d01ra: FL CL CPP AD PY MB

NAG CL Interface
d01rac (dim1_gen_vec_multi_rcomm)

Note: this function uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the specification of the optional parameters.

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NAG Library Manual, Mark 29.3

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

D01 (Quad) Chapter Contents

D01 (Quad) Chapter Introduction

d01ra: FL CL CPP AD PY MB

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

▸▿ 9 Further Comments

9.1 Details of the Computation

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

▸▿ 11 Optional Parameters

11.1 Description of the Optional Parameters

11.2 Diagnostic Options

Settings help

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

1 Purpose

d01rac is a general purpose adaptive integrator which calculates an approximation to a vector of definite integrals

F

over a finite range

[a, b]

, given the vector of integrands

f (x)

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

If the same subdivisions of the range are equally good for functions

f_{1} (x)

and

f_{2} (x)

, because

f_{1} (x)

and

f_{2} (x)

have common areas of the range where they vary slowly and where they vary quickly, then we say that

f_{1} (x)

and

f_{2} (x)

are ‘similar’. d01rac is particularly effective for the integration of a vector of similar functions.

2 Specification

#include <nag.h>

void

d01rac (Integer *irevcm, Integer ni, double a, double b, Integer *sid, Integer needi[], double x[], Integer lenx, Integer *nx, const double fm[], Integer ldfm, double dinest[], double errest[], const Integer iopts[], const double opts[], Integer icom[], Integer licom, double com[], Integer lcom, NagError *fail)

The function may be called by the names: d01rac, nag_quad_dim1_gen_vec_multi_rcomm or nag_quad_1d_gen_vec_multi_rcomm.

3 Description

d01rac is an extension to various QUADPACK routines, including QAG, QAGS and QAGP. The extensions made allow multiple integrands to be evaluated simultaneously, using a vectorized interface and reverse communication.

The quadrature scheme employed by d01rac can be chosen by you. Six Gauss–Kronrod schemes are available. The algorithm incorporates a global acceptance criterion (as defined by Malcolm and Simpson (1976)), optionally together with the ε-algorithm (see Wynn (1956)) to perform extrapolation. The local error estimation is described in Piessens et al. (1983).

d01rac is the integration function in the suite of functions d01rac and d01rcc. It also uses optional parameters, which can be set and queried using the functions d01zkc and d01zlc respectively. The options available are described in Section 11.

The algorithm used by d01rac is divided into an initial solve phase and an adaptive solve phase.

During the initial solve phase, the first estimation of the definite integral and error estimate is constructed over the interval

[a, b]

. This will have been divided into

s_{pri}

level

1

segments, where

s_{pri}

is the number of

Primary Divisions

, and will use any provided break-points if

Primary Division Mode = MANUAL

Once a complete integral estimate over

[a, b]

is available, i.e., after all the estimates for the level

1

segments have been evaluated, the function enters the adaptive phase. The estimated errors are tested against the requested tolerances

ε_{a}

and

ε_{r}

, corresponding to the

Absolute Tolerance

and

Relative Tolerance

respectively. Should this test fail, and additional subdivision be allowed, a segment is selected for subdivision, and is subsequently replaced by two new segments at the next level of refinement. How this segment is chosen may be altered by setting

Prioritize Error

to either favour the segment with the maximum error, or the segment with the lowest level supporting an unacceptable (although potentially non-maximal) error. Up to

\max_{sdiv}

subdivisions are allowed if sufficient memory is provided, where

\max_{sdiv}

is the value of

Maximum Subdivisions

Once a sufficient number of error estimates have been constructed for a particular integral, the function may optionally use

Extrapolation

to attempt to accelerate convergence. This may significantly lower the amount of work required for a given integration. To minimize the risk of premature convergence from extrapolation, a safeguard

ε_{safe}

can be set using

Extrapolation Safeguard

, and the extrapolated solution will only be considered if

ε_{safe} ε_{q} \leq ε_{ex}

, where

ε_{q}

and

ε_{ex}

are the estimated error directly from the quadrature and from the extrapolation respectively. If extrapolation is successful for the computation of integral

j

, the extrapolated solution will be returned in

dinest [j - 1]

on completion of d01rac. Otherwise the direct solution will be returned in

dinest [j - 1]

. This is indicated by the value of

needi [j - 1]

on completion.

4 References

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

Wynn P (1956) On a device for computing the

e_{m} (S_{n})

transformation Math. Tables Aids Comput. 10 91–96

5 Arguments

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than irevcm, needi and fm must remain unchanged.

where

FM (j, i)

appears in this document it refers to the array element

fm [(i - 1) \times ldfm + j - 1]

1: $irevcm$ – Integer * Input/Output

On initial entry:

irevcm = 1

$irevcm = 1$: Sets up data structures in icom and com and starts a new integration.

