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This chapter describes routines for performing numerical integration (quadrature) of a function in one dimension. There are routines for adaptive and non-adaptive integration of general functions, with specialised routines for specific cases. These include integration over infinite and semi-infinite ranges, singular integrals, including logarithmic singularities, computation of Cauchy principal values and oscillatory integrals. The library reimplements the algorithms used in QUADPACK, a numerical integration package written by Piessens, Doncker-Kapenga, Uberhuber and Kahaner. Fortran code for QUADPACK is available on Netlib.

The functions described in this chapter are declared in the header file
``gsl_integration.h'`.

Each algorithm computes an approximation to a definite integral of the form,

I = \int_a^b f(x) w(x) dx

where w(x) is a weight function (for general integrands w(x)=1). The user provides absolute and relative error bounds (epsabs, epsrel) which specify the following accuracy requirement,

|RESULT - I| <= max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT - I| in such a way that the following inequality holds,

|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)

The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.

The algorithms in QUADPACK use a naming convention based on the following letters,

`Q`

- quadrature routine`N`

- non-adaptive integrator`A`

- adaptive integrator`G`

- general integrand (user-defined)`W`

- weight function with integrand`S`

- singularities can be more readily integrated`P`

- points of special difficulty can be supplied`I`

- infinite range of integration`O`

- oscillatory weight function, cos or sin`F`

- Fourier integral`C`

- Cauchy principal value

The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order rule is used to compute the best approximation to an integral over a small range. The difference between the results of the higher order rule and the lower order rule gives an estimate of the error in the approximation.

The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules. A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of order m. This is extended with additional points between each of the abscissae to give a higher order Kronrod rule of order 2m+1. The Kronrod rule is efficient because it reuses existing function evaluations from the Gaussian rule. The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the two rules is used as an estimate of the error in the approximation.

For integrands with weight functions the algorithms use Clenshaw-Curtis
quadrature rules. A Clenshaw-Curtis rule begins with an n-th
order Chebyshev polynomial approximation to the integrand. This
polynomial can be integrated exactly to give an approximation to the
integral of the original function. The Chebyshev expansion can be
extended to higher orders to improve the approximation. The presence of
singularities (or other behavior) in the integrand can cause slow
convergence in the Chebyshev approximation. The modified
Clenshaw-Curtis rules used in QUADPACK separate out several common
weight functions which cause slow convergence. These weight functions
are integrated analytically against the Chebyshev polynomials to
precompute **modified Chebyshev moments**. Combining the moments with
the Chebyshev approximation to the function gives the desired
integral. The use of analytic integration for the singular part of the
function allows exact cancellations and substantially improves the
overall convergence behavior of the integration.

The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.

__Function:__int**gsl_integration_qng***(const gsl_function **`f`, double`a`, double`b`, double`epsabs`, double`epsrel`, double *`result`, double *`abserr`, size_t *`neval`)-
This function applies the Gauss-Kronrod 10-point, 21-point, 43-point and 87-point integration rules in succession 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`, an estimate of the absolute error,`abserr`and the number of function evaluations used,`neval`. The Gauss-Kronrod rules are designed in such a way that each rule uses all the results of its predecessors, in order to minimize the total number of function evaluations.

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.

__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.

The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types of integrable singularities.

__Function:__int**gsl_integration_qags***(const gsl_function **`f`, double`a`, double`b`, double`epsabs`, double`epsrel`, size_t`limit`, gsl_integration_workspace *`workspace`, double *`result`, double *`abserr`)-
This function applies the Gauss-Kronrod 21-point 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 results are extrapolated using the epsilon-algorithm, which accelerates the convergence of the integral in the presence of discontinuities and integrable singularities. The function returns the final approximation from the extrapolation,`result`, and an estimate of the absolute error,`abserr`. 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.

