Math::GSL : Perl GNU Scientific Library

Complex Eigenvalues with Math::GSL

Sat, 15 Nov 2008 17:09:27 -0800

After a question arose recently about finding complex eigenvalues of non-symmetric matrices, I wrote this example to show how it can be done with Math::GSL. The current example is called examples/matrix/nonsymmetric_eigen in the Math::GSL source, which will be in the 0.16 Math::GSL release. It uses the gsl_eigen_nonsymmv() function to do all the hard work, the rest is twiddling complex numbers. The documentation for this can be found with perldoc Math::GSL::Eigen or here.(The HTML rendering of the POD needs some work...) You can read the original C documentation for this function here.

It is a well known theorem of linear algebra that symmetric matrices can have only real eigenvalues. Thus it is much easier to find eigenvalues of symmetric matrices because there is no possibility of complex numbers arising. You can do this with Math::MatrixReal, which is written in pure-Perl.

The output of examples/matrix/nonsymmetric_eigen looks like:

Finding eigenvalue/eigenvectors for

[ 0 -1 ]

[ 1 0 ]

Eigenvectors:

u = (0.707106781186547,-0.707106781186547i)

v = (0.707106781186547,0.707106781186547i)

Eigenvalues:

lambda_0 = i

lambda_1 = -i

By the way, sqrt(2)/2 = 0.707106...

I am not quite happy with how long the program is, mainly because I have not written a Perlish interface around Complex Vectors and Matrices yet, so it is all in that script right now. The stringification could be built into the object and therefore reduce the length of that code dramatically.

Inside Look of Math::GSL

Fri, 14 Nov 2008 13:49:48 -0800

Here is a breakdown of how many lines of code (roughly) are in different languages in the Math::GSL codebase:

280889 lines - XS ( C-like Perl internals autogenerated by SWIG)
60181 lines - Total Perl (generated by SWIG)
22941 lines- Hand-written Perl

8617 lines - Tests

13731 lines - POD (this actually includes some Perl implementation, probably <5%). Much of this was autogenerated from the GSL C documentation and reformatted, but it stills needs work, especially in making the POD render to HTML in a more pleasant way.

1112 lines - SWIG

The reason why Math::GSL requires something like SWIG becomes very apparent. The SWIG interface files are roughly 1/280th the length of the raw XS source code. No reasonable person would be hacking on Perl internals, but not even a reasonable Perl hacker wants to write all that error checking boilerplate cruft. Thanks SWIG!

This was made possible by ripping all Perl code out of the SWIG interface files and putting them in the pod/ directory. It is not strictly POD but a simple naming scheme was key and the majority of the files do happen to be POD, so whatev. And now vim correctly syntax highlights the files! I think there is something to be said for sticking to one language per file.

Post-Mentor-Summit Wrap Up

Sat, 01 Nov 2008 06:20:20 -0800

I took a bit of a break after releasing Math::GSL 0.14 and tried not to stare at a screen too much during the GSOC08 conference, when I could be interacting with open source developers from around the world IRL. I met developers from Git, Xmms2, VLC, PHP, Zikula, OpenAFS, Boost, Jikes RVM, Squeak, MusicBrainz, Worldforge, Minix3, the list goes on. People shared many ideas about making sustainable open source projects, getting and keeping members in your community and dealing with "troublesome" members. Daniel Svensson from Xmms2 also showed me some git-fu relating to rebasing.

But that doesn't mean that Math::GSL development has come to a halt! Since 0.14 Thierry added operator overloading for addition in Vector. This sparked my interest and I then did subtraction. The current code will vectorize operations that you learn "don't make sense" in grade school, such as: $v - 5 , where v is a vector. Math::GSL will go ahead and subtract 5 from every element in $v. You can also do 2 + $v and you will get a new vector where 2 has been added to each element of $v.

Also, I reorganized the directory structure so that all SWIG interface files are now in the swig/ subdirectory, instead of clogging up the root directory. All SWIG-generated XS code and object files are thrown in there as well. This should make the Math::GSL codebase easier to navigate.

