Someone computes J= _0^1 x^ -4/5 x ,dx with a standard adaptive quadrature routine
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists
Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine
Problem
Someone computes J=∫01x−4/5lnxdx with a standard adaptive quadrature routine, asking for absolute and relative tolerances of 10−14. It returns J=−24.999999999999396 with an estimated error of 5.9×10−13, together with the warning 'The algorithm does not converge. Roundoff error is detected in the extrapolation table. It is assumed that the requested tolerance cannot be achieved, and that the returned result ... is the best which can be obtained.' They discard the number, arguing: 'a warning from the routine means the number cannot be trusted, so a different method must be used instead.'
((a)) Find the error in this argument.
((b)) Complete the audit: obtain the closed form of J by integration by parts, and state whether the quadrature routine's returned value should in fact be trusted, and to how many figures.