NAG FL Interface
d01auf (dim1_​fin_​osc_​vec)

Subroutine f(x,fv,nx)
Integer, Intent (In)         :: nx
Real (Kind=nag_wp), Intent (Out) :: fv(nx)
Real (Kind=nag_wp), Intent (In)  :: x(nx)

Subroutine f(x,nx,fv,iflag,iuser,ruser,cpuser)
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)
Type (c_ptr), Intent (In)          :: cpuser

Call d01auf(f,a,b,key,epsabs,epsrel,result,abserr,w,lw,iw,liw,ifail)

Call d01rkf(f,a,b,key,epsabs,epsrel,maxsub,result,abserr,rinfo,iinfo,iuser,ruser,cpuser,&
            ifail)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{f}$ – 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:

Fortran Interface copy

Subroutine f ( x, fv, n) Integer, Intent (In) :: n Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: fv(n)

C Header Interface copy

void  f (const double x[], double fv[], const Integer *n)

C++ Header Interface copy

#include <nag.h> extern "C" { void  f (const double x[], double fv[], const Integer &n) }

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

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

2: $\mathbf{fv}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

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: $\mathbf{n}$ – Integer Input

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. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01auf. If your code inadvertently does return any NaNs or infinities, d01auf 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. It is not necessary that $a<b$.

4: $\mathbf{key}$ – Integer Input

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

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

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

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

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

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

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

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

9: $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) array Output

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

10: $\mathbf{lw}$ – Integer Input

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: $\mathbf{iw}\left(\mathbf{liw}\right)$ – Integer array Output

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: $\mathbf{liw}$ – Integer Input

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: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when $\mathbf{ifail}\ne \mathbf{0}$ on exit. When the value $-\mathbf{1}$ 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).

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: $\mathbf{epsabs}=⟨\mathit{\text{value}}⟩$ and $\mathbf{epsrel}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=3$

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

$\mathbf{ifail}=4$

On entry, $\mathbf{key}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{key}\ge 1$ and $\mathbf{key}\le 6$.

$\mathbf{ifail}=5$

On entry, $\mathbf{liw}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{liw}\ge 1$.

On entry, $\mathbf{lw}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lw}\ge 4$.

$\mathbf{ifail}=-99$

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

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

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

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results