Numerical Integration¶

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, de Doncker-Kapenga, Ueberhuber and Kahaner. Fortran code for QUADPACK is available on Netlib. Also included are non-adaptive, fixed-order Gauss-Legendre integration routines with high precision coefficients, as well as fixed-order quadrature rules for a variety of weighting functions from IQPACK.

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

Introduction¶

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,

$|RESULT - I| \leq \max{(epsabs, epsrel |I|)}$

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,

$|RESULT - I| \leq ABSERR \leq \max{(epsabs, epsrel |I|)}$

In short, the routines return the first approximation which has an absolute error smaller than $epsabs$ or a relative error smaller than $epsrel$ .

Note that this is an either-or constraint, not simultaneous. To compute to a specified absolute error, set $epsrel$ to zero. To compute to a specified relative error, set $epsabs$ to zero. 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.

Integrands without weight functions¶

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.

Integrands with weight functions¶

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 and provide an estimate of the error.

Integrands with singular weight functions¶

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.

QNG non-adaptive Gauss-Kronrod integration¶

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

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.

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 the following struct,

type gsl_integration_workspace¶: This workspace handles the memory for the subinterval ranges, results and error estimates.

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.

void gsl_integration_workspace_free(gsl_integration_workspace *w)¶: This function frees the memory associated with the workspace w.

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,

Symbolic Name	Key
`GSL_INTEG_GAUSS15`	1
`GSL_INTEG_GAUSS21`	2
`GSL_INTEG_GAUSS31`	3
`GSL_INTEG_GAUSS41`	4
`GSL_INTEG_GAUSS51`	5
`GSL_INTEG_GAUSS61`	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.

QAGS adaptive integration with singularities¶

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.

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.

QAGP adaptive integration with known singular points¶

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 used:

pts[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 gsl_integration_qags().

QAGI adaptive integration on infinite intervals¶

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 semi-open 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.

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 semi-open 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.

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 semi-open interval $(0,1]$ 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.

QAWC adaptive integration for Cauchy principal values¶

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,

$I = \int_a^b dx\, {f(x) \over x - c} = \lim_{\epsilon \to 0} \left\{ \int_a^{c-\epsilon} dx\, {f(x) \over x - c} + \int_{c+\epsilon}^b dx\, {f(x) \over x - c} \right\}$

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.

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.

type gsl_integration_qaws_table¶: This structure contains precomputed quantities for the QAWS algorithm.

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 $\nu$ ,

Weight function $w(x)$	$(\mu,\nu)$
$(x - a)^\alpha (b - x)^\beta$	$(0,0)$
$(x - a)^\alpha (b - x)^\beta \log{(x-a)}$	$(1,0)$
$(x - a)^\alpha (b - x)^\beta \log{(b-x)}$	$(0,1)$
$(x - a)^\alpha (b - x)^\beta \log{(x-a)} \log{(b-x)}$	$(1,1)$

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.

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.

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.

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.

QAWO adaptive integration for oscillatory functions¶

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.

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 $(\omega, L)$ ,

$w(x) = \left\{ \begin{array}{c} \sin{(\omega x)} \\ \cos{(\omega x)} \\ \end{array} \right\}$

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.

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.

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.

void gsl_integration_qawo_table_free(gsl_integration_qawo_table *t)¶: This function frees all the memory associated with the workspace t.

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,

$I = \int_a^b dx f(x) \left\{ \begin{array}{c} \sin{(\omega x)} \\ \cos{(\omega x)} \end{array} \right\}$

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 $d$ where $d\omega > 4$ are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a “small” widths where $d\omega < 4$ are computed using a 15-point Gauss-Kronrod integration.

QAWF adaptive integration for Fourier integrals¶

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 $[a,+\infty)$

$I = \int_a^{+\infty} dx f(x) \left\{ \begin{array}{c} \sin{(\omega x)} \\ \cos{(\omega x)} \end{array} \right\}$

The parameter $\omega$ and choice of $\sin$ or $\cos$ 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,

$C_1 &= [a,a+c] \\ C_2 &= [a+c,a+2c] \\ \dots &= \dots \\ C_k &= [a+(k-1)c,a+kc]$

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 tolerance

$TOL_k = u_k abserr$

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.

CQUAD doubly-adaptive integration¶

CQUAD is a new doubly-adaptive general-purpose quadrature routine which can handle most types of singularities, non-numerical function values such as Inf or NaN, as well as some divergent integrals. It generally requires more function evaluations than the integration routines in QUADPACK, yet fails less often for difficult integrands.

