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 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152  GNU Scientific Library – Reference Manual: QAWS adaptive integration for singular functions

17.8 QAWS adaptive integration for singular functions

The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region. In order to work efficiently the algorithm requires a precomputed table of Chebyshev moments.

Function: gsl_integration_qaws_table * gsl_integration_qaws_table_alloc (double alpha, double beta, int mu, int nu)

This function allocates space for a gsl_integration_qaws_table struct describing a singular weight function W(x) with the parameters (\alpha, \beta, \mu, \nu),

W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)

where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1. The weight function can take four different forms depending on the values of \mu and \nu,

W(x) = (x-a)^alpha (b-x)^beta                   (mu = 0, nu = 0) W(x) = (x-a)^alpha (b-x)^beta log(x-a)          (mu = 1, nu = 0) W(x) = (x-a)^alpha (b-x)^beta log(b-x)          (mu = 0, nu = 1) W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)

The singular points (a,b) do not have to be specified until the integral is computed, where they are the endpoints of the integration range.

The function returns a pointer to the newly allocated table gsl_integration_qaws_table if no errors were detected, and 0 in the case of error.

Function: int gsl_integration_qaws_table_set (gsl_integration_qaws_table * t, double alpha, double beta, int mu, int nu)

This function modifies the parameters (\alpha, \beta, \mu, \nu) of an existing gsl_integration_qaws_table struct t.

Function: void gsl_integration_qaws_table_free (gsl_integration_qaws_table * t)

This function frees all the memory associated with the gsl_integration_qaws_table struct t.

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

This function computes the integral of the function f(x) over the interval (a,b) with the singular weight function (x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x). The parameters of the weight function (\alpha, \beta, \mu, \nu) are taken from the table t. The integral is,

I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).

The adaptive bisection algorithm of QAG is used. When a subinterval contains one of the endpoints then a special 25-point modified Clenshaw-Curtis rule is used to control the singularities. For subintervals which do not include the endpoints an ordinary 15-point Gauss-Kronrod integration rule is used.