NAG FL Interface
d01rlf (dim1_​fin_​brkpts)

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, 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

C Header Interface copy

void  f (const double x[], const Integer *nx, double fv[], Integer *iflag, Integer iuser[], double ruser[], void **cpuser)

C++ Header Interface copy

#include <nag.h> extern "C" { void  f (const double x[], const Integer &nx, double fv[], Integer &iflag, Integer iuser[], double ruser[], void *&cpuser) }

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

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

2: $\mathbf{nx}$ – Integer Input

On entry: the number of abscissae at which a function value is required. nx will be of size equal to the number of Kronrod points in the quadrature rule used, in this case $21$.

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

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: $\mathbf{iflag}$ – Integer Input/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 array User Workspace

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

7: $\mathbf{cpuser}$ – Type (c_ptr) User Workspace

f is called with the arguments iuser, ruser and cpuser as supplied to d01rlf. You should use the arrays iuser and ruser, and the data handle cpuser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01rlf 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 d01rlf. If your code inadvertently does return any NaNs or infinities, d01rlf 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 $a<b$.

Note: if $\mathbf{a}=\mathbf{b}$, the routine will immediately return with $\mathbf{result}=0.0$, $\mathbf{abserr}=0.0$, $\mathbf{rinfo}=0.0$ and $\mathbf{iinfo}=0$.

4: $\mathbf{npts}$ – Integer Input

On entry: the dimension of the array points as declared in the (sub)program from which d01rlf is called. The number of user-supplied break-points within the integration interval.

Constraint: $\mathbf{npts}\ge 0$.

5: $\mathbf{points}\left(\mathbf{npts}\right)$ – Real (Kind=nag_wp) array Input

On entry: the user-specified break-points.

Constraint: the break-points must all lie within the interval of integration (but may be supplied in any order).

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

On entry: ${\epsilon }_a$, the absolute accuracy required. If epsabs is negative, ${\epsilon }_a=\left|\mathbf{epsabs}\right|$. See Section 7.

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

On entry: ${\epsilon }_r$, the relative accuracy required. If epsrel is negative, ${\epsilon }_r=\left|\mathbf{epsrel}\right|$. See Section 7.

8: $\mathbf{maxsub}$ – Integer Input

On entry: ${\max }_{\mathit{sdiv}}$, the upper bound on the total number of subdivisions d01rlf may use to generate new segments. This does not include the initial segments of the interval generated from the supplied break-points. If ${\max }_{\mathit{sdiv}}=1$, only the initial segments will be evaluated.

Suggested value: a value in the range $200$ to $500$ is adequate for most problems.

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

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 $|I-\mathbf{result}|$.

11: $\mathbf{rinfo}\left(4\times \max (\mathbf{maxsub},\mathbf{npts})+\mathbf{npts}+6\right)$ – Real (Kind=nag_wp) array Output

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

12: $\mathbf{iinfo}\left(2\times \max (\mathbf{maxsub},\mathbf{npts})+\mathbf{npts}+4\right)$ – Integer array Output

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

13: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

14: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

15: $\mathbf{cpuser}$ – Type (c_ptr) User Workspace

iuser, ruser and cpuser are not used by d01rlf, but are passed directly to f and may be used to pass information to this routine. If you do not need to reference cpuser, it should be initialized to c_null_ptr.

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

On entry: ifail must be set to $0$, $-1$ 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 $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ 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 the position of a local difficulty within the interval can be determined (e.g., a singularity of the integrand or its derivative, a peak, a discontinuity, etc.) it should be supplied to the routine as an element of the vector points. 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 $\mathbf{ifail}=\mathbf{1}$.

$\mathbf{ifail}=4$

Round-off error is detected in the extrapolation table. The requested tolerance cannot be achieved because the extrapolation does not increase the accuracy satisfactorily; the returned result is the best that can be obtained. The same advice applies as in the case of $\mathbf{ifail}=\mathbf{1}$.

$\mathbf{ifail}=5$

The integral is probably divergent or slowly convergent.

$\mathbf{ifail}=41$

On entry, $\mathbf{npts}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{npts}\ge 0$.

$\mathbf{ifail}=51$

On entry, break-points are specified outside $(\mathbf{a},\mathbf{b})$: $\mathbf{a}=⟨\mathit{\text{value}}⟩$ and $\mathbf{b}=⟨\mathit{\text{value}}⟩$.

$\mathbf{ifail}=81$

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

$\mathbf{ifail}=-1$

Exit from f with $\mathbf{iflag}<0$.

$\mathbf{ifail}=-99$

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