__Function:__int**gsl_integration_qagp***(const gsl_function **`f`, double *`pts`, size_t`npts`, double`epsabs`, double`epsrel`, size_t`limit`, gsl_integration_workspace *`workspace`, double *`result`, double *`abserr`)-
This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array

`pts`of length`npts`should contain the endpoints of the integration ranges defined by the integration region and locations of the singularities. For example, to integrate over the region (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 < x_3 < b) the following`pts`array should be usedpts[0] = a pts[1] = x_1 pts[2] = x_2 pts[3] = x_3 pts[4] = b

with

`npts`= 5.If you know the locations of the singular points in the integration region then this routine will be faster than

`QAGS`

.

__Function:__int**gsl_integration_qagi***(gsl_function **`f`, double`epsabs`, double`epsrel`, size_t`limit`, gsl_integration_workspace *`workspace`, double *`result`, double *`abserr`)-
This function computes the integral of the function

`f`over the infinite interval (-\infty,+\infty). The integral is mapped onto the interval (0,1] using the transformation x = (1-t)/t,\int_{-\infty}^{+\infty} dx f(x) = \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.

It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case a lower-order rule is more efficient.

__Function:__int**gsl_integration_qagiu***(gsl_function **`f`, double`a`, double`epsabs`, double`epsrel`, size_t`limit`, gsl_integration_workspace *`workspace`, double *`result`, double *`abserr`)-
This function computes the integral of the function

`f`over the semi-infinite interval (a,+\infty). The integral is mapped onto the interval (0,1] using the transformation x = a + (1-t)/t,\int_{a}^{+\infty} dx f(x) = \int_0^1 dt f(a + (1-t)/t)/t^2

and then integrated using the QAGS algorithm.

__Function:__int**gsl_integration_qagil***(gsl_function **`f`, double`b`, double`epsabs`, double`epsrel`, size_t`limit`, gsl_integration_workspace *`workspace`, double *`result`, double *`abserr`)-
This function computes the integral of the function
`f`over the semi-infinite interval (-\infty,b). The integral is mapped onto the region (0,1] using the transformation x = b - (1-t)/t,\int_{+\infty}^{b} dx f(x) = \int_0^1 dt f(b - (1-t)/t)/t^2

and then integrated using the QAGS algorithm.

__Function:__int**gsl_integration_qawc***(gsl_function **`f`, double`a`, double`b`, double`c`, double`epsabs`, double`epsrel`, size_t`limit`, gsl_integration_workspace *`workspace`, double *`result`, double *`abserr`)-
This function computes the Cauchy principal value of the integral of f over (a,b), with a singularity at

`c`,I = \int_a^b dx f(x) / (x - c)

The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point x = c. When a subinterval contains the point x = c or is close to it then a special 25-point modified Clenshaw-Curtis rule is used to control the singularity. Further away from the singularity the algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.

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 and associated workspace 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

`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.

The QAWO algorithm is designed for integrands with an oscillatory factor, \sin(\omega x) or \cos(\omega x). In order to work efficiently the algorithm requires a table of Chebyshev moments which must be pre-computed with calls to the functions below.

__Function:__gsl_integration_qawo_table ***gsl_integration_qawo_table_alloc***(double*`omega`, double`L`, enum gsl_integration_qawo_enum`sine`, size_t`n`)-
This function allocates space for a

`gsl_integration_qawo_table`

struct and its associated workspace describing a sine or cosine weight function W(x) with the parameters (\omega, L),W(x) = sin(omega x) W(x) = cos(omega x)

The parameter

`L`must be the length of the interval over which the function will be integrated L = b - a. The choice of sine or cosine is made with the parameter`sine`which should be chosen from one of the two following symbolic values:GSL_INTEG_COSINE GSL_INTEG_SINE

The

`gsl_integration_qawo_table`

is a table of the trigonometric coefficients required in the integration process. The parameter`n`determines the number of levels of coefficients that are computed. Each level corresponds to one bisection of the interval L, so that`n`levels are sufficient for subintervals down to the length L/2^n. The integration routine`gsl_integration_qawo`

returns the error`GSL_ETABLE`

if the number of levels is insufficient for the requested accuracy.

