NAG Library Function Document
nag_quad_2d_fin (d01dac)
1 Purpose
nag_quad_2d_fin (d01dac) attempts to evaluate a double integral to a specified absolute accuracy by repeated applications of the method described by
Patterson (1968) and
Patterson (1969).
2 Specification
#include <nag.h> 
#include <nagd01.h> 
void 
nag_quad_2d_fin (double ya,
double yb,
double 
(*f)(double x,
double y,
Nag_Comm *comm),


double absacc,
double *ans,
Integer *npts,
Nag_Comm *comm,
NagError *fail) 

3 Description
nag_quad_2d_fin (d01dac) attempts to evaluate a definite integral of the form
where
$a$ and
$b$ are constants and
${\varphi}_{1}\left(y\right)$ and
${\varphi}_{2}\left(y\right)$ are functions of the variable
$y$.
The integral is evaluated by expressing it as
Both the outer integral
$I$ and the inner integrals
$F\left(y\right)$ are evaluated by the method, described by
Patterson (1968) and
Patterson (1969), of the optimum addition of points to Gauss quadrature formulae.
This method uses a family of interlacing common point formulae. Beginning with the $3$point Gauss rule, formulae using $7$, $15$, $31$, $63$, $127$ and finally $255$ points are derived. Each new formula contains all the pivots of the earlier formulae so that no function evaluations are wasted. Each integral is evaluated by applying these formulae successively until two results are obtained which differ by less than the specified absolute accuracy.
4 References
Patterson T N L (1968) On some Gauss and Lobatto based integration formulae Math. Comput. 22 877–881
Patterson T N L (1969) The optimum addition of points to quadrature formulae, errata Math. Comput. 23 892
5 Arguments
 1:
ya – doubleInput
On entry: $a$, the lower limit of the integral.
 2:
yb – doubleInput
On entry: $b$, the upper limit of the integral. It is not necessary that $a<b$.
 3:
phi1 – function, supplied by the userExternal Function
phi1 must return the lower limit of the inner integral for a given value of
$y$.
The specification of
phi1 is:
double 
phi1 (double y,
Nag_Comm *comm)


 1:
y – doubleInput
On entry: the value of $y$ for which the lower limit must be evaluated.
 2:
comm – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
phi1.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling nag_quad_2d_fin (d01dac) you may allocate memory and initialize these pointers with various quantities for use by
phi1 when called from nag_quad_2d_fin (d01dac) (see
Section 3.2.1 in the Essential Introduction).
 4:
phi2 – function, supplied by the userExternal Function
phi2 must return the upper limit of the inner integral for a given value of
$y$.
The specification of
phi2 is:
double 
phi2 (double y,
Nag_Comm *comm)


 1:
y – doubleInput
On entry: the value of $y$ for which the upper limit must be evaluated.
 2:
comm – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
phi2.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling nag_quad_2d_fin (d01dac) you may allocate memory and initialize these pointers with various quantities for use by
phi2 when called from nag_quad_2d_fin (d01dac) (see
Section 3.2.1 in the Essential Introduction).
 5:
f – function, supplied by the userExternal Function
f must return the value of the integrand
$f$ at a given point.
The specification of
f is:
double 
f (double x,
double y,
Nag_Comm *comm)


 1:
x – doubleInput
 2:
y – doubleInput
On entry: the coordinates of the point $\left(x,y\right)$ at which the integrand $f$ must be evaluated.
 3:
comm – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
f.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling nag_quad_2d_fin (d01dac) you may allocate memory and initialize these pointers with various quantities for use by
f when called from nag_quad_2d_fin (d01dac) (see
Section 3.2.1 in the Essential Introduction).
 6:
absacc – doubleInput
On entry: the absolute accuracy requested.
 7:
ans – double *Output
On exit: the estimated value of the integral.
 8:
npts – Integer *Output
On exit: the total number of function evaluations.
 9:
comm – Nag_Comm *Communication Structure

The NAG communication argument (see
Section 3.2.1.1 in the Essential Introduction).
 10:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
 NE_BAD_PARAM
On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_CONVERGENCE
The outer integral has converged, but
$n$ of the inner integrals have not converged with
$255$ points:
$n=\u2329\mathit{\text{value}}\u232a$.
ans may still contain an approximate estimate of the integral, but its reliability will decrease as
$n$ increases.
The outer integral has not converged, and
$n$ of the inner integrals have not converged with
$255$ points:
$n=\u2329\mathit{\text{value}}\u232a$.
ans may still contain an approximate estimate of the integral, but its reliability will decrease as
$n$ increases.
The outer integral has not converged with
$255$ points. However, all the inner integrals have converged, and
ans may still contain an approximate estimate of the integral.
 NE_INTERNAL_ERROR
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.
7 Accuracy
The absolute accuracy is specified by the variable
absacc. If, on exit,
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\text{}$NE_NOERROR then the result is most likely correct to this accuracy. Even if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\text{}$NE_CONVERGENCE
on exit, it is still possible that the calculated result could differ from the true value by less than the given accuracy.
The time taken by nag_quad_2d_fin (d01dac) depends upon the complexity of the integrand and the accuracy requested.
With Patterson's method accidental convergence may occasionally occur, when two estimates of an integral agree to within the requested accuracy, but both estimates differ considerably from the true result. This could occur in either the outer integral or in one or more of the inner integrals.
If it occurs in the outer integral then apparent convergence is likely to be obtained with considerably fewer integrand evaluations than may be expected. If it occurs in an inner integral, the incorrect value could make the function $F\left(y\right)$ appear to be badly behaved, in which case a very large number of pivots may be needed for the overall evaluation of the integral. Thus both unexpectedly small and unexpectedly large numbers of integrand evaluations should be considered as indicating possible trouble. If accidental convergence is suspected, the integral may be recomputed, requesting better accuracy; if the new request is more stringent than the degree of accidental agreement (which is of course unknown), improved results should be obtained. This is only possible when the accidental agreement is not better than machine accuracy. It should be noted that the function requests the same accuracy for the inner integrals as for the outer integral. In practice it has been found that in the vast majority of cases this has proved to be adequate for the overall result of the double integral to be accurate to within the specified value.
The function is not wellsuited to nonsmooth integrands, i.e., integrands having some kind of analytic discontinuity (such as a discontinuous or infinite partial derivative of some loworder) in, on the boundary of, or near, the region of integration.
Warning: such singularities may be induced by incautiously presenting an apparently smooth interval over the positive quadrant of the unit circle,
$R$
This may be presented to nag_quad_2d_fin (d01dac) as
but here the outer integral has an induced squareroot singularity stemming from the way the region has been presented to nag_quad_2d_fin (d01dac). This situation should be avoided by recasting the problem. For the example given, the use of polar coordinates would avoid the difficulty:
9 Example
This example evaluates the integral discussed in
Section 8, presenting it to nag_quad_2d_fin (d01dac) first as
and then as
Note the difference in the number of function evaluations.
9.1 Program Text
Program Text (d01dace.c)
9.2 Program Data
Program Data (d01dace.d)
9.3 Program Results
Program Results (d01dace.r)