NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01rm: FL CL CPP AD

NAG CL Interface
d01rmc (dim1_inf_general)

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NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01rm: FL CL CPP AD

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2021

CL Name Style:
Short (a00aac)
Long (impl_details)
Full (nag_info_impl_details)

1 Purpose

d01rmc calculates an approximation to the integral of a function

f (x)

over an infinite or semi-infinite interval

[a, b]

:

I = \int_{a}^{b} f (x) d x .

2 Specification

#include <nag.h>

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.

3 Description

d01rmc is based on the QUADPACK routine QAGI (see Piessens et al. (1983)). The entire infinite integration range is first transformed to

[0, 1]

using one of the identities:

\int_{- \infty}^{a} f (x) d x = \int_{0}^{1} f (a - \frac{1 - t}{t}) \frac{1}{t^{2}} d t

\int_{a}^{\infty} f (x) d x = \int_{0}^{1} f (a + \frac{1 - t}{t}) \frac{1}{t^{2}} d t

\int_{- \infty}^{\infty} f (x) d x = \int_{0}^{\infty} (f (x) + f (- x)) d x = \int_{0}^{1} ​ ​ [f (\frac{1 - t}{t}) + f (\frac{- 1 + t}{t})] \frac{1}{t^{2}} d t,

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

ε

-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.

4 References

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} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

5 Arguments

1: $f$ – 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: $x [\dim]$ – const double Input

On entry: the abscissae,

x_{i}

, for

i = 1, 2, \dots, nx

, at which function values are required.

2: $nx$ – 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: $fv [\dim]$ – double Output

On exit: fv must contain the values of the integrand

f

.

fv [i - 1] = f (x_{i})

for all

i = 1, 2, \dots, nx

.

4: $iflag$ – Integer * Input/Output

On entry:

iflag = 0

.

On exit: set

iflag < 0

to force an immediate exit with

fail . code =

5: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to f.

user – double *
iuser – Integer *
p – Pointer: 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: $bound$ – double Input

On entry: the finite limit of the integration range (if present). bound is not used if the interval is doubly infinite.

3: $\inf$ – Integer Input

On entry: indicates the kind of integration range.

$\inf = 1$: The range is $[bound, + \infty)$ .
$\inf = −1$: The range is $(- \infty, bound]$ .
$\inf = 2$: The range is $(- \infty, + \infty)$ .

Constraint:

\inf = −1

,

1

or

2

.

4: $epsabs$ – double Input

On entry:

ε_{a}

, the absolute accuracy required. If epsabs is negative,

ε_{a} = | epsabs |

. See Section 7.

5: $epsrel$ – double Input

On entry:

ε_{r}

, the relative accuracy required. If epsrel is negative,

ε_{r} = | epsrel |

. See Section 7.

6: $maxsub$ – Integer Input

On entry:

\max_{sdiv}

, the upper bound on the total number of subdivisions d01rmc may use to generate new segments. If

\max_{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:

maxsub \geq 1

.

7: $result$ – double * Output

On exit: the approximation to the integral

I

.

8: $abserr$ – double * Output

On exit: an estimate of the modulus of the absolute error, which should be an upper bound for

| I - result |

.

9: $rinfo [4 \times maxsub]$ – double Output

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

10: $iinfo [\max (maxsub, 4)]$ – Integer Output

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

11: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.

On entry, $\inf = ⟨ value ⟩$ .
Constraint: $\inf = −1$ , $1$ or $2$ .
NE_INT: On entry, $maxsub = ⟨ value ⟩$ .
Constraint: $maxsub \geq 1$ .
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.
NE_QUAD_BAD_SUBDIV: Extremely bad integrand behaviour occurs around the sub-interval $(⟨ value ⟩, ⟨ value ⟩)$ . The same advice applies as in the case of $fail . code =$ NE_QUAD_MAX_SUBDIV.
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.
NE_QUAD_NO_CONV: The integral is probably divergent or slowly convergent.
NE_QUAD_ROUNDOFF_EXTRAPL: 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 $fail . code =$ NE_QUAD_MAX_SUBDIV.
NE_QUAD_ROUNDOFF_TOL: Round-off error prevents the requested tolerance from being achieved: $epsabs = ⟨ value ⟩$ and $epsrel = ⟨ value ⟩$ .
NE_USER_STOP: Exit from f with $iflag < 0$ .

7 Accuracy

d01rmc cannot guarantee, but in practice usually achieves, the following accuracy:

| I - result | \leq tol,

where

tol = \max {| epsabs |, | epsrel | \times | 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 | \leq abserr \leq tol .

8 Parallelism and Performance

d01rmc is not threaded in any implementation.

9 Further Comments

The time taken by d01rmc depends on the integrand and the accuracy required.

If

fail . 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

fail . 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

[a_{i}, b_{i}]

in the partition of

[a, b]

and

e_{i}

be the corresponding absolute error estimate. Then,

\int_{a_{i}}^{b_{i}} f (x) d x ≃ r_{i}

and

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

iinfo [2] = 1

instead of

0

. The value of

n

is returned in

iinfo [0]

, and the values

a_{i}

,

b_{i}

,

e_{i}

and

r_{i}

are stored consecutively in the array rinfo, that is:

$a_{i} = rinfo [i - 1]$ ,
$b_{i} = rinfo [n + i - 1]$ ,
$e_{i} = rinfo [2 n + i - 1]$ and
$r_{i} = rinfo [3 n + i - 1]$ .

Note: this information applies to the integral transformed to

[0, 1]

as described in Section 3, not to the original integral.

The total number of abscissae at which the function was evaluated is returned in

iinfo [1]

.

10 Example

This example computes

\int_{0}^{\infty} \frac{1}{(x + 1) \sqrt{x}} d x .

The exact answer is

π

.

10.1 Program Text

Program Text (d01rmce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (d01rmce.r)

NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01rm: FL CL CPP AD

© The Numerical Algorithms Group Ltd. 2021