__Function:__int**gsl_integration_qawo_table_set***(gsl_integration_qawo_table **`t`, double`omega`, double`L`, enum gsl_integration_qawo_enum`sine`)-
This function changes the parameters
`omega`,`L`and`sine`of the existing workspace`t`.

__Function:__int**gsl_integration_qawo_table_set_length***(gsl_integration_qawo_table **`t`, double`L`)-
This function allows the length parameter
`L`of the workspace`t`to be changed.

__Function:__void**gsl_integration_qawo_table_free***(gsl_integration_qawo_table **`t`)-
This function frees all the memory associated with the workspace
`t`.

__Function:__int**gsl_integration_qawo***(gsl_function **`f`, const double`a`, const double`epsabs`, const double`epsrel`, const size_t`limit`, gsl_integration_workspace *`workspace`, gsl_integration_qawo_table *`wf`, double *`result`, double *`abserr`)-
This function uses an adaptive algorithm to compute the integral of f over (a,b) with the weight function \sin(\omega x) or \cos(\omega x) defined by the table

`wf`,I = \int_a^b dx f(x) sin(omega x) I = \int_a^b dx f(x) cos(omega x)

The results are extrapolated using the epsilon-algorithm to accelerate the convergence of the integral. The function returns the final approximation from the extrapolation,

`result`, and an estimate of the absolute error,`abserr`. 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.Those subintervals with "large" widths d, d\omega > 4 are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a "small" width d\omega < 4 are computed using a 15-point Gauss-Kronrod integration.

__Function:__int**gsl_integration_qawf***(gsl_function **`f`, const double`a`, const double`epsabs`, const size_t`limit`, gsl_integration_workspace *`workspace`, gsl_integration_workspace *`cycle_workspace`, gsl_integration_qawo_table *`wf`, double *`result`, double *`abserr`)-
This function attempts to compute a Fourier integral of the function

`f`over the semi-infinite interval [a,+\infty).I = \int_a^{+\infty} dx f(x) sin(omega x) I = \int_a^{+\infty} dx f(x) cos(omega x)

The parameter \omega is taken from the table

`wf`(the length`L`can take any value, since it is overridden by this function to a value appropriate for the fourier integration). The integral is computed using the QAWO algorithm over each of the subintervals,C_1 = [a, a + c] C_2 = [a + c, a + 2 c] ... = ... C_k = [a + (k-1) c, a + k c]

where c = (2 floor(|\omega|) + 1) \pi/|\omega|. The width c is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when

`f`is positive and monotonically decreasing. The sum of this sequence of contributions is accelerated using the epsilon-algorithm.This function works to an overall absolute tolerance of

`abserr`. The following strategy is used: on each interval C_k the algorithm tries to achieve the toleranceTOL_k = u_k abserr

where u_k = (1 - p)p^{k-1} and p = 9/10. The sum of the geometric series of contributions from each interval gives an overall tolerance of

`abserr`.If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is relaxed,

TOL_k = u_k max(abserr, max_{i<k}{E_i})

where E_k is the estimated error on the interval C_k.

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. The integration over each subinterval uses the memory provided by`cycle_workspace`as workspace for the QAWO algorithm.

In addition to the standard error codes for invalid arguments the functions can return the following values,

`GSL_EMAXITER`

- the maximum number of subdivisions was exceeded.
`GSL_EROUND`

- cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.
`GSL_ESING`

- a non-integrable singularity or other bad integrand behavior was found in the integration interval.
`GSL_EDIVERGE`

- the integral is divergent, or too slowly convergent to be integrated numerically.

The integrator `QAGS`

will handle a large class of definite
integrals. For example, consider the following integral, which has a
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); return 0; }

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

bash$ ./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.

The following book is the definitive reference for QUADPACK, and was written by the original authors. It provides descriptions of the algorithms, program listings, test programs and examples. It also includes useful advice on numerical integration and many references to the numerical integration literature used in developing QUADPACK.

- R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, D.K. Kahaner. QUADPACK A subroutine package for automatic integration Springer Verlag, 1983.

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