#8321: numerical integration with arbitrary precision
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Reporter: burcin | Owner:
Type: defect | Status: needs_work
Priority: major | Milestone: sage-4.7.2
Component: symbolics | Keywords: numerics,integration
Work_issues: | Upstream: N/A
Reviewer: | Author: Stefan Reiterer
Merged: | Dependencies:
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Comment(by benjaminfjones):
I've run some more comparisons which include Burcin's relative error for
mpmath's quad. Here are some results for 6 test functions. The code that
runs the tests is here: https://gist.github.com/1166436
In the output below:
- GSL means we call `numerical_integral`
- mpmath means we call `mpmath.call(mpmath.quad, mp_f, [a, b])` for an
appropriate function `mp_f`
- mpmath_rel means we call `mpmath.call(mpmath.quad, mp_f, [a, b],
method=GaussLegendreRel)`
All of the times are listed in seconds.
{{{
function: e^(-x^2)*log(x) on [17, 42]
GSL: 2.5657285007e-127 time:
0.001216173172
mpmath: 2.59225286296247e-127 time:
0.0308299064636
mpmath_rel (prec=64): 2.56572850056105156e-127 time:
0.7594602108
mpmath_rel (prec=100): 2.5657285005610514829173563974e-127 time:
0.894016981125
function: sqrt(-x^2 + 1) on [0, 1]
GSL: 0.785398167726 time:
0.000615119934082
mpmath: 0.785398163397448 time:
0.011482000351
mpmath_rel (prec=64): 0.785398183809260289 time:
0.50303196907
mpmath_rel (prec=100): 0.78539818380926028913861331848 time:
0.57284116745
function: sin(sin(x)) on [0, 1]
GSL: 0.430606103121 time:
0.00026798248291
mpmath: 0.430606103120691 time:
0.0110800266266
mpmath_rel (prec=64): 0.430606103120690605 time:
0.00665020942688
mpmath_rel (prec=100): 0.43060610312069060491237735525 time:
0.0202469825745
function: max(sin(x), cos(x)) on [0, pi]
GSL: 2.41421356237 time:
0.012158870697
mpmath: 2.41413598800040 time:
0.102693796158
mpmath_rel (prec=64): 2.41424024561759656 time:
0.514449834824
mpmath_rel (prec=100): 2.4142402456175965601829446506 time:
0.583966970444
function: e^cos(x)*sin(x) on [0, pi]
GSL: 2.35040238729 time:
0.000401973724365
mpmath: 2.35040238728760 time:
0.103391170502
mpmath_rel (prec=64): 2.35040238728760291 time:
0.0510699748993
mpmath_rel (prec=100): 2.3504023872876029137647637012 time:
0.132436037064
function: e^(-x^100) on [0, 1.1]
GSL: 0.994325851192 time:
0.000550031661987
mpmath: 0.994325851192472 time:
0.390738010406
mpmath_rel (prec=64): 0.994325851192555258 time:
0.75753903389
mpmath_rel (prec=100): 0.99432585119255525754634686152 time:
0.875532865524
}}}
It looks like mpmath is now just as accurate as GSL but the times are
obviously a lot longer. I agree with @kcrisman 's suggestion for calling
GSL when the precision is default (float or RDF or 53 bits?) and calling
mpmath with relative errors when the precision is higher. Perhaps adding
the mpmath relative errors should be on a different ticket which this one
will depend on?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8321#comment:37>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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