NAG Library Routine Document

D01AUF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     F – SUBROUTINE, supplied by the user.External Procedure

F must return the values of the integrand $f$ at a set of points.

The specification of F is:

SUBROUTINE F ( X, FV, N) INTEGER  N REAL (KIND=nag_wp)  X(N), FV(N)

1:     X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the points at which the integrand $f$ must be evaluated.

2:     FV(N) – REAL (KIND=nag_wp) arrayOutput

On exit: $\mathbf{FV}\left(\mathit{j}\right)$ must contain the value of $f$ at the point $\mathbf{X}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{N}$.

3:     N – INTEGERInput

On entry: the number of points at which the integrand is to be evaluated. The actual value of N is equal to the number of points in the Kronrod rule (see specification of KEY).

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D01AUF is called. Parameters denoted as Input must not be changed by this procedure.

2:     A – REAL (KIND=nag_wp)Input

On entry: $a$, the lower limit of integration.

3:     B – REAL (KIND=nag_wp)Input

On entry: $b$, the upper limit of integration. It is not necessary that $a<b$.

4:     KEY – INTEGERInput

On entry: indicates which integration rule is to be used.

$\mathbf{KEY}=1$

For the Gauss $7$-point and Kronrod $15$-point rule.

$\mathbf{KEY}=2$

For the Gauss $10$-point and Kronrod $21$-point rule.

$\mathbf{KEY}=3$

For the Gauss $15$-point and Kronrod $31$-point rule.

$\mathbf{KEY}=4$

For the Gauss $20$-point and Kronrod $41$-point rule.

$\mathbf{KEY}=5$

For the Gauss $25$-point and Kronrod $51$-point rule.

$\mathbf{KEY}=6$

For the Gauss $30$-point and Kronrod $61$-point rule.

Suggested value: $\mathbf{KEY}=6$.

Constraint: $\mathbf{KEY}=1$, $2$, $3$, $4$, $5$ or $6$.

5:     EPSABS – REAL (KIND=nag_wp)Input

On entry: the absolute accuracy required. If EPSABS is negative, the absolute value is used. See Section 7.

6:     EPSREL – REAL (KIND=nag_wp)Input

On entry: the relative accuracy required. If EPSREL is negative, the absolute value is used. See Section 7.

7:     RESULT – REAL (KIND=nag_wp)Output

On exit: the approximation to the integral $I$.

8:     ABSERR – REAL (KIND=nag_wp)Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $\left|I-\mathbf{RESULT}\right|$.

9:     W(LW) – REAL (KIND=nag_wp) arrayOutput

On exit: details of the computation see Section 8 for more information.

10:   LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which D01AUF is called. The value of LW (together with that of LIW) imposes a bound on the number of sub-intervals into which the interval of integration may be divided by the routine. The number of sub-intervals cannot exceed $\mathbf{LW}/4$. The more difficult the integrand, the larger LW should be.

Suggested value: $\mathbf{LW}=800$ to $2000$ is adequate for most problems.

Constraint: $\mathbf{LW}\ge 4$.

11:   IW(LIW) – INTEGER arrayOutput

On exit: $\mathbf{IW}\left(1\right)$ contains the actual number of sub-intervals used. The rest of the array is used as workspace.

12:   LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which D01AUF is called.

The number of sub-intervals into which the interval of integration may be divided cannot exceed LIW.

Suggested value: $\mathbf{LIW}=\mathbf{LW}/4$.

Constraint: $\mathbf{LIW}\ge 1$.

13:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ 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).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

The maximum number of subdivisions allowed with the given workspace has been reached without the accuracy requirements being achieved. Look at the integrand in order to determine the integration difficulties. If necessary, another integrator, which is designed for handling the type of difficulty involved, must be used. Alternatively, consider relaxing the accuracy requirements specified by EPSABS and EPSREL, or increasing the amount of workspace.

$\mathbf{IFAIL}=2$

Round-off error prevents the requested tolerance from being achieved. Consider requesting less accuracy.

$\mathbf{IFAIL}=3$

Extremely bad local integrand behaviour causes a very strong subdivision around one (or more) points of the interval. The same advice applies as in the case of $\mathbf{IFAIL}=\mathbf{1}$.

$\mathbf{IFAIL}=4$

On entry, $\mathbf{KEY}\ne 1$, $2$, $3$, $4$, $5$ or $6$.

$\mathbf{IFAIL}=5$

On entry,	$\mathbf{LW}<4$,
or	$\mathbf{LIW}<1$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d01aufe.f90)

9.2  Program Data

9.3  Program Results

Program Results (d01aufe.r)