Math::GSL 0.14

Wed, 22 Oct 2008 08:09:31 -0800

Get it here: http://search.cpan.org/~leto/Math-GSL-0.14/

Changes in 0.14:

- Chebyshev Series Approximation, with tests and docs

- Examples and new documentation in Statistics, as well as tests (Thierry)

- Bugfixes relating to compiling on 64bit Red Hat Linux, reported by Michael Roberge

- Improved FFT, Deriv, Integration, Chebyshev, Combination and Roots docs

- Documentation for Math::GSL::Test

- Improved introduction examples in Math::GSL

- Improved Minimization subsystem tests, but it is not functional

- Added examples/fft/forward_real

- Added examples/deriv/basic

This shows the simple fact that d/dx(sin(x)) = cos(x)

- Added examples/sf/erfc_check (thanks to Keith Lofstrom)

Script which tests the erfc() special function against computing

the integral definition with gsl_integration_qagiu()

- Added examples/vector/speed

This shows a considerable performance boost using Math::GSL::Vectors

instead of List::Util when searching for the min and max elements of

large sets of random numbers.

- Fixed return signature of gsl_deriv_* functions to return a flat list

- Fixed some failing tests relating to -Duselongdouble (Sisyphus)

- Fix location of shared objects (Sisyphus)

- Added raw() method to RNG objects

Math::GSL on GitHub

Tue, 21 Oct 2008 21:30:32 -0800

I have created a GitHub mirror of the Math::GSL git repo, hopefully this will make it easier for developers to join the project and I will get to see why people like GitHub so much.

Public Clone URL:

git://github.com/leto/math--gsl.git

I like that it automatically creates a "donate" link via Pledgie. Please consider donating to Math::GSL today if you enjoy using it!

Math::GSL Podcast from PDX.pm

Sat, 18 Oct 2008 09:47:05 -0800

Here you can find my Scientfic Computing with Math::GSL Podcast from PDX.pm.( Stream, Download) In the talk I compare PDL and Math::GSL, talk about the Google Summer of Code 2008, and show off some of the subsystems that are currently working. I field some questions from the audience about whether Math::GSL does symbolic computing or parrallelized/vectorized computing (it doesn't, at least yet) and how exactly I turn Perl subroutine references into C function pointers. If you look at the presentation slides here while you listen, it is almost like being there!

Math::GSL 0.13_03 - More stuff!

Thu, 16 Oct 2008 20:16:12 -0800

I've just sent another dev release off to CPAN, 0.13_03 to test the recent Build changes that were necessary to get AMD64 boxen to correctly compile. It turns out that they really need the "-fPIC" flag at compilation and link time or else stuff blows up:

usr/bin/ld: BLAS_wrap.o: relocation R_X86_64_32S against `a local
symbol' can not be used when making a shared object; recompile with -fPIC
BLAS_wrap.o: could not read symbols: Bad value
collect2: ld returned 1 exit status
error building blib/arch/auto/Math/GSL/BLAS/BLAS.so file from
'BLAS_wrap.o' at inc/GSLBuilder.pm line 99.

This link with go active sometime tomorrow:

http://search.cpan.org/~leto/Math-GSL-0.13_03/

New stuff since 0.13_02:

AMD64 Build fixes
Improved FFT docs
Improved Math::GSL::Test docs

New Developer Release : Math::GSL 0.13_02

Sun, 12 Oct 2008 13:54:08 -0800

Please test this out and let me know if anything goes *boom*:

http://search.cpan.org/~leto/Math-GSL-0.13_02/

Changes since 0.12:

- Chebyshev Series Approximation, with tests and docs
- Improved Deriv, Integration, Chebyshev, Combination and Roots docs
- Improved introduction examples in Math::GSL
- Improved Minimization subsystem tests, but it is not functional
- Added example/deriv/basic

This shows the simple fact that d/dx(sin(x)) = cos(x)

- Added example/sf/erfc_check (thanks to Keith Lofstrom)

Script which tests the erfc() special function against computing
the integral definition with gsl_integration_qagiu()

- Added example/vector/speed

This shows a considerable performance boost using Math::GSL::Vectors
instead of List::Util when searching for the min and max elements of
large sets of random numbers.