Constraint:

irevcm = 1

on initial entry.

On intermediate exit:

irevcm = 11

12

irevcm requests the integrands

f_{j} (x_{i})

be evaluated for all required

j \in 1, \dots, n_{i}

as indicated by needi, and at all the points

x_{i}

, for

i = 1, 2, \dots, n_{x}

. Abscissae

x_{i}

are provided in

x [i - 1]

and

f_{j} (x_{i})

must be returned in

FM (j, i)

During the initial solve phase:

$irevcm = 11$: Function values are required to construct the initial estimates of the definite integrals.
If $needi [j - 1] = 1$ , $f_{j} (x_{i})$ must be supplied in $FM (j, i)$ . This will be the case unless you have abandoned the evaluation of specific integrals on a previous call.

If $needi [j - 1] = 0$ , you have previously abandoned the evaluation of integral $j$ , and hence should not supply the value of $f_{j} (x_{i})$ .

dinest and errest contain incomplete information during this phase. As such you should not abandon the evaluation of any integrals during this phase unless you do not require their estimate.

If irevcm is set to a negative value during this phase, $needi [j - 1]$ , for $j = 1, 2, \dots, n_{i}$ , will be set to this negative value and $fail . code =$ NE_USER_STOP will be returned.

During the adaptive solve phase:

$irevcm = 12$: Function values are required to improve the estimates of the definite integrals.
If $needi [j - 1] = 0$ , any evaluation of $f_{j} (x_{i})$ will be discarded, so there is no need to provide them.

If $needi [j - 1] = 1$ , $f_{j} (x_{i})$ must be provided in $FM (j, i)$ .

If $needi [j - 1] = 2$ , $3$ or $4$ , the current error estimate of integral $j$ does not require integrand $j$ to be evaluated and provided in $FM (j, i)$ . Should you choose to, integrand $j$ can be evaluated in which case $needi [j - 1]$ must be set to $1$ .

dinest and errest contain complete information during this phase.

If irevcm is set to a negative value during this phase $fail . code =$ NE_ACCURACY or NE_QUAD_BAD_SUBDIV_INT will be returned and the elements of needi will reflect the current state of the adaptive process.

On intermediate re-entry: irevcm should normally be left unchanged. However, if irevcm is set to a negative value, d01rac will terminate, (see

irevcm = 11

and

irevcm = 12

above).

On final exit:

irevcm = 0

$irevcm = 0$: Indicates the algorithm has completed.

Note: any values you return to d01rac as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by d01rac. If your code inadvertently does return any NaNs or infinities, d01rac is likely to produce unexpected results.

2: $ni$ – Integer Input

On entry:

n_{i}

, the number of integrands.

3: $a$ – double Input

On entry:

a

, the lower bound of the domain.

4: $b$ – double Input

On entry:

b

, the upper bound of the domain.

| b - a | < 10 ε

, where

ε

is the machine precision (see X02AJC), d01rac will return

dinest [j - 1] = errest [j - 1] = 0.0

, for

j = 1, 2, \dots, n_{i}

5: $sid$ – Integer * Output

For advanced users.

On intermediate exit: sid identifies a specific set of abscissae,

x

, returned during the integration process. When a new set of abscissae are generated the value of sid is incremented by

1

. Advanced users may store calculations required for an identified set

x

, and reuse them should d01rac return the same value of sid, i.e., the same set of abscissae was used.

6: $needi [ni]$ – Integer Input/Output

On initial entry: need not be set.

On intermediate exit:

needi [j - 1]

indicates what action must be taken for integral

j = 1, 2, \dots n_{i}

(see irevcm).

