@cindex quadrature @cindex numerical integration (quadrature) @cindex integration, numerical (quadrature) @cindex QUADPACK 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 @sc{quadpack}, a numerical integration package written by Piessens, Doncker-Kapenga, Uberhuber and Kahaner. Fortran code for @sc{quadpack} is available on Netlib. The functions described in this chapter are declared in the header file @file{gsl_integration.h}. @menu * Numerical Integration Introduction:: * QNG non-adaptive Gauss-Kronrod integration:: * QAG adaptive integration:: * QAGS adaptive integration with singularities:: * QAGP adaptive integration with known singular points:: * QAGI adaptive integration on infinite intervals:: * QAWC adaptive integration for Cauchy principal values:: * QAWS adaptive integration for singular functions:: * QAWO adaptive integration for oscillatory functions:: * QAWF adaptive integration for Fourier integrals:: * Numerical integration error codes:: * Numerical integration examples:: * Numerical integration References and Further Reading:: @end menu @node Numerical Integration Introduction @section Introduction Each algorithm computes an approximation to a definite integral of the form, @tex \beforedisplay $$ I = \int_a^b f(x) w(x) \,dx $$ \afterdisplay @end tex @ifinfo @example I = \int_a^b f(x) w(x) dx @end example @end ifinfo @noindent where @math{w(x)} is a weight function (for general integrands @math{w(x)=1}). The user provides absolute and relative error bounds @c{$(\hbox{\it epsabs}, \hbox{\it epsrel}\,)$} @math{(epsabs, epsrel)} which specify the following accuracy requirement, @tex \beforedisplay $$ |\hbox{\it RESULT} - I| \leq \max(\hbox{\it epsabs}, \hbox{\it epsrel}\, |I|) $$ \afterdisplay @end tex @ifinfo @example |RESULT - I| <= max(epsabs, epsrel |I|) @end example @end ifinfo @noindent where @c{$\hbox{\it RESULT}$} @math{RESULT} is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error @c{$\hbox{\it ABSERR} = |\hbox{\it RESULT} - I|$} @math{ABSERR = |RESULT - I|} in such a way that the following inequality holds, @tex \beforedisplay $$ |\hbox{\it RESULT} - I| \leq \hbox{\it ABSERR} \leq \max(\hbox{\it epsabs}, \hbox{\it epsrel}\,|I|) $$ \afterdisplay @end tex @ifinfo @example |RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|) @end example @end ifinfo @noindent 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 @sc{quadpack} use a naming convention based on the following letters, @display @code{Q} - quadrature routine @code{N} - non-adaptive integrator @code{A} - adaptive integrator @code{G} - general integrand (user-defined) @code{W} - weight function with integrand @code{S} - singularities can be more readily integrated @code{P} - points of special difficulty can be supplied @code{I} - infinite range of integration @code{O} - oscillatory weight function, cos or sin @code{F} - Fourier integral @code{C} - Cauchy principal value @end display @noindent 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. @menu * Integrands without weight functions:: * Integrands with weight functions:: * Integrands with singular weight functions:: @end menu @node Integrands without weight functions @subsection Integrands without weight functions @cindex Gauss-Kronrod quadrature 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 @math{m}. This is extended with additional points between each of the abscissae to give a higher order Kronrod rule of order @math{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. @node Integrands with weight functions @subsection Integrands with weight functions @cindex Clenshaw-Curtis quadrature @cindex Modified Clenshaw-Curtis quadrature For integrands with weight functions the algorithms use Clenshaw-Curtis quadrature rules. A Clenshaw-Curtis rule begins with an @math{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. @node Integrands with singular weight functions @subsection 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 @sc{quadpack} separate out several common weight functions which cause slow convergence. These weight functions are integrated analytically against the Chebyshev polynomials to precompute @dfn{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. @node QNG non-adaptive Gauss-Kronrod integration @section QNG non-adaptive Gauss-Kronrod integration @cindex QNG quadrature algorithm 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. @deftypefun int gsl_integration_qng (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, double * @var{result}, double * @var{abserr}, size_t * @var{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 @math{f} over @math{(a,b)} is achieved within the desired absolute and relative error limits, @var{epsabs} and @var{epsrel}. The function returns the final approximation, @var{result}, an estimate of the absolute error, @var{abserr} and the number of function evaluations used, @var{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. @end deftypefun @node QAG adaptive integration @section QAG adaptive integration @cindex QAG quadrature algorithm 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 @code{gsl_integration_workspace} struct, which