#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, sd32
Work_issues: | Upstream: N/A
Reviewer: | Author: Stefan Reiterer
Merged: | Dependencies:
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Comment(by kcrisman):
Replying to [comment:47 kcrisman]:
> Just FYI - the Maxima list has a similar discussion going on right now,
starting [http://www.math.utexas.edu/pipermail/maxima/2011/026230.html
here] in the archives. We should continue to allow access to their
methods as optional, of course :)
I especially like this quote. I think it is a better way forward than
trying to guess what the user needs - maybe just allowing many options is
better.
{{{
In my opinion, an interface simplifying the use of packages like QUADPACK
should not attempt to guess which method is more appropriate for a given
problem. This should be left to the user. Otherwise we are going to have
an
"all-purpose" Maxima function supposed to do everything, like
Mathematica's
function "NIntegrate", which, however, not only fails to do what is
supposed
to do in many cases, but also encourages the "blind use" of Numerical
Analysis. There is no "perfect" numerical method, able to solve any
problem
efficiently; this is true for any numerical problem, such as numerical
quadrature, solution of initial or boundary value problems etc.
Two simple examples:
(1) consider integrating a "sawtooth" function. Romberg integration, which
is generally very fast and accurate for smooth functions would have a hard
time integrating a sawtooth function, while a simple trapezoidal method
would be much faster in that particular case.
(2) Most people use "natural" cubic splines for interpolation, just
because
of their name I guess, or maybe because Computer Algebra Systems use
natural
cubic splines by default for interpolation. However, "natural" splines are
not natural at all, and should be used only if there is a reason to assume
second derivative is zero at the end points of the interpolation interval.
Otherwise, "not-a-knot" splines should be used instead.
I doubt there is a way to make a function which automatically selects the
best method to solve a numerical problem. Mathematica tries that and the
result is disappointing. I am very suspicious about such attempts, and I
believe none should trust them, no matter how sophisticated they are.
Everyone who needs to use numerical methods should have a basic knowledge
of
what (s)he is doing, and should be able to pick the numerical method most
appropriate method for a given problem.
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8321#comment:48>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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