$needi [j - 1] = 0$: Do not provide $f_{j} (x_{i})$ . Any provided values will be ignored.
$needi [j - 1] = 1$: The values $f_{j} (x_{i})$ must be provided in $FM (j, i)$ , for $i = 1, 2, \dots, n_{x}$ .
$needi [j - 1] = 2$: The values $f_{j} (x_{i})$ are not required, however the error estimate for integral $j$ is still above the requested tolerance. If you wish to provide values for the evaluation of integral $j$ , set $needi [j - 1] = 1$ , and supply $f_{j} (x_{i})$ in $FM (j, i)$ , for $i = 1, 2, \dots, n_{x}$ .
$needi [j - 1] = 3$: The error estimate for integral $j$ cannot be improved to below the requested tolerance directly, either because no more new splits may be performed due to exhaustion, or due to the detection of extremely bad integrand behaviour. However, providing the values $f_{j} (x_{i})$ may still lead to some improvement, and may lead to an acceptable error estimate indirectly using Wynn's epsilon algorithm. If you wish to provide values for the evaluation of integral $j$ , set $needi [j - 1] = 1$ , and supply $f_{j} (x_{i})$ in $FM (j, i)$ , for $i = 1, 2, \dots, n_{x}$ .
$needi [j - 1] = 4$: The error estimate of integral $j$ is below the requested tolerance. If you believe this to be false, if for example the result in $dinest [j - 1]$ is greatly different to what you may expect, you may force the algorithm to re-evaluate this conclusion by including the evaluations of integrand $j$ at $x_{i}$ , for $i = 1, 2, \dots, n_{x}$ , and setting $needi [j - 1] = 1$ . Integral and error estimation will be performed again during the next iteration.

On intermediate re-entry:

needi [j - 1]

may be used to indicate what action you have taken for integral

j

$needi [j - 1] = 1$: You have provided the values $f_{j} (x_{i})$ in $FM (j, i)$ , for $i = 1, 2, \dots, n_{x}$ .
$needi [j - 1] < 0$: You are abandoning the evaluation of integral $j$ . The current values of $dinest [j - 1]$ and $errest [j - 1]$ will be returned on final completion.

Otherwise you have not provided the value

f_{j} (x_{i})

On final exit:

needi [j - 1]

indicates the final state of integral

j

$needi [j - 1] = 0$: The error estimate for $F_{j}$ is below the requested tolerance.
$needi [j - 1] = 1$: The error estimate for $F_{j}$ is below the requested tolerance after extrapolation.
$needi [j - 1] = 2$: The error estimate for $F_{j}$ is above the requested tolerance.
$needi [j - 1] = 3$: The error estimate for $F_{j}$ is above the requested tolerance, and extremely bad behaviour of integral $j$ has been detected.
$needi [j - 1] < 0$: You prohibited further evaluation of integral $j$ .

7: $x [lenx]$ – double Input/Output

On initial entry: if

Primary Division Mode = AUTOMATIC

, x need not be set. This is the default behaviour.

Primary Division Mode = MANUAL

, x is used to supply a set of initial ‘break-points’ inside the domain of integration. Specifically,

x [i - 1]

must contain a break-point

x_{i}^{0}

, for

i = 1, 2, \dots, (s_{pri} - 1)

, where

s_{pri}

is the number of

Primary Divisions

Constraint: if break-points are supplied,

x_{i}^{0} \in (a, b)

| x_{i}^{0} - a | > 10.0 ε

| x_{i}^{0} - b | > 10.0 ε

, for

i = 1, 2, \dots, (s_{pri} - 1)

On intermediate exit:

x [i - 1]

is the abscissa

x_{i}

, for

i = 1, 2, \dots, n_{x}

, at which the appropriate integrals must be evaluated.

8: $lenx$ – Integer Input

On entry: the dimension of the array x. Currently

lenx = \max (122, s_{pri} - 1)

will be sufficient for all cases.

Constraint:

lenx \geq lenxrq

, where

lenxrq

is dependent upon the options currently set (see Section 11).

lenxrq

is returned as lenxrq from d01rcc.

9: $nx$ – Integer * Input/Output

On initial entry: need not be set.

On intermediate exit:

n_{x}

, the number of abscissae at which integrands are required.

On intermediate re-entry: must not be changed.

10: $fm [\dim]$ – const double Input

Note: the dimension, dim, of the array fm must be at least

ldfm \times sdfmrq

, where

sdfmrq

is dependent upon

n_{i}

and the options currently set.

sdfmrq

is returned as sdfmrq from d01rcc. If default options are chosen,

sdfmrq = lenxrq

On initial entry: need not be set.

On intermediate re-entry: if indicated by

needi [j - 1]

you must supply the values

f_{j} (x_{i})

FM (j, i)

, for

i = 1, 2, \dots, n_{x}

and

j = 1, 2, \dots, n_{i}

11: $ldfm$ – Integer Input

On entry: the stride separating matrix row elements in the array fm.

Constraint:

ldfm \geq ldfmrq

, where

ldfmrq

is dependent upon

n_{i}

and the options currently set.

ldfmrq

is returned as ldfmrq from d01rcc. If default options are chosen,

ldfmrq = n_{i}

, implying

ldfm \geq ni

12: $dinest [ni]$ – double Input/Output

dinest [j - 1]

contains the current estimate of the definite integral

F_{j}

On initial entry: need not be set.

