.m,
chebfun/chebtests/maxtest.m,
chebfun/chebtests/misclnttests1.m,
chebfun/chebtests/NaNandInf.m,
chebfun/chebtests/normtests.m,
chebfun/chebtests/operarith.m,
chebfun/chebtests/orrsommerfeld.m,
chebfun/chebtests/orthosincos.m,
chebfun/chebtests/outerprod.m,
chebfun/chebtests/plotrow.m,
chebfun/chebtests/plusquasimatrix.m,
chebfun/chebtests/pointevals.m,
chebfun/chebtests/qrtest.m,
chebfun/chebtests/realize_eye.m,
chebfun/chebtests/remeztest.m,
chebfun/chebtests/resampletest.m,
chebfun/chebtests/restrictimps.m,
chebfun/chebtests/restrictscl.m,
chebfun/chebtests/rootspol.m,
chebfun/chebtests/scaleinvariance.m,
chebfun/chebtests/scaleinvariance2.m,
chebfun/chebtests/sinx.m,
chebfun/chebtests/smallintervals.m,
chebfun/chebtests/splittingtest.m,
chebfun/chebtests/subspacetest.m,
chebfun/chebtests/sumcos20x.m,
chebfun/chebtests/systemapply.m,
chebfun/chebtests/systemeig.m,
chebfun/chebtests/systemexpm.m,
chebfun/chebtests/systemsolve1.m,
chebfun/chebtests/systemsolve2.m,
chebfun/chebtests/testpoly.m,
chebfun/chebtests/testpolyfit.m,
chebfun/chebtests/vectornorms.m,
chebfun/Contents.m,
chebfun/cumsummat.m,
chebfun/diffmat.m,
chebfun/guide/guide1.m,
chebfun/guide/guide2.m,
chebfun/guide/guide3.m,
chebfun/guide/guide4.m,
chebfun/guide/guide5.m,
chebfun/guide/guide6.m,
chebfun/guide/guide7.m,
chebfun/guide/guide8.m,
chebfun/guide/guidedefaults.m,
chebfun/guide/html/cheblogo.png,
chebfun/guide/html/guide1.png,
chebfun/guide/html/guide1_01.png,
chebfun/guide/html/guide1_02.png,
chebfun/guide/html/guide1_03.png,
chebfun/guide/html/guide1_04.png,
chebfun/guide/html/guide1_05.png,
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chebfun/guide/html/guide5.png,
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chebfun/guide/html/guide6.png,
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chebfun/guide/html/guide7_eq50266.png,
chebfun/guide/html/guide7_eq65936.png,
chebfun/guide/html/guide7_eq94515.png,
chebfun/guide/html/guide8.png,
chebfun/guide/html/guide8_01.png,
chebfun/guide/html/guide8_02.png,
chebfun/guide/html/guide8_03.png,
chebfun/guide/html/guide8_04.png,
chebfun/guide/html/guide8_05.png,
chebfun/guide/html/guide8_06.png,
chebfun/guide/html/guide8_07.png,
chebfun/guide/html/guide8_08.png,
chebfun/guide/html/guide8_09.png,
chebfun/guide/html/guide8_10.png,
chebfun/guide/html/guide8_11.png,
chebfun/guide/html/guide8_12.png,
chebfun/guide/html/guide8_13.png,
chebfun/guide/html/helptoc.xml,
chebfun/info.xml,
chebfun/INSTALL.txt,
chebfun/legpoly.m,
chebfun/legpts.m,
chebfun/resampling.m,
chebfun/splitting.m,
chebfun/vec.m,
license.txt

[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

[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.

[Tobin 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]

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.

[Erdal Bizkevelci]

 

[Siyi Deng]

this toolbox is awesome.

[Erdal Bizkevelci]