# NAG FL Interfaced01akf (dim1_​fin_​osc)

Note: this routine is deprecated and will be withdrawn at Mark 31.3. Replaced by d01rkf.
d01rkf provides thread safety in passing of data to user-supplied function. d01rkf also requires the user-supplied subroutine f to calculate a vector of abscissae at once for greater efficiency, has an improved interface for setting the maximum number of subdivisions allowed (maxsub), and returns additional information on the computation (in the arrays rinfo and iinfo rather than w and iw previously).

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## 1Purpose

d01akf is an adaptive integrator, especially suited to oscillating, nonsingular integrands, which calculates an approximation to the integral of a function $f\left(x\right)$ over a finite interval $\left[a,b\right]$:
 $I= ∫ab f(x) dx .$

## 2Specification

Fortran Interface
 Subroutine d01akf ( f, a, b, w, lw, iw, liw,
 Integer, Intent (In) :: lw, liw Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iw(liw) Real (Kind=nag_wp), External :: f Real (Kind=nag_wp), Intent (In) :: a, b, epsabs, epsrel Real (Kind=nag_wp), Intent (Out) :: result, abserr, w(lw)
#include <nag.h>
 void d01akf_ (double (NAG_CALL *f)(const double *x),const double *a, const double *b, const double *epsabs, const double *epsrel, double *result, double *abserr, double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)
The routine may be called by the names d01akf or nagf_quad_dim1_fin_osc.

## 3Description

d01akf is based on the QUADPACK routine QAG (see Piessens et al. (1983)). It is an adaptive routine, using the Gauss $30$-point and Kronrod $61$-point rules. A ‘global’ acceptance criterion (as defined by Malcolm and Simpson (1976)) is used. The local error estimation is described in Piessens et al. (1983).
Because d01akf is based on integration rules of high order, it is especially suitable for nonsingular oscillating integrands.
d01akf requires you to supply a function to evaluate the integrand at a single point.
The routine d01auf uses an identical algorithm but requires you to supply a subroutine to evaluate the integrand at an array of points. Therefore, d01auf will be more efficient if the evaluation can be performed in vector mode on a vector-processing machine.

## 4References

Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
Piessens R (1973) An algorithm for automatic integration Angew. Inf. 15 399–401
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag

## 5Arguments

1: $\mathbf{f}$real (Kind=nag_wp) Function, supplied by the user. External Procedure
f must return the value of the integrand $f$ at a given point.
The specification of f is:
Fortran Interface
 Function f ( x)
 Real (Kind=nag_wp) :: f Real (Kind=nag_wp), Intent (In) :: x
 double f (const double *x)
1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the point at which the integrand $f$ must be evaluated.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d01akf 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 d01akf. If your code inadvertently does return any NaNs or infinities, d01akf 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.
4: $\mathbf{epsabs}$Real (Kind=nag_wp) Input
On entry: the absolute accuracy required. If epsabs is negative, the absolute value is used. See Section 7.
5: $\mathbf{epsrel}$Real (Kind=nag_wp) Input
On entry: the relative accuracy required. If epsrel is negative, the absolute value is used. See Section 7.
6: $\mathbf{result}$Real (Kind=nag_wp) Output
On exit: the approximation to the integral $I$.
7: $\mathbf{abserr}$Real (Kind=nag_wp) Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I-{\mathbf{result}}|$.
8: $\mathbf{w}\left({\mathbf{lw}}\right)$Real (Kind=nag_wp) array Output
On exit: details of the computation see Section 9 for more information.
9: $\mathbf{lw}$Integer Input
On entry: the dimension of the array w as declared in the (sub)program from which d01akf 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: ${\mathbf{lw}}\ge 4$.
10: $\mathbf{iw}\left({\mathbf{liw}}\right)$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.
11: $\mathbf{liw}$Integer Input
On entry: the dimension of the array iw as declared in the (sub)program from which d01akf 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: ${\mathbf{liw}}\ge 1$.
12: $\mathbf{ifail}$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 $0$ is recommended. 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
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 $\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$. The same advice applies as in the case of ${\mathbf{ifail}}={\mathbf{1}}$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{liw}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{liw}}\ge 1$.
On entry, ${\mathbf{lw}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lw}}\ge 4$.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

d01akf cannot guarantee, but in practice usually achieves, the following accuracy:
 $|I-result| ≤ tol ,$
where
 $tol=max{|epsabs|,|epsrel|×|I|} ,$
and epsabs and epsrel are user-specified absolute and relative error tolerances. Moreover, it returns the quantity abserr which, in normal circumstances, satisfies
 $|I-result|≤abserr≤tol.$

## 8Parallelism and Performance

d01akf is not threaded in any implementation.

The time taken by d01akf depends on the integrand and the accuracy required.
If ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, then you may wish to examine the contents of the array w, which contains the end points of the sub-intervals used by d01akf along with the integral contributions and error estimates over these sub-intervals.
Specifically, for $i=1,2,\dots ,n$, let ${r}_{i}$ denote the approximation to the value of the integral over the sub-interval $\left[{a}_{i},{b}_{i}\right]$ in the partition of $\left[a,b\right]$ and ${e}_{i}$ be the corresponding absolute error estimate. Then, $\underset{{a}_{i}}{\overset{{b}_{i}}{\int }}f\left(x\right)dx\simeq {r}_{i}$ and ${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$. The value of $n$ is returned in ${\mathbf{iw}}\left(1\right)$, and the values ${a}_{i}$, ${b}_{i}$, ${e}_{i}$ and ${r}_{i}$ are stored consecutively in the array w, that is:
• ${a}_{i}={\mathbf{w}}\left(i\right)$,
• ${b}_{i}={\mathbf{w}}\left(n+i\right)$,
• ${e}_{i}={\mathbf{w}}\left(2n+i\right)$ and
• ${r}_{i}={\mathbf{w}}\left(3n+i\right)$.

## 10Example

This example computes

### 10.1Program Text

Program Text (d01akfe.f90)

None.

### 10.3Program Results

Program Results (d01akfe.r)