handles the memory for the subinterval ranges, results and error estimates. @deftypefun {gsl_integration_workspace *} gsl_integration_workspace_alloc (size_t @var{n}) This function allocates a workspace sufficient to hold @var{n} double precision intervals, their integration results and error estimates. @end deftypefun @deftypefun void gsl_integration_workspace_free (gsl_integration_workspace * @var{w}) This function frees the memory associated with the workspace @var{w}. @end deftypefun @deftypefun int gsl_integration_qag (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, int @var{key}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function applies an integration rule adaptively until an estimate of the integral of @math{f} over @math{(a,b)} is achieved within the desired absolute and relative error limits, @var{epsabs} and @var{epsrel}. The function returns the final approximation, @var{result}, and an estimate of the absolute error, @var{abserr}. The integration rule is determined by the value of @var{key}, which should be chosen from the following symbolic names, @example 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) @end example @noindent 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 @var{workspace}. The maximum number of subintervals is given by @var{limit}, which may not exceed the allocated size of the workspace. @end deftypefun @node QAGS adaptive integration with singularities @section QAGS adaptive integration with singularities @cindex QAGS quadrature algorithm 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. @deftypefun int gsl_integration_qags (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function applies the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the integral of @math{f} over @math{(a,b)} is achieved within the desired absolute and relative error limits, @var{epsabs} and @var{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, @var{result}, and an estimate of the absolute error, @var{abserr}. The subintervals and their results are stored in the memory provided by @var{workspace}. The maximum number of subintervals is given by @var{limit}, which may not exceed the allocated size of the workspace. @end deftypefun @node QAGP adaptive integration with known singular points @section QAGP adaptive integration with known singular points @cindex QAGP quadrature algorithm @cindex singular points, specifying positions in quadrature @deftypefun int gsl_integration_qagp (const gsl_function * @var{f}, double * @var{pts}, size_t @var{npts}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array @var{pts} of length @var{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 @math{(a,b)} with break-points at @math{x_1, x_2, x_3} (where @math{a < x_1 < x_2 < x_3 < b}) the following @var{pts} array should be used @example pts[0] = a pts[1] = x_1 pts[2] = x_2 pts[3] = x_3 pts[4] = b @end example @noindent with @var{npts} = 5. @noindent If you know the locations of the singular points in the integration region then this routine will be faster than @code{QAGS}. @end deftypefun @node QAGI adaptive integration on infinite intervals @section QAGI adaptive integration on infinite intervals @cindex QAGI quadrature algorithm @deftypefun int gsl_integration_qagi (gsl_function * @var{f}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function computes the integral of the function @var{f} over the infinite interval @math{(-\infty,+\infty)}. The integral is mapped onto the semi-open interval @math{(0,1]} using the transformation @math{x = (1-t)/t}, @tex \beforedisplay $$ \int_{-\infty}^{+\infty} dx \, f(x) = \int_0^1 dt \, (f((1-t)/t) + f(-(1-t)/t))/t^2. $$ \afterdisplay @end tex @ifinfo @example \int_@{-\infty@}^@{+\infty@} dx f(x) = \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2. @end example @end ifinfo @noindent 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. @end deftypefun @deftypefun int gsl_integration_qagiu (gsl_function * @var{f}, double @var{a}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function computes the integral of the function @var{f} over the semi-infinite interval @math{(a,+\infty)}. The integral is mapped onto the semi-open interval @math{(0,1]} using the transformation @math{x = a + (1-t)/t}, @tex \beforedisplay $$ \int_{a}^{+\infty} dx \, f(x) = \int_0^1 dt \, f(a + (1-t)/t)/t^2 $$ \afterdisplay @end tex @ifinfo @example \int_@{a@}^@{+\infty@} dx f(x) = \int_0^1 dt f(a + (1-t)/t)/t^2 @end example @end ifinfo @noindent and then integrated using the QAGS algorithm. @end deftypefun @deftypefun int gsl_integration_qagil (gsl_function * @var{f}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function computes the integral of the function @var{f} over the semi-infinite interval @math{(-\infty,b)}. The integral is mapped onto the semi-open interval @math{(0,1]} using the transformation @math{x = b - (1-t)/t}, @tex \beforedisplay $$ \int_{-\infty}^{b} dx \, f(x) = \int_0^1 dt \, f(b - (1-t)/t)/t^2 $$ \afterdisplay @end tex @ifinfo @example \int_@{+\infty@}^@{b@} dx f(x) = \int_0^1 dt f(b - (1-t)/t)/t^2 @end example @end ifinfo @noindent and then integrated