Diff for "symbolics" - Sagemath Wiki

Differences between revisions 9 and 11 (spanning 2 versions)

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
  - Stefan Reiterer ([email protected]) is working on this

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 9 as of 2009-06-13 17:06:53 → 
  Size: 3527
  Editor: BurcinErocal
  Comment: added transforms and orthogonal polys to todo list, removed pynac notice
+   ← Revision 11 as of 2010-07-20 00:00:39 → ⇥
  Size: 3599
  Editor: StefanReiterer
  Comment: I wanted just mention that I work on the Ortho. Poly issues....
-Deletions are marked like this.
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+   * Stefan Reiterer ([email protected]) is working on this
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-$$\sum_{s \ge m} \binom{s}{m} \frac{(n)_s}{(\frac{n}{2} + 1)_s 2^{s}} = \frac{(n)_m}{2^{m}(\frac{n}{2}+1)_m} \,_2 F_1 \left( \begin{array}{cc} m+1, m+n \\ m+ \frac{n}{2} +1 \end{array} ; \frac{1}{2} \right) = \frac{2^{n-1} \Gamma(\frac{n}{2} +1) \Gamma(\frac{m}{2} + \frac{n}{2})}{\Gamma(\frac{m}{2} + 1)\Gamma(n)}$$
+$$\sum_{s \ge m} {s \choose m} \frac{(n)_s}{(\frac{n}{2} + 1)_s 2^{s}} = \frac{(n)_m}{2^{m}(\frac{n}{2}+1)_m} \,_2 F_1 \left( \begin{array}{cc} m+1, m+n \\ m+ \frac{n}{2} +1 \end{array} ; \frac{1}{2} \right) = \frac{2^{n-1} \Gamma(\frac{n}{2} +1) \Gamma(\frac{m}{2} + \frac{n}{2})}{\Gamma(\frac{m}{2} + 1)\Gamma(n)}$$