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numerical_integration returns two numbers. What is the second?

asked 2016-06-10 18:23:44 -0500

Alan gravatar image

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?

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answered 2016-06-11 16:16:38 -0500

numerical_integral calls routines in the GNU Scientific Library. As described in the relevant 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 to 1e-6 in SageMath.

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answered 2016-06-11 02:40:35 -0500

The second number is an estimation of an error bound. You could have seen it looking at the documentation as follows

sage: numerical_integral?
   Returns the numerical integral of the function on the interval from
   a to b and an error bound.
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Asked: 2016-06-10 18:23:44 -0500

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Last updated: Jun 11 '16