Diff for "symbolics" - Sagemath Wiki

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Symbolics in Sage

These pages are aimed at developers of symbolics functionality in Sage. If you're interested in helping out with any of the items below please contact the sage-devel googlegroup or the people already working on your item of interest.

TODO

Integration
- Poor man's integrator
  - Minh Van Nguyen is working on this.
- A summary of heuristics used in Maple from Algorithms in Computer Algebra by Geddes, Czapor and Labahn
- A survey by Keith Geddes: Algorithms for Indefinite and Definite Integration in Maple
- Heuristics from maxima (relevant parts of the maxima manual have some documentation)
- Transcendental Risch (from Bronstein's book)
Solve
- Talk by Barton Willis on Maxima's to_poly solve function
- test suite for Maxima's to_poly solver
Limits
Differential equation solver
- documentation of Sum^it library might be useful
Basic simplification routines
- trig
- radical
- rational
- binomial/factorial?
  - maxima documentation
Transforms
- laplace, inverse_laplace, hilbert, mellin, etc.
- Some discussion on the sage-support list in this thread.
Orthogonal polynomials
- The orthogonal polynomials defined by sage in the module sage.functions.orthogonal_polys are wrappers to maxima, we should provide native implementations of these, preferably with an argument to specify the parent of the resulting polynomial

Some of the functionality listed above is provided by Maxima wrappers at the moment.

Summation
- Burcin Erocal ([email protected]) is working on this
Hypergeometric functions
- HYP from Christian Krattenthaler for MMA
- HYPERG from Bruno Gauthier for Maple
  - This should let us do the following:

\sum_{s \ge m} {s \choose m} \frac{(n)_s}{(\frac{n}{2} + 1)_s 2^{s}} = \frac{(n)_m}{2^{m}(\frac{n}{2}+1)_m} \,_2 F_1 \left( \begin{array}{cc} m+1, m+n \\ m+ \frac{n}{2} +1 \end{array} ; \frac{1}{2} \right) = \frac{2^{n-1} \Gamma(\frac{n}{2} +1) \Gamma(\frac{m}{2} + \frac{n}{2})}{\Gamma(\frac{m}{2} + 1)\Gamma(n)}

Meijer G-Functions
Generating functions
- This is a building block for many things. A prerequisite for this is linear algebra over polynomial rings, Burcin Erocal is working on this.
- gfun by Bruno Salvy and Paul Zimmermann included in Maple
- GeneratingFunctions by Christian Mallinger for MMA

-  ⇤ ← Revision 1 as of 2009-02-22 16:49:15 → 
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+   ← Revision 10 as of 2009-06-13 17:14:23 → ⇥
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  Comment: trying to get jsmath to work
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-=== Pynac and Maxima based symbolics ===

Most symbolics functionality in Sage relies on [[http://maxima.sf.net|Maxima]]. While Maxima is a very mature and capable software package, there are some problems with using it in Sage. Namely,
  * the pexpect interface to Maxima is slow, and
  * Sage developers cannot fix problems in Maxima or enhance its functionality.

To provide a better framework for symbolics in Sage, and to make implementation of new symbolics functionality that relies on components of Sage easier we created a fork of the C++ library [[http://ginac.de|GiNaC]], called Pynac, which works with Python types. There are still some low level changes required in Pynac before it can be used as a basis for symbolics in Sage, we keep track of these at the [[symbolics/pynac_todo|Pynac todo list]].

With the switch to Pynac based symbolics, it will be possible to implement native functionality in Sage. Here is a brief list of areas that need work.
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+   * Minh Van Nguyen is working on this.
  * [[http://books.google.at/books?id=9fOUwkkRxT4C&pg=PR2&hl=en#PPA473,M1|A summary of heuristics used in Maple]] from Algorithms in Computer Algebra by Geddes, Czapor and Labahn
  * A survey by Keith Geddes: [[http://www.cs.uwaterloo.ca/~kogeddes/papers/Integration/IntSurvey.html|Algorithms for Indefinite and Definite Integration in Maple]]
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+ * Solve
  * [[http://www.unk.edu/uploadedFiles/facstaff/profiles/willisb/solve-talk(3).pdf|Talk by Barton Willis]] on Maxima's to_poly solve function
  * [[http://maxima.cvs.sourceforge.net/viewvc/maxima/maxima/share/contrib/rtest_to_poly_solver.mac?view=markup|test suite]] for Maxima's to_poly solver
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+ * Transforms
  * laplace, inverse_laplace, hilbert, mellin, etc.
  * Some discussion on the sage-support list in [[http://groups.google.com/group/sage-support/browse_thread/thread/7b48e5dbd06f9f46|this thread]].

 * Orthogonal polynomials
  * The orthogonal polynomials defined by sage in the module `sage.functions.orthogonal_polys` are wrappers to maxima, we should provide native implementations of these, preferably with an argument to specify the parent of the resulting polynomial
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-  * [[http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html|HYP]] from Christian Krattenthaler
+  * [[http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html|HYP]] from Christian Krattenthaler for MMA
  * [[http://www-igm.univ-mlv.fr/~gauthier/HYPERG.html|HYPERG]] from Bruno Gauthier for Maple
   This should let us do the following:
$$\sum_{s \ge m} {s \choose m} \frac{(n)_s}{(\frac{n}{2} + 1)_s 2^{s}} = \frac{(n)_m}{2^{m}(\frac{n}{2}+1)_m} \,_2 F_1 \left( \begin{array}{cc} m+1, m+n \\ m+ \frac{n}{2} +1 \end{array} ; \frac{1}{2} \right) = \frac{2^{n-1} \Gamma(\frac{n}{2} +1) \Gamma(\frac{m}{2} + \frac{n}{2})}{\Gamma(\frac{m}{2} + 1)\Gamma(n)}$$
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+ * Generating functions
  This is a building block for many things. A prerequisite for this is linear algebra over polynomial rings, Burcin Erocal is working on this.
  * [[http://algo.inria.fr/libraries/|gfun]] by Bruno Salvy and Paul Zimmermann included in Maple
  * [[http://www.risc.uni-linz.ac.at/research/combinat/software/GeneratingFunctions/index.php|GeneratingFunctions]] by Christian Mallinger for MMA