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The integrator `QAGS`

will handle a large class of definite
integrals. For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,

\int_0^1 x^{-1/2} log(x) dx = -4

The program below computes this integral to a relative accuracy bound of
`1e-7`

.

#include <stdio.h> #include <math.h> #include <gsl/gsl_integration.h> double f (double x, void * params) { double alpha = *(double *) params; double f = log(alpha*x) / sqrt(x); return f; } int main (void) { gsl_integration_workspace * w = gsl_integration_workspace_alloc (1000); double result, error; double expected = -4.0; double alpha = 1.0; gsl_function F; F.function = &f; F.params = α gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000, w, &result, &error); printf ("result = % .18f\n", result); printf ("exact result = % .18f\n", expected); printf ("estimated error = % .18f\n", error); printf ("actual error = % .18f\n", result - expected); printf ("intervals = %d\n", w->size); gsl_integration_workspace_free (w); return 0; }

The results below show that the desired accuracy is achieved after 8 subdivisions.

$ ./a.out

result = -3.999999999999973799 exact result = -4.000000000000000000 estimated error = 0.000000000000246025 actual error = 0.000000000000026201 intervals = 8

In fact, the extrapolation procedure used by `QAGS`

produces an
accuracy of almost twice as many digits. The error estimate returned by
the extrapolation procedure is larger than the actual error, giving a
margin of safety of one order of magnitude.