# NAG CL Interfaced01rmc (dim1_​inf_​general)

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## 1Purpose

d01rmc calculates an approximation to the integral of a function $f\left(x\right)$ over an infinite or semi-infinite interval $\left[a,b\right]$:
 $I= ∫ab f(x) dx .$

## 2Specification

 #include
void  d01rmc (
 void (*f)(const double x[], Integer nx, double fv[], Integer *iflag, Nag_Comm *comm),
double bound, Integer inf, 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: d01rmc or nag_quad_dim1_inf_general.

## 3Description

d01rmc is based on the QUADPACK routine QAGI (see Piessens et al. (1983)). The entire infinite integration range is first transformed to $\left[0,1\right]$ using one of the identities:
 $∫ -∞ a f(x) dx = ∫01 f (a-1-tt) 1t2 dt$
 $∫a∞ f(x) dx = ∫01 f (a+1-tt) 1t2 dt$
 $∫ -∞ ∞ f(x) dx = ∫0∞ (f(x)+f(-x)) dx = ∫01 ​ ​ [f(1-tt)+f( -1+t t )] 1t2 dt ,$
where $a$ represents a finite integration limit. An adaptive procedure, based on the Gauss $7$-point and Kronrod $15$-point rules, is then employed on the transformed integral. 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).
d01rmc requires you to supply a function to evaluate the integrand at an array of points.
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 $15$.
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 d01rmc you may allocate memory and initialize these pointers with various quantities for use by f when called from d01rmc (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 d01rmc. If your code inadvertently does return any NaNs or infinities, d01rmc is likely to produce unexpected results.
2: $\mathbf{bound}$double Input
On entry: the finite limit of the integration range (if present). bound is not used if the interval is doubly infinite.
3: $\mathbf{inf}$Integer Input
On entry: indicates the kind of integration range.
${\mathbf{inf}}=1$
The range is $\left[{\mathbf{bound}},+\infty \right)$.
${\mathbf{inf}}=-1$
The range is $\left(-\infty ,{\mathbf{bound}}\right]$.
${\mathbf{inf}}=2$
The range is $\left(-\infty ,+\infty \right)$.
Constraint: ${\mathbf{inf}}=-1$, $1$ or $2$.
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 d01rmc 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.
On entry, ${\mathbf{inf}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{inf}}=-1$, $1$ or $2$.
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 d01rmc on the infinite subrange and an appropriate integrator on the finite subrange. 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

d01rmc 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

d01rmc is not threaded in any implementation.

The time taken by d01rmc 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 d01rmc 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 d01rmc 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
• ${r}_{i}={\mathbf{rinfo}}\left[3n+i-1\right]$.
Note:  this information applies to the integral transformed to $\left[0,1\right]$ as described in Section 3, not to the original integral.
The total number of abscissae at which the function was evaluated is returned in ${\mathbf{iinfo}}\left[1\right]$.

## 10Example

This example computes
 $∫ 0 ∞ 1 (x+1) x dx .$
The exact answer is $\pi$.

### 10.1Program Text

Program Text (d01rmce.c)

None.

### 10.3Program Results

Program Results (d01rmce.r)