- Fixed return signature of gsl_deriv_* functions to return a flat list
- Fix location of shared objects (Sisyphus)
- Added raw() method to RNG objects

The Riemann Zeta Function

Sun, 07 Sep 2008 14:08:18 -0800

Here is a view of the Riemann Zeta function graphed from x=1.2 to 10. You will notice a sharp spike as x goes toward 1, where it shoots off to infinity. The Riemann Zeta function at x=1 is the harmonic series. Since *everybody* knows the harmonic series diverges, so does the Riemann Zeta function at x=1. As x gets larger, the function approaches 1 quickly. This function directly determines the statistical properties of the distribution of prime numbers, so mathematician go wild studying everything about it.

If you can prove that the only solutions to the equation Zeta(z) = 0 occur on the line Re(z) = 1/2 (aka The Riemann Hypothesis), then you get a million bucks!

Sexy Bessel Functions with Chart::Clicker

Sun, 07 Sep 2008 12:12:19 -0800

These charts are created with Perl and Math::GSL via the great new charting module Chart::Clicker, which uses the Cairo library to produce very nice alpha transparency and other visual eye candy. Here are the first two Bessel Functions of the First Kind (J0(x), J1(x)) and the first two Spherical Bessel Functions (j0(x), j1(x)). Bessel Functions of the First Kind usually arise in a problem that has circular or cylindrical symmetry, while the the latter occur when spherical symmetry is present. These functions where discovered when German mathematician/astronomer Friedrich Wilhelm Bessel was studying the motion of the planets and comets in the early 19th century.

The script to generate these charts is examples/clicker/chart in the latest bleed branch of Math::GSL. A release to CPAN is coming soon...

Google Summer of Code 2008 Recap

Mon, 01 Sep 2008 11:18:41 -0800

What an amazing summer! I really couldn't have imagined all the things that I would learn or teach, the obstacles that would make it challenging or the people that I would meet.

Thierry Moisan submitted the Math::GSL proposal that was chosen by The Perl Foundation for one of the five slots that we received. It was well-written and organized, but I reordered some of the milestones, pushing more difficult subsystems to the end so that they wouldn't stall development. Thierry made 226 git commits during GSOC08, which ranged from tests, POD documentation, SWIG typemaps and Perl implementation. He is mostly responsible for an immense amount of tests and documentation in the BLAS, Linalg, Matrix and Fit subsystems. Thierry also wrote a function graphing program using perl/Tk, which is in the "examples" directory of the distribution.

I live in Portland and Thierry is in Quebec City, so we mostly communicated through gchat, email and git commit messages, which seemed pretty effective. Public mailing lists like pdx-pm and swig-user were also helpful, as well as all the feedback I got when I gave a talk featuring Math::GSL at the Advanced Topics Portland Linux User Group (ATPLUG).

GSOC08 really provided the impetus to make Math::GSL a reality. I had a prototype (read: hackish script) that implemented only the Special Functions of GSL in 2000 and then it just sat around until Thierry's proposal.

As of the end of GSOC08, 41/48 of the Math::GSL subsystems were fully functional. The few remaining subsystems required callbacks, which were only partially implemented. As of this writing though, I have cracked the case and started to fully implement the remaining subsystems with Thierry. Numerical derivatives and integration (Deriv and Integration) are already supported.

This summer has also spawned two mailing lists, the math-gsl-dev google group and the perl-scientific-computing group. The first is for Math::GSL development and usage in particular, the second is for a broader audience. I envision Math::GSL as only the beginning of a suite of modules (or a custom Perl distribution) that enable scientists to "do science" in a quick and efficient way with Perl. If this sounds interesting, please come join and start some gossip!

I believe that Thierry grew tremendously as a developer, he became proficient using git and doing advanced vim hackery as well being introduced to new programming concepts (like typemaps) and new libraries (Tk). Knowing that I helped introduce a new member to the open source community while hacking on something cool (and making it work!) is the best payment that I could get. That and maybe a tshirt.