using the QAGS algorithm. @end deftypefun @node QAWC adaptive integration for Cauchy principal values @section QAWC adaptive integration for Cauchy principal values @cindex QAWC quadrature algorithm @cindex Cauchy principal value, by numerical quadrature @deftypefun int gsl_integration_qawc (gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{c}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function computes the Cauchy principal value of the integral of @math{f} over @math{(a,b)}, with a singularity at @var{c}, @tex \beforedisplay $$ I = \int_a^b dx\, {f(x) \over x - c} = \lim_{\epsilon \to 0} \left\{ \int_a^{c-\epsilon} dx\, {f(x) \over x - c} + \int_{c+\epsilon}^b dx\, {f(x) \over x - c} \right\} $$ \afterdisplay @end tex @ifinfo @example I = \int_a^b dx f(x) / (x - c) @end example @end ifinfo @noindent The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point @math{x = c}. When a subinterval contains the point @math{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. @end deftypefun @node QAWS adaptive integration for singular functions @section QAWS adaptive integration for singular functions @cindex QAWS quadrature algorithm @cindex singular functions, numerical integration of 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. @deftypefun {gsl_integration_qaws_table *} gsl_integration_qaws_table_alloc (double @var{alpha}, double @var{beta}, int @var{mu}, int @var{nu}) This function allocates space for a @code{gsl_integration_qaws_table} struct describing a singular weight function @math{W(x)} with the parameters @math{(\alpha, \beta, \mu, \nu)}, @tex \beforedisplay $$ W(x) = (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x) $$ \afterdisplay @end tex @ifinfo @example W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x) @end example @end ifinfo @noindent where @math{\alpha > -1}, @math{\beta > -1}, and @math{\mu = 0, 1}, @math{\nu = 0, 1}. The weight function can take four different forms depending on the values of @math{\mu} and @math{\nu}, @tex \beforedisplay $$ \matrix{ W(x) = (x - a)^\alpha (b - x)^\beta \hfill~ (\mu = 0, \nu = 0) \cr W(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) \hfill~ (\mu = 1, \nu = 0) \cr W(x) = (x - a)^\alpha (b - x)^\beta \log(b - x) \hfill~ (\mu = 0, \nu = 1) \cr W(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) \log(b - x) \hfill~ (\mu = 1, \nu = 1) } $$ \afterdisplay @end tex @ifinfo @example 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) @end example @end ifinfo @noindent The singular points @math{(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 @code{gsl_integration_qaws_table} if no errors were detected, and 0 in the case of error. @end deftypefun @deftypefun int gsl_integration_qaws_table_set (gsl_integration_qaws_table * @var{t}, double @var{alpha}, double @var{beta}, int @var{mu}, int @var{nu}) This function modifies the parameters @math{(\alpha, \beta, \mu, \nu)} of an existing @code{gsl_integration_qaws_table} struct @var{t}. @end deftypefun @deftypefun void gsl_integration_qaws_table_free (gsl_integration_qaws_table * @var{t}) This function frees all the memory associated with the @code{gsl_integration_qaws_table} struct @var{t}. @end deftypefun @deftypefun int gsl_integration_qaws (gsl_function * @var{f}, const double @var{a}, const double @var{b}, gsl_integration_qaws_table * @var{t}, const double @var{epsabs}, const double @var{epsrel}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr}) This function computes the integral of the function @math{f(x)} over the interval @math{(a,b)} with the singular weight function @math{(x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x)}. The parameters of the weight function @math{(\alpha, \beta, \mu, \nu)} are taken from the table @var{t}. The integral is, @tex \beforedisplay $$ I = \int_a^b dx\, f(x) (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x). $$ \afterdisplay @end tex @ifinfo @example I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x). @end example @end ifinfo @noindent 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. @end deftypefun @node QAWO adaptive integration for oscillatory functions @section QAWO adaptive integration for oscillatory functions @cindex QAWO quadrature algorithm @cindex oscillatory functions, numerical integration of The QAWO algorithm is designed for integrands with an oscillatory factor, @math{\sin(\omega x)} or @math{\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. @deftypefun {gsl_integration_qawo_table *} gsl_integration_qawo_table_alloc (double @var{omega}, double @var{L}, enum gsl_integration_qawo_enum @var{sine}, size_t @var{n}) This function allocates space for a @code{gsl_integration_qawo_table} struct and its associated workspace describing a sine or cosine weight function @math{W(x)} with the parameters @math{(\omega, L)}, @tex \beforedisplay $$ \eqalign{ W(x) & = \left\{\matrix{\sin(\omega x) \cr \cos(\omega x)} \right\} } $$ \afterdisplay @end tex @ifinfo @example W(x) = sin(omega x) W(x) = cos(omega x) @end example @end ifinfo @noindent The parameter @var{L} must be the length of the interval over which the function will be