Hi, is there any command i can use (or link I can refer to) on calculating the Gauss-Legendre error term? For example in PARI, we have these "intnumgaussinit({n})" as well as "intnumgauss(X = a, b, expr, {tab})". thank you.
Edited to take into account suggestions and observations by @FrédéricC and @nbruin.
Sage's numerical_integral function uses a heuristic to guess an error bound,
provide no certified error bound.
The good way to go is to use Arb, which is easy from Sage.
Example (from http://fredrikj.net/math/scan2018.pdf):
sage: C = ComplexBallField(100)
sage: C.integral(lambda x, _: cos(x) * sin(x), 0, 1)
[0.35403670913678559674939205737 +/- 8.68e-30]Side observation: the file gauss_legendre.pyx
in src/sage/numerical/
strangely does not result in a gauss_legendre entry in the reference manual
under sage/numerical.
That implementation doesn't do anything to compute a guaranteed error term. It does extrapolate one heuristically, and uses that as termination condition. It would be worth looking at "arb" for certified numerical integration (with Gauss-Legendre available).
See Sage Trac ticket 30698 for better documenting Gauss-Legendre integration in the Sage documentation.
Asked: 6 years ago
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Last updated: Dec 04 '18
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Did you look at the doc: numerical_integral?