Math::GSL 0.10 released to CPAN

Thu, 28 Aug 2008 19:25:27 -0800

Math::GSL has been kicking around on CPAN for a few days now, and seems to be doing well with the CPANtesters , except for multi-threaded Perls. This is a very important piece of information to learn, since I develop on FreeBSD 6.x and Ubuntu 8.x, which come with single-threaded Perls, and I had of course recompiled my own Perl from source, but single-threaded is again the default in Configure.

The Build fails like this:

gcc -I/usr/lib/perl/5.8/CORE -c -shared -Wall -fpic -Dbool=char -Wno-strict-aliasing -Wno-unused-function -Wno-unused-value -Wno-unused-function -Wno-unused-variable -o
BLAS_wrap.o BLAS_wrap.c
In file included from /usr/lib/perl/5.8/CORE/op.h:

497,
from /usr/lib/perl/5.8/CORE/perl.h:2754,
from BLAS_wrap.c:716:
/usr/lib/perl/5.8/CORE/reentr.h:612: error: field '_crypt_struct' has
incomplete type
In file included from /usr/lib/perl/5.8/CORE/perl.h:3950,
from BLAS_wrap.c:716:
/usr/lib/perl/5.8/CORE/proto.h:297: error: expected declaration
specifiers or '...' before 'off64_t'
/usr/lib/perl/5.8/CORE/proto.h:299: error: expected '=', ',', ';', 'asm'
or '__attribute__' before 'Perl_do_sysseek'
/usr/lib/perl/5.8/CORE/proto.h:300: error: expected '=', ',', ';', 'asm'
or '__attribute__' before 'Perl_do_tell'
/usr/lib/perl/5.8/CORE/proto.h:2005: error:
expected '=', ',', ';', 'asm' or '__attribute__'
before 'Perl_PerlIO_tell'
/usr/lib/perl/5.8/CORE/proto.h:2006: error: expected declaration
specifiers or '...' before 'off64_t'
error building .o file from 'BLAS_wrap.c'
at /tmp/Math-GSL-0.08/_build/lib/MyModuleBuilder.pm line 167.

Does anyone know of extra build flags which are necessary when building XS modules on multi-threaded Perls?

Math::GSL 0.08 released to CPAN

Wed, 13 Aug 2008 00:51:45 -0800

Coming to a CPAN mirror near you, a brand spanking new Math::GSL, now with Cygwin support!

Enjoy!

week 11 report

Mon, 11 Aug 2008 10:55:21 -0800

There's only one week left to the google summer of code part of the project! Time passed really fast this summer. Last week I started a overall review of all the subsystems of the project to make sure that everything is fine for the pencil down date which is this Friday, august 18. I preferred adding more documentation to the subsystems than adding code to the Tk example since I think it's more important at the moment. I'll touch the Tk example this week if I have enough time but I plan to first finish my overall review before.

I've learned a lot this summer on various aspect of programming : using git, vim and Tk, Object-Oriented programming, and better Perl programming. I plan to continue to work on this project even after the end of google summer of code. Of course I'd like to thank Jonathan Leto who gave me his time and knowledge. Without him, this project wouldn't exist.

Week 10 report

Mon, 04 Aug 2008 15:33:48 -0800

Last week I worked on making the documentation more HTML friendly. I added basic documentation to the remaining subsystems not yet implemented. We also released a public version on CPAN. I fixed various bugs that we received from the CPANT tester. There are still little issues to look at for this release like the META.yml file which doesn't output exactly the right things but this is not anything major. I worked on the example using Tk::TableDataset to add a menu and various additions to it. I've learn a lot about Tk and GUI while working on this example. I also discovered the Perl::Tidy module to make the perl code easier to read. Finally, I finished the Sum subsystem, it is now fully functional.

In the next week I will complete the overall documentation by adding more concrete examples to all the functional subsystems. These examples will probably mainly come from the tests we've already wrote for those subsystems. I also wish to add more functions and change a bit more the code of the Tk::TableDataset example. Depending on the time left, I might also add more Object-Oriented wrappers.