The underlying algorithm uses a doubly-adaptive scheme in which Clenshaw-Curtis quadrature rules of increasing degree are used to compute the integral in each interval. The $L_2$ -norm of the difference between the underlying interpolatory polynomials of two successive rules is used as an error estimate. The interval is subdivided if the difference between two successive rules is too large or a rule of maximum degree has been reached.

gsl_integration_cquad_workspace *gsl_integration_cquad_workspace_alloc(size_t n)¶: This function allocates a workspace sufficient to hold the data for n intervals. The number n is not the maximum number of intervals that will be evaluated. If the workspace is full, intervals with smaller error estimates will be discarded. A minimum of 3 intervals is required and for most functions, a workspace of size 100 is sufficient.

void gsl_integration_cquad_workspace_free(gsl_integration_cquad_workspace *w)¶: This function frees the memory associated with the workspace w.

int gsl_integration_cquad(const gsl_function *f, double a, double b, double epsabs, double epsrel, gsl_integration_cquad_workspace *workspace, double *result, double *abserr, size_t *nevals)¶

This function computes the integral of $f$ over $(a,b)$ within the desired absolute and relative error limits, epsabs and epsrel using the CQUAD algorithm. The function returns the final approximation, result, an estimate of the absolute error, abserr, and the number of function evaluations required, nevals.

The CQUAD algorithm divides the integration region into subintervals, and in each iteration, the subinterval with the largest estimated error is processed. The algorithm uses Clenshaw-Curtis quadrature rules of degree 4, 8, 16 and 32 over 5, 9, 17 and 33 nodes respectively. Each interval is initialized with the lowest-degree rule. When an interval is processed, the next-higher degree rule is evaluated and an error estimate is computed based on the $L_2$ -norm of the difference between the underlying interpolating polynomials of both rules. If the highest-degree rule has already been used, or the interpolatory polynomials differ significantly, the interval is bisected.

The subintervals and their results are stored in the memory provided by workspace. If the error estimate or the number of function evaluations is not needed, the pointers abserr and nevals can be set to NULL.

Romberg integration¶

The Romberg integration method estimates the definite integral

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

by applying Richardson extrapolation on the trapezoidal rule, using equally spaced points with spacing

$h_k = (b - a) 2^{-k}$

for $k = 1, \dots, n$ . For each $k$ , Richardson extrapolation is used $k-1$ times on previous approximations to improve the order of accuracy as much as possible. Romberg integration typically works well (and converges quickly) for smooth integrands with no singularities in the interval or at the end points.

gsl_integration_romberg_workspace *gsl_integration_romberg_alloc(const size_t n)¶: This function allocates a workspace for Romberg integration, specifying a maximum of $n$ iterations, or divisions of the interval. Since the number of divisions is $2^n + 1$ , $n$ can be kept relatively small (i.e. $10$ or $20$ ). It is capped at a maximum value of $30$ to prevent overflow. The size of the workspace is $O(2n)$ .

void gsl_integration_romberg_free(gsl_integration_romberg_workspace *w)¶: This function frees the memory associated with the workspace w.

int gsl_integration_romberg(const gsl_function *f, const double a, const double b, const double epsabs, const double epsrel, double *result, size_t *neval, gsl_integration_romberg_workspace *w)¶

This function integrates $f(x)$ , specified by f, from a to b, storing the answer in result. At each step in the iteration, convergence is tested by checking:

$| I_k - I_{k-1} | \le \textrm{max} \left( epsabs, epsrel \times |I_k| \right)$

where $I_k$ is the current approximation and $I_{k-1}$ is the approximation of the previous iteration. If the method does not converge within the previously specified $n$ iterations, the function stores the best current estimate in result and returns GSL_EMAXITER. If the method converges, the function returns GSL_SUCCESS. The total number of function evaluations is returned in neval.

Gauss-Legendre integration¶

The fixed-order Gauss-Legendre integration routines are provided for fast integration of smooth functions with known polynomial order. The $n$ -point Gauss-Legendre rule is exact for polynomials of order $2n - 1$ or less. For example, these rules are useful when integrating basis functions to form mass matrices for the Galerkin method. Unlike other numerical integration routines within the library, these routines do not accept absolute or relative error bounds.

gsl_integration_glfixed_table *gsl_integration_glfixed_table_alloc(size_t n)¶: This function determines the Gauss-Legendre abscissae and weights necessary for an $n$ -point fixed order integration scheme. If possible, high precision precomputed coefficients are used. If precomputed weights are not available, lower precision coefficients are computed on the fly.

double gsl_integration_glfixed(const gsl_function *f, double a, double b, const gsl_integration_glfixed_table *t)¶: This function applies the Gauss-Legendre integration rule contained in table t and returns the result.

int gsl_integration_glfixed_point(double a, double b, size_t i, double *xi, double *wi, const gsl_integration_glfixed_table *t)¶: For i in $[0, \dots, n - 1]$ , this function obtains the i-th Gauss-Legendre point xi and weight wi on the interval [a, b]. The points and weights are ordered by increasing point value. A function $f$ may be integrated on [a, b] by summing $wi * f(xi)$ over i.

void gsl_integration_glfixed_table_free(gsl_integration_glfixed_table *t)¶: This function frees the memory associated with the table t.

