d01da 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 points 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.

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

Note: d01da may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

The absolute accuracy is specified by the variable absacc. If, on exit, $\mathbf{ifail}=0$ then the result is most likely correct to this accuracy. Even if ifail is nonzero on exit, it is still possible that the calculated result could differ from the true value by less than the given accuracy.

None.

The time taken by d01da 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 method 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 method is not well-suited to non-smooth integrands, i.e., integrands having some kind of analytic discontinuity (such as a discontinuous or infinite partial derivative of some low-order) 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 d01da as but here the outer integral has an induced square-root singularity stemming from the way the region has been presented to d01da. This situation should be avoided by re-casting the problem. For the example given, the use of polar coordinates would avoid the difficulty:

This example evaluates the integral discussed in [Further Comments], presenting it to d01da first as and then as Note the difference in the number of function evaluations.