NAG Library Routine Document

d01apf (dim1_fin_wsing)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{g}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

g must return the value of the function $g$ at a given point x.

The specification of g is:

Fortran Interface

Function g ( x) Real (Kind=nag_wp) :: g Real (Kind=nag_wp), Intent (In) :: x

C Header Interface

#include <nagmk26.h> double  g (const double *x)

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

On entry: the point at which the function $g$ must be evaluated.

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01apf is called. Arguments denoted as Input must not be changed by this procedure.

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01apf. If your code inadvertently does return any NaNs or infinities, d01apf is likely to produce unexpected results.

2:     $\mathbf{a}$ – Real (Kind=nag_wp)Input

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

3:     $\mathbf{b}$ – Real (Kind=nag_wp)Input

On entry: $b$, the upper limit of integration.

Constraint: $\mathbf{b}>\mathbf{a}$.

4:     $\mathbf{alfa}$ – Real (Kind=nag_wp)Input

On entry: the argument $\alpha$ in the weight function.

Constraint: $\mathbf{alfa}>-1.0$.

5:     $\mathbf{beta}$ – Real (Kind=nag_wp)Input

On entry: the argument $\beta$ in the weight function.

Constraint: $\mathbf{beta}>-1.0$.

6:     $\mathbf{key}$ – IntegerInput

On entry: indicates which weight function is to be used.

$\mathbf{key}=1$

$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }$.

$\mathbf{key}=2$

$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }\ln \left(x-a\right)$.

$\mathbf{key}=3$

$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }\ln \left(b-x\right)$.

$\mathbf{key}=4$

$w\left(x\right)={\left(x-a\right)}^{\alpha }{\left(b-x\right)}^{\beta }\ln \left(x-a\right)\ln \left(b-x\right)$.

Constraint: $\mathbf{key}=1$, $2$, $3$ or $4$.

7:     $\mathbf{epsabs}$ – Real (Kind=nag_wp)Input

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

8:     $\mathbf{epsrel}$ – Real (Kind=nag_wp)Input

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

9:     $\mathbf{result}$ – Real (Kind=nag_wp)Output

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

10:   $\mathbf{abserr}$ – Real (Kind=nag_wp)Output

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

11:   $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: details of the computation see Section 9 for more information.

12:   $\mathbf{lw}$ – IntegerInput

On entry: the dimension of the array w as declared in the (sub)program from which d01apf 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: $\mathbf{lw}\ge 8$.

13:   $\mathbf{iw}\left(\mathbf{liw}\right)$ – 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.

14:   $\mathbf{liw}$ – IntegerInput

On entry: the dimension of the array iw as declared in the (sub)program from which d01apf 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: $\mathbf{liw}\ge 2$.

15:   $\mathbf{ifail}$ – 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, because for this routine the values of the output arguments 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 the position of a discontinuity or a singularity of algebraico-logarithmic type within the interval can be determined, the interval must be split up at this point and the integrator called on the subranges. 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: $\mathbf{epsabs}=〈\mathit{\text{value}}〉$ and $\mathbf{epsrel}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=3$

Extremely bad integrand behaviour occurs around the sub-interval $\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$. The same advice applies as in the case of $\mathbf{ifail}=\mathbf{1}$.

$\mathbf{ifail}=4$

On entry, $\mathbf{alfa}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{alfa}>-1.0$.

On entry, $\mathbf{b}=〈\mathit{\text{value}}〉$ and $\mathbf{a}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{b}>\mathbf{a}$.

On entry, $\mathbf{beta}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{beta}>-1.0$.

On entry, $\mathbf{key}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{key}\le 4$.

On entry, $\mathbf{key}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{key}\ge 1$.

$\mathbf{ifail}=5$

On entry, $\mathbf{liw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{liw}\ge 2$.

On entry, $\mathbf{lw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lw}\ge 8$.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d01apfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (d01apfe.r)