### 17.3 QAG adaptive integration

The QAG algorithm is a simple adaptive integration procedure. The integration region is divided into subintervals, and on each iteration the subinterval with the largest estimated error is bisected. This reduces the overall error rapidly, as the subintervals become concentrated around local difficulties in the integrand. These subintervals are managed by a `gsl_integration_workspace` struct, which handles the memory for the subinterval ranges, results and error estimates.

Function: gsl_integration_workspace * gsl_integration_workspace_alloc (size_t n)

  This function allocates a workspace sufficient to hold n double precision intervals, their integration results and error estimates. One workspace may be used multiple times as all necessary reinitialization is performed automatically by the integration routines.

Function: void gsl_integration_workspace_free (gsl_integration_workspace * w)

  This function frees the memory associated with the workspace w.

Function: int gsl_integration_qag (const gsl_function * f, double a, double b, double epsabs, double epsrel, size_t limit, int key, gsl_integration_workspace * workspace, double * result, double * abserr)

  This function applies an integration rule adaptively until an estimate of the integral of f over (a,b) is achieved within the desired absolute and relative error limits, epsabs and epsrel. The function returns the final approximation, result, and an estimate of the absolute error, abserr. The integration rule is determined by the value of key, which should be chosen from the following symbolic names,

      GSL_INTEG_GAUSS15  (key = 1)
      GSL_INTEG_GAUSS21  (key = 2)
      GSL_INTEG_GAUSS31  (key = 3)
      GSL_INTEG_GAUSS41  (key = 4)
      GSL_INTEG_GAUSS51  (key = 5)
      GSL_INTEG_GAUSS61  (key = 6)

  corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod rules. The higher-order rules give better accuracy for smooth functions, while lower-order rules save time when the function contains local difficulties, such as discontinuities.

  On each iteration the adaptive integration strategy bisects the interval with the largest error estimate. The subintervals and their results are stored in the memory provided by workspace. The maximum number of subintervals is given by limit, which may not exceed the allocated size of the workspace.