Fixed point quadratures¶

The routines in this section approximate an integral by the sum

$\int_a^b w(x) f(x) dx = \sum_{i=1}^n w_i f(x_i)$

where $f(x)$ is the function to be integrated and $w(x)$ is a weighting function. The $n$ weights $w_i$ and nodes $x_i$ are carefully chosen so that the result is exact when $f(x)$ is a polynomial of degree $2n - 1$ or less. Once the user chooses the order $n$ and weighting function $w(x)$ , the weights $w_i$ and nodes $x_i$ can be precomputed and used to efficiently evaluate integrals for any number of functions $f(x)$ .

This method works best when $f(x)$ is well approximated by a polynomial on the interval $(a,b)$ , and so is not suitable for functions with singularities. Since the user specifies ahead of time how many quadrature nodes will be used, these routines do not accept absolute or relative error bounds. The table below lists the weighting functions currently supported.

Name	Interval	Weighting function $w(x)$	Constraints
Legendre	$(a,b)$	$1$	$b > a$
Chebyshev Type 1	$(a,b)$	$1 / \sqrt{(b - x) (x - a)}$	$b > a$
Gegenbauer	$(a,b)$	$((b - x) (x - a))^{\alpha}$	$\alpha > -1, b > a$
Jacobi	$(a,b)$	$(b - x)^{\alpha} (x - a)^{\beta}$	$\alpha,\beta > -1, b > a$
Laguerre	$(a,\infty)$	$(x-a)^{\alpha} \exp{( -b (x - a) )}$	$\alpha > -1, b > 0$
Hermite	$(-\infty,\infty)$	$\|x-a\|^{\alpha} \exp{( -b (x-a)^2 )}$	$\alpha > -1, b > 0$
Exponential	$(a,b)$	$\|x - (a + b)/2\|^{\alpha}$	$\alpha > -1, b > a$
Rational	$(a,\infty)$	$(x - a)^{\alpha} (x + b)^{\beta}$	$\alpha > -1, \alpha + \beta + 2n < 0, a + b > 0$
Chebyshev Type 2	$(a,b)$	$\sqrt{(b - x) (x - a)}$	$b > a$

The fixed point quadrature routines use the following workspace to store the nodes and weights, as well as additional variables for intermediate calculations:

type gsl_integration_fixed_workspace¶

This workspace is used for fixed point quadrature rules and looks like this:

typedef struct
{
  size_t n;        /* number of nodes/weights */
  double *weights; /* quadrature weights */
  double *x;       /* quadrature nodes */
  double *diag;    /* diagonal of Jacobi matrix */
  double *subdiag; /* subdiagonal of Jacobi matrix */
  const gsl_integration_fixed_type * type;
} gsl_integration_fixed_workspace;

gsl_integration_fixed_workspace *gsl_integration_fixed_alloc(const gsl_integration_fixed_type *T, const size_t n, const double a, const double b, const double alpha, const double beta)¶

This function allocates a workspace for computing integrals with interpolating quadratures using n quadrature nodes. The parameters a, b, alpha, and beta specify the integration interval and/or weighting function for the various quadrature types. See the table above for constraints on these parameters. The size of the workspace is $O(4n)$ .

type gsl_integration_fixed_type¶

The type of quadrature used is specified by T which can be set to the following choices:

gsl_integration_fixed_type *gsl_integration_fixed_legendre¶: This specifies Legendre quadrature integration. The parameters alpha and beta are ignored for this type.

gsl_integration_fixed_type *gsl_integration_fixed_chebyshev¶: This specifies Chebyshev type 1 quadrature integration. The parameters alpha and beta are ignored for this type.

gsl_integration_fixed_type *gsl_integration_fixed_gegenbauer¶: This specifies Gegenbauer quadrature integration. The parameter beta is ignored for this type.

gsl_integration_fixed_type *gsl_integration_fixed_jacobi¶: This specifies Jacobi quadrature integration.

gsl_integration_fixed_type *gsl_integration_fixed_laguerre¶: This specifies Laguerre quadrature integration. The parameter beta is ignored for this type.

