make nintegrate/nintegral top-level functions

[jasongrout]

We just need to add them to sage/devel/sage/misc/functional.py. Modifying the versions of integrate/integral in that file would probably be easiest.

CC: @kcrisman @jasongrout @sagetrac-mhampton

Component: calculus

Author: Gagan Sekhon

Reviewer: Karl-Dieter Crisman

Issue created by migration from https://trac.sagemath.org/ticket/7763

[sagetrac-ryan]

comment:3

Is this supposed to be a numerical approximation of the integral function?

[kcrisman]

comment:4

Replying to @sagetrac-ryan:

Is this supposed to be a numerical approximation of the integral function?

No, this is an existing method of symbolic functions, which uses a different algorithm to get a numerical integral. But you can only call it in

sage: f(x)=some_formula_with_x
sage: f.nintegrate(...)

not as a top-level

nintegral(f,...)

type function.

We should also probably make the syntax like that of numerical_integral and make that syntax consistent (at least as an option) with that of integral itself (see ask.sagemath.org for various problems this causes).

[kcrisman]

Reviewer: Karl-Dieter Crisman

[kcrisman]

comment:6

There are two issues with this. First, because of the extreme possibility for confusion with the already top-level numerical_integralfunction, this will need some work in documentation. See some of the documentation in

sage: sage.calculus.calculus.nintegral??

for what I mean. I don't think all of that needs to be there, but there should then be a reference for how to access the rest of it. Basically, Maxima numerical integration and GSL numerical integration are different. In particular, one would want to use Maxima for symbolic expressions - in case there is an exact answer known! - and then have a variety of options for numerical integration if that fails.

Anyway, that was a longish digression. More importantly, this patch doesn't exactly do what is asked. The point isn't to be able to approximate exact answers to integrals, but rather to expose to the top level the Maxima integration.

sage: a = e^(-x^4 + x)
sage: a.nintegral(x,0,1)
(1.3638178766496709, 1.5141420080518571e-14, 21, 0)
sage: integral(a,(x,0,1)).n()
1.3638178766496716

Note the slight difference in output, incidentally - presumably within the error tolerance, of course. This is because we apparently have a THIRD way to evaluate integrals - Pynac!

sage: R = RealField(53)
sage: integral(a,(x,0,1))._convert(R)
1.3638178766496716

If you check the code for _convert, it turns out this goes to Pynac. See here for some of how this happens in Ginac... crazy. We really need to unify this. But at any rate, we shouldn't use two different algorithms and do two different things for the same name nintegrate.

Anyone have ideas for what the best resolution on this would be?

Oh, and just for comparison:

sage: numerical_integral(a,0,1)
(1.3638178766496716, 1.5141420080518571e-14)

[kcrisman]

Author: Gagan Sekhon

[kcrisman]

comment:7

And of course a more informative commit message :)

Don't worry, this is on the way to being a good contribution; we just have to figure out what the right thing to do is.