NAG CL Interfaced01rjc (dim1_​fin_​general)

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1Purpose

d01rjc is a general purpose integrator 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

 #include
void  d01rjc (
 void (*f)(const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm),
double a, double b, double epsabs, double epsrel, Integer maxsub, double *result, double *abserr, double rinfo[], Integer iinfo[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: d01rjc or nag_quad_dim1_fin_general.

3Description

d01rjc is based on the QUADPACK routine QAGS (see Piessens et al. (1983)). It is an adaptive function, using the Gauss $10$-point and Kronrod $21$-point rules. The algorithm, described in de Doncker (1978), incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)) together with the $\epsilon$-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described in Piessens et al. (1983).
The function is suitable as a general purpose integrator, and can be used when the integrand has singularities, especially when these are of algebraic or logarithmic type.
d01rjc requires you to supply a function to evaluate the integrand at an array of points.

4References

de Doncker E (1978) An adaptive extrapolation algorithm for automatic integration ACM SIGNUM Newsl. 13(2) 12–18
Malcolm M A and Simpson R B (1976) Local versus global strategies for adaptive quadrature ACM Trans. Math. Software 1 129–146
Piessens R, de Doncker–Kapenga E, Überhuber C and Kahaner D (1983) QUADPACK, A Subroutine Package for Automatic Integration Springer–Verlag
Wynn P (1956) On a device for computing the ${e}_{m}\left({S}_{n}\right)$ transformation Math. Tables Aids Comput. 10 91–96

5Arguments

1: $\mathbf{f}$function, supplied by the user External Function
f must return the values of the integrand $f$ at a set of points.
The specification of f is:
 void f (const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm)
1: $\mathbf{x}\left[\mathit{dim}\right]$const double 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: $\mathbf{fv}\left[\mathit{dim}\right]$double Output
On exit: fv must contain the values of the integrand $f$. ${\mathbf{fv}}\left[i-1\right]=f\left({x}_{i}\right)$ 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
5: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d01rjc you may allocate memory and initialize these pointers with various quantities for use by f when called from d01rjc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d01rjc. If your code inadvertently does return any NaNs or infinities, d01rjc is likely to produce unexpected results.
2: $\mathbf{a}$double Input
On entry: $a$, the lower limit of integration.
3: $\mathbf{b}$double Input
On entry: $b$, the upper limit of integration. It is not necessary that $a.
Note: if ${\mathbf{a}}={\mathbf{b}}$, the function will immediately return with ${\mathbf{result}}=0.0$, ${\mathbf{abserr}}=0.0$, ${\mathbf{rinfo}}=0.0$ and ${\mathbf{iinfo}}=0$.
4: $\mathbf{epsabs}$double Input
On entry: ${\epsilon }_{a}$, the absolute accuracy required. If epsabs is negative, ${\epsilon }_{a}=|{\mathbf{epsabs}}|$. See Section 7.
5: $\mathbf{epsrel}$double Input
On entry: ${\epsilon }_{r}$, the relative accuracy required. If epsrel is negative, ${\epsilon }_{r}=|{\mathbf{epsrel}}|$. See Section 7.
6: $\mathbf{maxsub}$Integer Input
On entry: ${\mathrm{max}}_{\mathit{sdiv}}$, the upper bound on the total number of subdivisions d01rjc may use to generate new segments. If ${\mathrm{max}}_{\mathit{sdiv}}=1$, only the initial segment will be evaluated.
Suggested value: a value in the range $200$ to $500$ is adequate for most problems.
Constraint: ${\mathbf{maxsub}}\ge 1$.
7: $\mathbf{result}$double * Output
On exit: the approximation to the integral $I$.
8: $\mathbf{abserr}$double * Output
On exit: an estimate of the modulus of the absolute error, which should be an upper bound for $|I-{\mathbf{result}}|$.
9: $\mathbf{rinfo}\left[4×{\mathbf{maxsub}}\right]$double Output
On exit: details of the computation. See Section 9 for more information.
10: $\mathbf{iinfo}\left[\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{maxsub}},4\right)\right]$Integer Output
On exit: details of the computation. See Section 9 for more information.
11: $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
12: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{maxsub}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{maxsub}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_QUAD_MAX_SUBDIV.
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.) you will probably gain from splitting up the interval at this point and calling the integrator 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.
The integral is probably divergent or slowly convergent.
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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_QUAD_MAX_SUBDIV.
Round-off error prevents the requested tolerance from being achieved: ${\mathbf{epsabs}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{epsrel}}=⟨\mathit{\text{value}}⟩$.
NE_USER_STOP
Exit from f with ${\mathbf{iflag}}<0$.

7Accuracy

d01rjc 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

d01rjc is not threaded in any implementation.

The time taken by d01rjc depends on the integrand and the accuracy required.
If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, NE_QUAD_BAD_SUBDIV, NE_QUAD_MAX_SUBDIV, NE_QUAD_NO_CONV, NE_QUAD_ROUNDOFF_EXTRAPL or NE_QUAD_ROUNDOFF_TOL, or if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP and at least one complete vector evaluation of f was completed, result and abserr will contain computed results. If these results are unacceptable, or if otherwise required, then you may wish to examine the contents of the array rinfo, which contains the end points of the sub-intervals used by d01rjc along with the integral contributions and error estimates over the 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 ${\mathbf{result}}=\sum _{i=1}^{n}{r}_{i}$, unless d01rjc terminates while testing for divergence of the integral (see Section 3.4.3 of Piessens et al. (1983)). In this case, result (and abserr) are taken to be the values returned from the extrapolation process. This will be indicated by ${\mathbf{iinfo}}\left[2\right]=1$ instead of $0$. The value of $n$ is returned in ${\mathbf{iinfo}}\left[0\right]$, and the values ${a}_{i}$, ${b}_{i}$, ${e}_{i}$ and ${r}_{i}$ are stored consecutively in the array rinfo, that is:
• ${a}_{i}={\mathbf{rinfo}}\left[i-1\right]$,
• ${b}_{i}={\mathbf{rinfo}}\left[n+i-1\right]$,
• ${e}_{i}={\mathbf{rinfo}}\left[2n+i-1\right]$ and
• ${r}_{i}={\mathbf{rinfo}}\left[3n+i-1\right]$.
The total number of abscissae at which the function was evaluated is returned in ${\mathbf{iinfo}}\left[1\right]$.

10Example

This example computes
 $∫ 0 2π xsin(30x) 1- (x/2π) 2 dx .$

10.1Program Text

Program Text (d01rjce.c)

None.

10.3Program Results

Program Results (d01rjce.r)