#9706: New Version of orthogonal Polynomials
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Reporter: maldun | Owner: burcin
Type: defect | Status: new
Priority: minor | Milestone:
Component: symbolics | Keywords: orthogonal polynomials, symbolics
Author: Stefan Reiterer | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by maldun):
I added in the latest patch (and orthogonal_polys.4.py contains these
changes also) a new symbolic evaluation method for the orthogonal
polynomials: Instead of call Maxima or use of the recursion, the
polynomial is evaluated just using explicit formulas from Abramowitz and
Stegun. This is an O(n) algorithm of course.
a little comparison on my machine:
old version:
sage: time chebyshev_T(10,x);
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.04 s
sage: time chebyshev_T(100,x);
CPU times: user 0.13 s, sys: 0.01 s, total: 0.14 s
Wall time: 0.23 s
sage: time chebyshev_T(1000,x);
CPU times: user 5.01 s, sys: 0.01 s, total: 5.02 s
Wall time: 6.98 s
sage time chebyshev_T(5000,x);
??? (I got no output her after 2min)
sage: time gegenbauer(10,5,x);
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.05 s
sage: time gegenbauer(100,5,x);
CPU times: user 0.19 s, sys: 0.00 s, total: 0.19 s
Wall time: 0.29 s
sage: time gegenbauer(1000,5,x);
CPU times: user 5.46 s, sys: 0.02 s, total: 5.48 s
Wall time: 7.79 s
New Version
sage: time chebyshev_T(10,x);
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s
sage: time chebyshev_T(100,x);
CPU times: user 0.06 s, sys: 0.00 s, total: 0.06 s
Wall time: 0.08 s
sage: time chebyshev_T(1000,x);
CPU times: user 1.22 s, sys: 0.00 s, total: 1.22 s
Wall time: 1.22 s
sage: time chebyshev_T(5000,x);
CPU times: user 27.17 s, sys: 0.15 s, total: 27.32 s
Wall time: 27.46 s
sage: time gegenbauer(10,5,x);
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01 s
sage: time gegenbauer(100,5,x);
CPU times: user 0.03 s, sys: 0.00 s, total: 0.03 s
Wall time: 0.04 s
sage: time gegenbauer(1000,5,x);
CPU times: user 1.08 s, sys: 0.01 s, total: 1.09 s
Wall time: 1.11 s
A little bit faster :) I also don't need to spawn an instance of maxima
which makes the initialisation faster.
And now also wider symbolic evaluation is possible:
old version:
sage: var('a')
a
sage: gegenbauer(3,a,x)
...
NameError: name 'a' is not defined
new version:
sage: var('a')
a
sage: gegenbauer(3,a,x)
4/3*x^3*gamma(a + 3) - 2*x*gamma(a + 2)
The code needs now some cleanup, especially the documentations.
The complete versions for legendre_Q, gen_legendre_P, and gen_legendre_Q
will not be finished
soon since the mpmath functions, don't seem to work correctly...
I only provide a call function for maxima for them now.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9706#comment:4>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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