gsl_integration_fixed_type *gsl_integration_fixed_hermite¶: This specifies Hermite quadrature integration. The parameter beta is ignored for this type.

gsl_integration_fixed_type *gsl_integration_fixed_exponential¶: This specifies exponential quadrature integration. The parameter beta is ignored for this type.

gsl_integration_fixed_type *gsl_integration_fixed_rational¶: This specifies rational quadrature integration.

gsl_integration_fixed_type *gsl_integration_fixed_chebyshev2¶: This specifies Chebyshev type 2 quadrature integration. The parameters alpha and beta are ignored for this type.

void gsl_integration_fixed_free(gsl_integration_fixed_workspace *w)¶: This function frees the memory assocated with the workspace w

size_t gsl_integration_fixed_n(const gsl_integration_fixed_workspace *w)¶: This function returns the number of quadrature nodes and weights.

double *gsl_integration_fixed_nodes(const gsl_integration_fixed_workspace *w)¶: This function returns a pointer to an array of size n containing the quadrature nodes $x_i$ .

double *gsl_integration_fixed_weights(const gsl_integration_fixed_workspace *w)¶: This function returns a pointer to an array of size n containing the quadrature weights $w_i$ .

int gsl_integration_fixed(const gsl_function *func, double *result, const gsl_integration_fixed_workspace *w)¶

This function integrates the function $f(x)$ provided in func using previously computed fixed quadrature rules. The integral is approximated as

$\sum_{i=1}^n w_i f(x_i)$

where $w_i$ are the quadrature weights and $x_i$ are the quadrature nodes computed previously by gsl_integration_fixed_alloc(). The sum is stored in result on output.

Integrating on the unit sphere¶

This section contains routines to calculate the surface integral of a function over the unit sphere,

$I[f] = \int d\Omega f(\Omega) = \int_0^{\pi} \sin{\theta} d\theta \int_0^{2\pi} d\phi f(\theta,\phi)$

Lebedev developed a quadrature scheme to approximate this integral using a single sum,

$I[f] \approx 4 \pi \sum_{i=1}^n w_i f(\theta_i, \phi_i)$

for appropriately chosen weights $w_i$ and nodes $(\theta_i,\phi_i)$ . The Lebedev nodes are chosen to lie on the unit sphere and be invariant under the octahedral rotation group with inversion.

The number of quadrature nodes $n$ is often chosen in order to exactly integrate a certain degree spherical harmonic function $Y_l^m$ . A general rule of thumb for integrating spherical harmonics up to degree and order $L$ is to choose the number of nodes as,

$n \approx \frac{(L+1)^2}{3}$

Calculating the Lebedev weights and nodes requires solving a set of nonlinear equations. These equations have been solved, and the nodes and weights have been tabulated for integrating spherical harmonics up to degree and order 131. GSL offers a smaller subset of 32 quadrature rules, which are listed in the table below.

Spherical Harmonic degree $L$	Quadrature weights and nodes $n$
3	6
5	14
7	26
9	38
11	50
13	74
15	86
17	110
19	146
21	170
23	194
25	230
27	266
29	302
31	350
35	434
41	590
47	770
53	974
59	1202
65	1454
71	1730
77	2030
83	2354
89	2702
95	3074
101	3470
107	3890
113	4334
119	4802
125	5294
131	5810

type gsl_integration_lebedev_workspace¶

This workspace is used for Lebedev quadrature rules and looks like this:

typedef struct
{
  size_t n;        /* number of nodes/weights */
  double *weights; /* quadrature weights */
  double *x;       /* x quadrature nodes */
  double *y;       /* y quadrature nodes */
  double *z;       /* z quadrature nodes */
  double *theta;   /* theta quadrature nodes */
  double *phi;     /* phi quadrature nodes */
} gsl_integration_lebedev_workspace;

The arrays x, y, z of length $n$ contain the Cartesian coordinates $(x_i,y_i,z_i)$ of the Lebedev nodes which lie on the unit sphere. The arrays theta, phi contain the spherical coordinates $(\theta_i,\phi_i)$ of the same nodes on the unit sphere.

gsl_integration_lebedev_workspace *gsl_integration_lebedev_alloc(const size_t n)¶

This function allocates a workspace for a Lebedev quadrature rule of size n and computes the nodes and weights. The size of the workspace is $O(6n)$ .

If the input n does not match one of the rules in the table above, the error code GSL_EDOM is returned.

