ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 11 Jun 2016 23:16:38 +0200numerical_integration returns two numbers. What is the second?https://ask.sagemath.org/question/33747/numerical_integration-returns-two-numbers-what-is-the-second/The command numerical_integral returns two numbers. In the documentation, under "numerical integration", the second is called an "error bound", and later on it is called an "error estimate". What is this number, and precisely how is it defined?Sat, 11 Jun 2016 01:23:44 +0200https://ask.sagemath.org/question/33747/numerical_integration-returns-two-numbers-what-is-the-second/Answer by vdelecroix for <p>The command numerical_integral returns two numbers. In the documentation, under "numerical integration", the second is called an "error bound", and later on it is called an "error estimate". What is this number, and precisely how is it defined?</p>
https://ask.sagemath.org/question/33747/numerical_integration-returns-two-numbers-what-is-the-second/?answer=33748#post-id-33748The second number is an estimation of an error bound. You could have seen it looking at the documentation as follows
sage: numerical_integral?
Docstring:
Returns the numerical integral of the function on the interval from
a to b and an error bound.
...Sat, 11 Jun 2016 09:40:35 +0200https://ask.sagemath.org/question/33747/numerical_integration-returns-two-numbers-what-is-the-second/?answer=33748#post-id-33748Answer by paulmasson for <p>The command numerical_integral returns two numbers. In the documentation, under "numerical integration", the second is called an "error bound", and later on it is called an "error estimate". What is this number, and precisely how is it defined?</p>
https://ask.sagemath.org/question/33747/numerical_integration-returns-two-numbers-what-is-the-second/?answer=33754#post-id-33754`numerical_integral` calls routines in the GNU Scientific Library. As described in the relevant [introduction](https://www.gnu.org/software/gsl/manual/html_node/Numerical-Integration-Introduction.html#Numerical-Integration-Introduction), the library returns a numerical integration result and an absolute error that satisfy
|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
where `I` is the exact value of the integral. The absolute and relative error inputs on the right both [default](https://github.com/sagemath/sage/blob/master/src/sage/gsl/integration.pyx#L62) to 1e-6 in SageMath.Sat, 11 Jun 2016 23:16:38 +0200https://ask.sagemath.org/question/33747/numerical_integration-returns-two-numbers-what-is-the-second/?answer=33754#post-id-33754