NAG C Library Function Document

nag_quad_1d_fin_gonnet_vec (d01rgc)


nag_quad_1d_fin_gonnet_vec (d01rgc) is a general purpose integrator which calculates an approximation to the integral of a function fx over a finite interval a,b:
I= ab fx dx .  
The function is suitable as a general purpose integrator, and can be used when the integrand has singularities and infinities. In particular, the function can continue if the function f explicitly returns a quiet or signalling NaN or a signed infinity.


#include <nag.h>
#include <nagd01.h>
void  nag_quad_1d_fin_gonnet_vec (double a, double b,
void (*f)(const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm),
double epsabs, double epsrel, double *dinest, double *errest, Integer *nevals, Nag_Comm *comm, NagError *fail)


nag_quad_1d_fin_gonnet_vec (d01rgc) uses the algorithm described in Gonnet (2010). It is an adaptive algorithm, similar to the QUADPACK routine QAGS (see Piessens et al. (1983), see also nag_quad_1d_gen_vec_multi_rcomm (d01rac)) but includes significant differences regarding how the integrand is represented, how the integration error is estimated and how singularities and divergent integrals are treated. The local error estimation is described in Gonnet (2010).
nag_quad_1d_fin_gonnet_vec (d01rgc) requires a function to evaluate the integrand at an array of different points and is therefore amenable to parallel execution.


Gonnet P (2010) Increasing the reliability of adaptive quadrature using explicit interpolants ACM Trans. Math. software 37 26
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag


1:     a doubleInput
On entry: a, the lower limit of integration.
2:     b doubleInput
On entry: b, the upper limit of integration. It is not necessary that a<b.
Note: if a=b, the function will immediately return dinest=0.0, errest=0.0 and nevals=0.
3:     f function, supplied by the userExternal Function
f must return the value of the integrand f at a set of points.
The specification of f is:
void  f (const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm)
1:     x[dim] const doubleInput
On entry: the abscissae, xi, for i=1,2,,nx, at which function values are required.
2:     nx IntegerInput
On entry: the number of abscissae at which a function value is required.
3:     fv[dim] doubleOutput
On exit: fv must contain the values of the integrand f. fv[i-1]=fxi for all i=1,2,,nx.
4:     iflag Integer *Input/Output
On entry: iflag=0.
On exit: set iflag<0 to force an immediate exit with fail.code= NE_USER_STOP.
5:     comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
The type Pointer will be void *. Before calling nag_quad_1d_fin_gonnet_vec (d01rgc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_quad_1d_fin_gonnet_vec (d01rgc) (see Section in How to Use the NAG Library and its Documentation).
4:     epsabs doubleInput
On entry: the absolute accuracy required.
If epsabs is negative, epsabs is used. See Section 7.
If epsabs=0.0, only the relative error will be used.
5:     epsrel doubleInput
On entry: the relative accuracy required.
If epsrel is negative, epsrel is used. See Section 7.
If epsrel=0.0, only the absolute error will be used otherwise the actual value of epsrel used by nag_quad_1d_fin_gonnet_vec (d01rgc) is maxmachine precision,epsrel.
Constraint: at least one of epsabs and epsrel must be nonzero.
6:     dinest double *Output
On exit: the estimate of the definite integral f.
7:     errest double *Output
On exit: the error estimate of the definite integral f.
8:     nevals Integer *Output
On exit: the total number of points at which the integrand, f, has been evaluated.
9:     comm Nag_Comm *
The NAG communication argument (see Section in How to Use the NAG Library and its Documentation).
10:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

The requested accuracy was not achieved. Consider using larger values of epsabs and epsrel.
Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
The integral is probably divergent or slowly convergent.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
Both epsabs=0.0 and epsrel=0.0.
Exit requested from f with iflag=value.


nag_quad_1d_fin_gonnet_vec (d01rgc) cannot guarantee, but in practice usually achieves, the following accuracy:
I-dinest tol ,  
tol = max epsabs , epsrel × I ,  
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity errest which, in normal circumstances, satisfies
I-dinest errest tol .  

Parallelism and Performance

nag_quad_1d_fin_gonnet_vec (d01rgc) is currently neither directly nor indirectly threaded. In particular, the user-supplied function f is not called from within a parallel region initialized inside nag_quad_1d_fin_gonnet_vec (d01rgc).
The user-supplied function f uses a vectorized interface, allowing for the required vector of function values to be evaluated in parallel; for example by placing appropriate OpenMP directives in the code for the user-supplied function f.

Further Comments

The time taken by nag_quad_1d_fin_gonnet_vec (d01rgc) depends on the integrand and the accuracy required.
nag_quad_1d_fin_gonnet_vec (d01rgc) is suitable for evaluating integrals that have singularities within the requested interval.
In particular, nag_quad_1d_fin_gonnet_vec (d01rgc) accepts non-finite values on return from the user-supplied function f, and will adapt the integration rule accordingly to eliminate such points. Non-finite values include NaNs and infinities.


This example computes
-1 1 sinx x ln101 - x .  

Program Text

Program Text (d01rgce.c)

Program Data


Program Results

Program Results (d01rgce.r)