Here is some example code for integrating a function $f(\theta,\phi)$ with Lebedev quadrature:

const size_t n = 230; /* integrate exactly up to spherical harmonic degree 25 */
gsl_integration_lebedev_workspace * w = gsl_integration_lebedev_alloc(n);
double result = 0.0;
size_t i;

for (i = 0; i < n; ++i)
  result += w->weights[i] * f(w->theta[i], w->phi[i]);

result *= 4.0 * M_PI;
gsl_integration_lebedev_free(w);

void gsl_integration_lebedev_free(gsl_integration_lebedev_workspace *w)¶: This function frees the memory associated with the workspace w.

size_t gsl_integration_lebedev_n(const gsl_integration_lebedev_workspace *w)¶: This function returns the number of quadrature nodes associated with the workspace w.

Error codes¶

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.
`GSL_EDOM`	error in the values of the input arguments

Examples¶

Adaptive integration example¶

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 = &alpha;

  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       = %zu\n", w->size);

  gsl_integration_workspace_free (w);

  return 0;
}

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

result          = -4.000000000000085265
exact result    = -4.000000000000000000
estimated error =  0.000000000000135447
actual error    = -0.000000000000085265
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.

Fixed-point quadrature example¶

In this example, we use a fixed-point quadrature rule to integrate the integral

$\int_{-\infty}^{\infty} e^{-x^2} \left( x^m + 1 \right) dx = \left\{ \begin{array}{cc} \sqrt{\pi} + \Gamma{\left( \frac{m+1}{2} \right)}, & m \textrm{ even} \\ \sqrt{\pi}, & m \textrm{ odd} \end{array} \right.$

for integer $m$ . Consulting our table of fixed point quadratures, we see that this integral can be evaluated with a Hermite quadrature rule, setting $\alpha = 0, a = 0, b = 1$ . Since we are integrating a polynomial of degree $m$ , we need to choose the number of nodes $n \ge (m+1)/2$ to achieve the best results.

First we will try integrating for $m = 10, n = 5$ , which does not satisfy our criteria above:

$ ./integration2 10 5

The output is,

m               = 10
intervals       = 5
result          =  47.468529694563351029
exact result    =  54.115231635459025483
actual error    = -6.646701940895674454

So, we find a large error. Now we try integrating for $m = 10, n = 6$ which does satisfy the criteria above:

$ ./integration2 10 6

The output is,

m               = 10
intervals       = 6
result          =  54.115231635459096537
exact result    =  54.115231635459025483
actual error    =  0.000000000000071054

The program is given below.

#include <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>
#include <gsl/gsl_sf_gamma.h>

double
f(double x, void * params)
{
  int m = *(int *) params;
  double f = gsl_pow_int(x, m) + 1.0;
  return f;
}

int
main (int argc, char *argv[])
{
  gsl_integration_fixed_workspace * w;
  const gsl_integration_fixed_type * T = gsl_integration_fixed_hermite;
  int m = 10;
  int n = 6;
  double expected, result;
  gsl_function F;

  if (argc > 1)
    m = atoi(argv[1]);

  if (argc > 2)
    n = atoi(argv[2]);

  w = gsl_integration_fixed_alloc(T, n, 0.0, 1.0, 0.0, 0.0);

  F.function = &f;
  F.params = &m;

  gsl_integration_fixed(&F, &result, w);

  if (m % 2 == 0)
    expected = M_SQRTPI + gsl_sf_gamma(0.5*(1.0 + m));
  else
    expected = M_SQRTPI;

  printf ("m               = %d\n", m);
  printf ("intervals       = %zu\n", gsl_integration_fixed_n(w));
  printf ("result          = % .18f\n", result);
  printf ("exact result    = % .18f\n", expected);
  printf ("actual error    = % .18f\n", result - expected);

  gsl_integration_fixed_free (w);

  return 0;
}

References and Further Reading¶

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. Ueberhuber, D.K. Kahaner. QUADPACK A subroutine package for automatic integration Springer Verlag, 1983.

The CQUAD integration algorithm is described in the following paper:

P. Gonnet, “Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants”, ACM Transactions on Mathematical Software, Volume 37 (2010), Issue 3, Article 26.

The fixed-point quadrature routines are based on IQPACK, described in the following papers:

S. Elhay, J. Kautsky, Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature, ACM Transactions on Mathematical Software, Volume 13, Number 4, December 1987, pages 399-415.
J. Kautsky, S. Elhay, Calculation of the Weights of Interpolatory Quadratures, Numerische Mathematik, Volume 40, Number 3, October 1982, pages 407-422.

The Lebedev quadrature routines are based on the paper:

Lebedev, V. I. and Laikov, D. N. (1999). A quadrature formula for the sphere of the 131st algebraic order of accuracy. In Doklady Mathematics (Vol. 59, No. 3, pp. 477-481).