I need to evaluate some (one-variable) integrals that neither SAGE nor Mathematica can do symbolically. As far as I can tell, I have two options:
(a) Use GSL (via SAGE), Maxima or Mathematica to do numerical integration. This is really a non-option, since, if I understand correctly, the "error bound" they give is not really a guarantee.
(b) Cobble together my own programs using the trapezoidal rule, Simpson's rule, etc., and get rigorous error bounds using bounds I have for the second (or fourth, or what have you) derivative of the function I am integrating. This is what I have been doing.
Is there a third option? Is there standard software that does (b) for me?
Best Answer
Interval arithmetic methods will permit rigorous bounds. You might try INTLAB. There are various books on rigorous numerics, e.g., Warwick Tucker's Validated Numerics, and the journal Reliable Computing is dedicated to such things.