d01gaf integrates a function which is specified numerically at four or more points, over the whole of its specified range, using thirdorder finite difference formulae with error estimates, according to a method due to
Gill and Miller (1972).
d01gaf evaluates the definite integral
where the function
$y$ is specified at the
$n$points
${x}_{1},{x}_{2},\dots ,{x}_{n}$, which should be all distinct, and in either ascending or descending order. The integral between successive points is calculated by a fourpoint finite difference formula centred on the interval concerned, except in the case of the first and last intervals, where fourpoint forward and backward difference formulae respectively are employed. If
$n$ is less than
$4$, the routine fails. An approximation to the truncation error is integrated and added to the result. It is also returned separately to give an estimate of the uncertainty in the result. The method is due to
Gill and Miller (1972).
Gill P E and Miller G F (1972) An algorithm for the integration of unequally spaced data Comput. J. 15 80–83
 1: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the values of the independent variable, i.e., the ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
Constraint:
either${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{n}}\right)$ or ${\mathbf{x}}\left(1\right)>{\mathbf{x}}\left(2\right)>\cdots >{\mathbf{x}}\left({\mathbf{n}}\right)$.
 2: $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the values of the dependent variable
${y}_{\mathit{i}}$ at the points ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
 3: $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of points.
Constraint:
${\mathbf{n}}\ge 4$.
 4: $\mathbf{ans}$ – Real (Kind=nag_wp)Output

On exit: the estimated value of the integral.
 5: $\mathbf{er}$ – Real (Kind=nag_wp)Output

On exit: an estimate of the uncertainty in
ans.
 6: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
No error is reported arising from the relative magnitudes of
ans and
er on return, due to the difficulty when the true answer is zero.
No accuracy level is specified by you before calling
d01gaf but on return the absolute value of
er is an approximation to, but not necessarily a bound for,
$\leftI{\mathbf{ans}}\right$. If on exit
${\mathbf{ifail}}>{\mathbf{0}}$, both
ans and
er are returned as zero.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
In their paper,
Gill and Miller (1972) do not add the quantity
er to
ans before return. However, extensive tests have shown that a dramatic reduction in the error often results from such addition. In other cases, it does not make an improvement, but these tend to be cases of low accuracy in which the modified answer is not significantly inferior to the unmodified one. You have the option of recovering the Gill–Miller answer by subtracting
er from
ans on return from the routine.
This example evaluates the integral
reading in the function values at
$21$ unequally spaced points.