#10550: integration not working
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Reporter: mariah | Owner: burcin
Type: defect | Status: new
Priority: minor | Milestone: sage-4.6.2
Component: calculus | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by gbe):
Getting this right will be tricky. While simply changing from a blind
application of gsl's numerical_integral() to mpmath's quad() may be often
be an improvement, in this situation it makes things worse:
{{{
sage: f = abs(x^2-1)^(-2/3)
sage: g = fast_callable(f, RealField(100), [x])
sage: import mpmath as mp
sage: mp.mp.prec = 100
sage: for i in [mp.quad(g, [-10^i, 10^1]) for i in range(15)]:
....: print i
....:
+inf
8.8396932208278888892057527426
8.3969760821447628003659891605
9.7953936385023669571121865442
9.2339519516756815076612878988
8.2441701346282922281629165563
8.044969834509411578891250552
11.635099287297832304011545535
11.084646393563409097851890886
7.5468622876138167706495612921
7.0790386862087011969090472793
8.4443867135724817987750671703
10.690050592136777734927819129
32.336415464969432639295368434
10.533897172238729354640752465
sage: mp.quad(g, [mp.mpf('-inf'),mp.mpf('+inf')])
mpf('+inf')
}}}
Switching from tanh-sinh quadrature to Gauss-Legendre gives
{{{
sage: for i in [mp.quadgl(g, [-10^i, 10^1]) for i in range(15)]:
....: print i
....:
6.3780650727028379121031555789
7.9564100202151404330674115745
8.3052587750738550159021915092
11.156320092824073496435254839
9.5630214580799381652111853848
2.8857506804320681789970931663
1.7298436143374586397937735747
0.63762636717628812549878116562
0.29045600775700733200460854022
0.13456948161705169202354793393
0.062450129901992433601218328054
0.02898624937362903007167714756
0.013454200394425034174020875196
0.0062448854829133373771884652354
0.002898619019151035832588794256
sage: mp.quadgl(g, [mp.mpf('-inf'),mp.mpf('+inf')])
mpf('11.350773575585495163385287559838')
}}}
Hand-tuning with Gauss-Legendre:
{{{
sage: mp.quadgl(g, [mp.mpf('-inf'),-1,1,mp.mpf('+inf')])
mpf('12.350413467045610425723223632513')
sage: mp.quadgl(g, [mp.mpf('-inf'),-1,1,mp.mpf('+inf')], maxdegree=12)
mpf('12.592883474959440301710672862944')
}}}
And with tanh-sinh:
{{{
sage: mp.quad(g, [mp.mpf('-inf'),-1,1,mp.mpf('+inf')])
mpf('+inf')
sage: mp.quad(g, [mp.mpf('-inf'),-1,1,mp.mpf('+inf')], maxdegree=12)
mpf('+inf')
}}}
It seems this integral ''requires'' at least some hand tuning to be
reasonable.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10550#comment:10>
Sage <http://www.sagemath.org>
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