NAG Library Routine Document

d01rgf (dim1_fin_gonnet_vec)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

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

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

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

Note: if $\mathbf{a}=\mathbf{b}$, the routine will immediately return $\mathbf{dinest}=0.0$, $\mathbf{errest}=0.0$ and $\mathbf{nevals}=0$.

3:     $\mathbf{f}$ – Subroutine, supplied by the user.External Procedure

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

The specification of f is:

Fortran Interface

Subroutine f ( x, nx, fv, iflag, iuser, ruser) Integer, Intent (In) :: nx Integer, Intent (Inout) :: iflag, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(nx) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: fv(nx)

C Header Interface

#include <nagmk26.h> void  f (const double x[], const Integer *nx, double fv[], Integer *iflag, Integer iuser[], double ruser[])

1:     $\mathbf{x}\left(\mathbf{nx}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the abscissae, $x_i$, for $\mathit{i}=1,2,\dots ,\mathbf{nx}$, at which function values are required.

2:     $\mathbf{nx}$ – IntegerInput

On entry: the number of abscissae at which a function value is required.

3:     $\mathbf{fv}\left(\mathbf{nx}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: fv must contain the values of the integrand $f$. $\mathbf{fv}\left(i\right)=f\left(x_i\right)$ for all $i=1,2,\dots ,\mathbf{nx}$.

4:     $\mathbf{iflag}$ – IntegerInput/Output

On entry: $\mathbf{iflag}=0$.

On exit: set $\mathbf{iflag}<0$ to force an immediate exit with $\mathbf{ifail}=-\mathbf{1}$.

5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

f is called with the arguments iuser and ruser as supplied to d01rgf. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01rgf is called. Arguments denoted as Input must not be changed by this procedure.

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

On entry: the absolute accuracy required.

If epsabs is negative, $\left|\mathbf{epsabs}\right|$ is used. See Section 7.

If $\mathbf{epsabs}=0.0$, only the relative error will be used.

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

On entry: the relative accuracy required.

If epsrel is negative, $\left|\mathbf{epsrel}\right|$ is used. See Section 7.

If $\mathbf{epsrel}=0.0$, only the absolute error will be used otherwise the actual value of epsrel used by d01rgf is $\max \left(\mathit{machine precision},\left|\mathbf{epsrel}\right|\right)$.

Constraint: at least one of epsabs and epsrel must be nonzero.

6:     $\mathbf{dinest}$ – Real (Kind=nag_wp)Output

On exit: the estimate of the definite integral f.

7:     $\mathbf{errest}$ – Real (Kind=nag_wp)Output

On exit: the error estimate of the definite integral f.

8:     $\mathbf{nevals}$ – IntegerOutput

On exit: the total number of points at which the integrand, $f$, has been evaluated.

9:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

10:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by d01rgf, but are passed directly to f and may be used to pass information to this routine.

11:   $\mathbf{ifail}$ – 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 requested accuracy was not achieved. Consider using larger values of epsabs and epsrel.

$\mathbf{ifail}=2$

The integral is probably divergent or slowly convergent.

$\mathbf{ifail}=14$

Both $\mathbf{epsabs}=0.0$ and $\mathbf{epsrel}=0.0$.

$\mathbf{ifail}=-1$

Exit requested from f with $\mathbf{iflag}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=-99$

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 (d01rgfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (d01rgfe.r)