integrated @math{L = b - a}. The choice of sine or cosine is made with the parameter @var{sine} which should be chosen from one of the two following symbolic values: @example GSL_INTEG_COSINE GSL_INTEG_SINE @end example @noindent The @code{gsl_integration_qawo_table} is a table of the trigonometric coefficients required in the integration process. The parameter @var{n} determines the number of levels of coefficients that are computed. Each level corresponds to one bisection of the interval @math{L}, so that @var{n} levels are sufficient for subintervals down to the length @math{L/2^n}. The integration routine @code{gsl_integration_qawo} returns the error @code{GSL_ETABLE} if the number of levels is insufficient for the requested accuracy. @end deftypefun @deftypefun int gsl_integration_qawo_table_set (gsl_integration_qawo_table * @var{t}, double @var{omega}, double @var{L}, enum gsl_integration_qawo_enum @var{sine}) This function changes the parameters @var{omega}, @var{L} and @var{sine} of the existing workspace @var{t}. @end deftypefun @deftypefun int gsl_integration_qawo_table_set_length (gsl_integration_qawo_table * @var{t}, double @var{L}) This function allows the length parameter @var{L} of the workspace @var{t} to be changed. @end deftypefun @deftypefun void gsl_integration_qawo_table_free (gsl_integration_qawo_table * @var{t}) This function frees all the memory associated with the workspace @var{t}. @end deftypefun @deftypefun int gsl_integration_qawo (gsl_function * @var{f}, const double @var{a}, const double @var{epsabs}, const double @var{epsrel}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, gsl_integration_qawo_table * @var{wf}, double * @var{result}, double * @var{abserr}) This function uses an adaptive algorithm to compute the integral of @math{f} over @math{(a,b)} with the weight function @math{\sin(\omega x)} or @math{\cos(\omega x)} defined by the table @var{wf}, @tex \beforedisplay $$ \eqalign{ I & = \int_a^b dx\, f(x) \left\{ \matrix{\sin(\omega x) \cr \cos(\omega x)}\right\} } $$ \afterdisplay @end tex @ifinfo @example I = \int_a^b dx f(x) sin(omega x) I = \int_a^b dx f(x) cos(omega x) @end example @end ifinfo @noindent The results are extrapolated using the epsilon-algorithm to accelerate the convergence of the integral. The function returns the final approximation from the extrapolation, @var{result}, and an estimate of the absolute error, @var{abserr}. The subintervals and their results are stored in the memory provided by @var{workspace}. The maximum number of subintervals is given by @var{limit}, which may not exceed the allocated size of the workspace. Those subintervals with ``large'' widths @math{d} where @math{d\omega > 4} are computed using a 25-point Clenshaw-Curtis integration rule, which handles the oscillatory behavior. Subintervals with a ``small'' widths where @math{d\omega < 4} are computed using a 15-point Gauss-Kronrod integration. @end deftypefun @node QAWF adaptive integration for Fourier integrals @section QAWF adaptive integration for Fourier integrals @cindex QAWF quadrature algorithm @cindex Fourier integrals, numerical @deftypefun int gsl_integration_qawf (gsl_function * @var{f}, const double @var{a}, const double @var{epsabs}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, gsl_integration_workspace * @var{cycle_workspace}, gsl_integration_qawo_table * @var{wf}, double * @var{result}, double * @var{abserr}) This function attempts to compute a Fourier integral of the function @var{f} over the semi-infinite interval @math{[a,+\infty)}. @tex \beforedisplay $$ \eqalign{ I & = \int_a^{+\infty} dx\, f(x) \left\{ \matrix{ \sin(\omega x) \cr \cos(\omega x) } \right\} } $$ \afterdisplay @end tex @ifinfo @example I = \int_a^@{+\infty@} dx f(x) sin(omega x) I = \int_a^@{+\infty@} dx f(x) cos(omega x) @end example @end ifinfo The parameter @math{\omega} and choice of @math{\sin} or @math{\cos} is taken from the table @var{wf} (the length @var{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, @tex \beforedisplay $$ \eqalign{ C_1 & = [a, a + c] \cr C_2 & = [a + c, a + 2c] \cr \dots & = \dots \cr C_k & = [a + (k-1) c, a + k c] } $$ \afterdisplay @end tex @ifinfo @example C_1 = [a, a + c] C_2 = [a + c, a + 2 c] ... = ... C_k = [a + (k-1) c, a + k c] @end example @end ifinfo @noindent where @c{$c = (2 \,\hbox{floor}(|\omega|) + 1) \pi/|\omega|$} @math{c = (2 floor(|\omega|) + 1) \pi/|\omega|}. The width @math{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 @var{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 @var{abserr}. The following strategy is used: on each interval @math{C_k} the algorithm tries to achieve the tolerance @tex \beforedisplay $$ TOL_k = u_k \hbox{\it abserr} $$ \afterdisplay @end tex @ifinfo @example TOL_k = u_k abserr @end example @end ifinfo @noindent where @c{$u_k = (1 - p)p^{k-1}$} @math{u_k = (1 - p)p^@{k-1@}} and @math{p = 9/10}. The sum of the geometric series of contributions from each interval gives an overall tolerance of @var{abserr}. If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is relaxed, @tex \beforedisplay $$ TOL_k = u_k \max(\hbox{\it abserr}, \max_{i