#8321: numerical integration with arbitrary precision
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Reporter: burcin | Owner: burcin
Type: defect | Status: needs_work
Priority: major | Milestone: sage-4.7.1
Component: symbolics | Keywords: numerics,integration
Work_issues: | Upstream: N/A
Reviewer: | Author: Stefan Reiterer
Merged: | Dependencies:
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Comment(by burcin):
I hadn't actually done the computation in Sage. I just wanted to note that
web site here. :)
With the patch attached to this ticket:
{{{
sage: sigma = 1e-4
sage: ff = exp(-x**2/(2*sigma)) / sqrt(2*pi*sigma)
sage: from sage.symbolic.integration.integral import definite_integral
sage: definite_integral(ff, x, -20, 10, hold=True)
integrate(70.7106781186548*e^(-5000.00000000000*x^2)/sqrt(pi), x, -20, 10)
sage: definite_integral(ff, x, -20, 10, hold=True).n()
5.38249970415053e-939
}}}
Without the patch:
{{{
sage: definite_integral(ff, x, -20, 10, hold=True).n()
2.1074458151264474e-45
}}}
We get a better result if we allow maxima to evaluate the integral
symbolically:
{{{
sage: definite_integral(ff, x, -20, 10).n()
1.00000000000000
sage: definite_integral(ff, x, -20, 10)
0.353553390593*(sqrt(2)*sqrt(pi)*erf(500*sqrt(2)) +
sqrt(2)*sqrt(pi)*erf(1000*sqrt(2)))/sqrt(pi)
}}}
The web site I linked to in comment:24 links to this page, where i got the
example from:
http://trac.openopt.org/openopt/browser/PythonPackages/FuncDesigner/FuncDesigner/examples/integrate1.py
Here is an excerpt from that file:
{{{
'''interalg result: 1.000006 (usually solution, obtained by interalg, has
real residual 10-100-1000 times less
than required tolerance, because interalg works with "most worst case"
that extremely rarely occurs.
Unfortunately, real obtained residual cannot be revealed).
Now let's ensure scipy.integrate quad fails to solve the problem and mere
lies about obtained residual: '''
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8321#comment:26>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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