d01tac computes an estimate of the definite integral of a function of known analytical form, using a Gaussian quadrature formula with a specified number of abscissae. Formulae are provided for a finite interval (Gauss–Legendre), a semi-infinite interval (Gauss–Laguerre, rational Gauss), and an infinite interval (Gauss–Hermite).
The formula is appropriate for functions which can be well approximated by such a polynomial over $[a,b]$. It is inappropriate for functions with algebraic singularities at one or both ends of the interval, such as $\frac{1}{\sqrt{(1+x)}}$ on $[-1,1]$.
3.3One Limit Infinite
$${\int}_{a}^{\infty}f\left(x\right)dx\text{\hspace{1em} or \hspace{1em}}{\int}_{-\infty}^{a}f\left(x\right)dx\text{.}$$
Two quadrature formulae are available for these integrals:
(a)The Gauss–Laguerre formula is exact for any function of the form:
(b)This formula is appropriate for functions decaying exponentially at infinity; the argument $b$ should be chosen if possible to match the decay rate of the function.
(c)The rational Gauss formula is exact for any function of the form:
(d)This formula is likely to be more accurate for functions having only an inverse power rate of decay for large $x$. Here the choice of a suitable value of $b$ may be more difficult; unfortunately a poor choice of $b$ can make a large difference to the accuracy of the computed integral.
On entry: the point at which the integrand $f$ must be evaluated.
2: $\mathbf{comm}$ – Nag_User *
Pointer to a structure of type Nag_User with the following member:
p – Pointer
On entry/exit: the pointer $\mathbf{comm}\mathbf{\to}\mathbf{p}$ should be cast to the required type, e.g.,
struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01tac. If your code inadvertently does return any NaNs or infinities, d01tac is likely to produce unexpected results.
Some points to bear in mind when coding f are mentioned in Section 9.
3: $\mathbf{a}$ – doubleInput
4: $\mathbf{b}$ – doubleInput
On entry: the arguments $a$ and $b$ which occur in the integration formulae:
Gauss–Legendre: $a$ is the lower limit and $b$ is the upper limit of the integral. It is not necessary that $a<b$.
Rational Gauss: $b$ must be chosen so as to make the integrand match as closely as possible the exact form given in Section 3. The interval of integration is $[a,\infty )$ if $a+b>0$, and $(-\infty ,a]$ if $a+b<0$.
Gauss–Laguerre: $b$ must be chosen so as to make the integrand match as closely as possible the exact form given in Section 3. The interval of integration is $[a,\infty )$ if $b>0$, and $(-\infty ,a]$ if $b<0$.
Gauss–Hermite: $a$ and $b$ must be chosen so as to make the integrand match as closely as possible the exact form given in Section 3.
Pointer to a structure of type Nag_User with the following member:
p – Pointer
On entry/exit: the pointer $\mathbf{comm}\mathbf{\to}\mathbf{p}$, of type Pointer, allows you to communicate information to and from f(). An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer $\mathbf{comm}\mathbf{\to}\mathbf{p}$ by means of a cast to Pointer in the calling program, e.g., comm.p = (Pointer)&s. The type pointer will be void * with a C compiler that defines void * and char * otherwise.
7: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
Gauss–Hermite input is invalid with ${\mathbf{b}}\le 0.0$.
Constraint: ${\mathbf{b}}>0.0$.
Gauss–Laguerre input is invalid with ${\mathbf{b}}=0.0$.
Constraint: ${\mathbf{b}}\ne 0.0$.
Rational Gauss input is invalid with ${\mathbf{a}}+{\mathbf{b}}=0.0$.
Constraint: ${\mathbf{a}}+{\mathbf{b}}\ne 0.0$.
The answer is returned as zero.
NE_QUAD_GAUSS_NPTS_RULE
The $N$-point rule is not among those stored.
The answer is evaluated for $\u27e8\mathit{\text{value}}\u27e9$, the largest possible value of npts less than the requested value, $\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
The accuracy depends on the behaviour of the integrand, and on the number of abscissae used. No tests are carried out in d01tac to estimate the accuracy of the result. If such an estimate is required, the function may be called more than once, with a different number of abscissae each time, and the answers compared. It is to be expected that for sufficiently smooth functions a larger number of abscissae will give improved accuracy.
Alternatively, the interval of integration may be subdivided, the integral estimated separately for each sub-interval, and the sum of these estimates compared with the estimate over the whole interval.
The coding of the function f may also have a bearing on the accuracy. For example, if a high-order Gauss–Laguerre formula is used, and the integrand is of the form
$$f\left(x\right)={e}^{-bx}g\left(x\right)$$
it is possible that the exponential term may underflow for some large abscissae. Depending on the machine, this may produce an error, or simply be assumed to be zero. In any case, it would be better to evaluate the expression as:
The integrand here assumes very large values; for example, for $m=63$, the peak value exceeds $3\times {10}^{33}$. Now, if the machine holds floating-point numbers to an accuracy of $k$ significant decimal digits, we could not expect such terms to cancel in the summation leaving an answer of much less than ${10}^{33-k}$ (the weights being of order unity); that is instead of zero, we obtain a rather large answer through rounding error. Fortunately, such situations are characterised by great variability in the answers returned by formulae with different values of $n$. In general, you should be aware of the order of magnitude of the integrand, and should judge the answer in that light.
8Parallelism and Performance
d01tac is not threaded in any implementation.
9Further Comments
The time taken by d01tac depends on the complexity of the expression for the integrand and on the number of abscissae required.