On intermediate re-entry: must not be altered.

On exit: contains the current estimates of the ni integrals. If

irevcm = 0

, this will be the final solution.

13: $errest [ni]$ – double Input/Output

errest [j - 1]

contains the current error estimate of the definite integral

F_{j}

On initial entry: need not be set.

On intermediate re-entry: must not be altered.

On exit: contains the current error estimates for the ni integrals. If

irevcm = 0

, errest contains the final error estimates of the ni integrals.

14: $iopts [100]$ – const Integer Communication Array

15: $opts [100]$ – const double Communication Array

The arrays iopts and opts MUST NOT be altered between calls to any of the functions d01rac, d01rcc, d01zkc and d01zlc.

16: $icom [licom]$ – Integer Communication Array

icom contains details of the integration procedure, including information on the integration of the

n_{i}

integrals over individual segments. This data is stored sequentially in the order that segments are created. For further information see Section 9.1.

17: $licom$ – Integer Input

On entry: the dimension of the array icom.

Constraint:

licom \geq licmin

, where

licmin

is dependent upon ni and the current options set.

licmin

is returned as licmin from d01rcc. If the default options are set,

licmin = 55 + 6 \times ni

. Larger values than

licmin

are recommended if you anticipate that any integrals will require the domain to be further subdivided.

The maximum value that may be required,

licmax

, is returned as licmax from d01rcc. If default options are chosen, except for possibly increasing the value of

s_{pri}

licmax = 50 + 5 \times ni + (s_{pri} + 100) \times (5 + ni)

18: $com [lcom]$ – double Communication Array

com contains details of the integration procedure, including information on the integration of the

n_{i}

integrals over individual segments. This data is stored sequentially in the order that segments are created. For further information see Section 9.1.

19: $lcom$ – Integer Input

On entry: the dimension of the array com.

Constraint:

lcom > lcmin

, where

lcmin

is dependent upon ni,

s_{pri}

and the current options set.

lcmin

is returned as lcmin from d01rcc. If default options are set,

lcmin = 96 + 12 \times ni

. Larger values are recommended if you anticipate that any integrals will require the domain to be further subdivided.

Given the current options and arguments, the maximum value,

lcmax

, of lcom that may be required, is returned as lcmax from d01rcc. If default options are chosen,

lcmax = 94 + 9 \times ni + ⌈ ni / 2 ⌉ + (s_{pri} + 100) \times (2 + ⌈ ni / 2 ⌉ + 2 \times ni)

20: $fail$ – 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_ACCURACY: At least one error estimate exceeded the requested tolerances.
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.
NE_INT: irevcm had an illegal value.
On entry, $irevcm = ⟨ value ⟩$ .

On entry, $ni = ⟨ value ⟩$ .
Constraint: $ni \geq 1$ .
NE_INT_2: $ldfm < ldfmrq$ . If default options are chosen, this implies $ldfm < ni$ .
On entry, $ldfm = ⟨ value ⟩$ .
Constraint: $ldfm \geq ⟨ value ⟩$ .
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_INVALID_ARRAY: On entry, one of icom and com has become corrupted.
NE_INVALID_OPTION: Either the option arrays iopts and opts have not been initialized for d01rac, or they have become corrupted.
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_INT: Extremely bad behaviour was detected for at least one integral.

Extremely bad behaviour was detected for at least one integral. At least one other integral error estimate was above the requested tolerance.
NE_QUAD_BRKPTS_INVAL: On entry, $Primary Division Mode = MANUAL$ and at least one supplied break-point in x is outside of the domain of integration.
NE_TOO_SMALL: lcom is insufficient for additional subdivision.
On entry, $lcom = ⟨ value ⟩$ .
Constraint: $lcom \geq ⟨ value ⟩$ .

lenx is insufficient for the chosen options.
On entry, $lenx = ⟨ value ⟩$ .
Constraint: $lenx \geq ⟨ value ⟩$ .

licom is insufficient for additional subdivision.
On entry, $licom = ⟨ value ⟩$ .
Constraint: $licom \geq ⟨ value ⟩$ .
NE_USER_STOP: Evaluation of all integrals has been stopped during the initial phase.

7 Accuracy

d01rac cannot guarantee, but in practice usually achieves, the following accuracy for each integral

F_{j}

| F_{j} - dinest [j - 1] | \leq tol

where

tol = \max (ε_{a}, ε_{r} \times | F_{j} |)

ε_{a}

and

ε_{r}

are the error tolerances

Absolute Tolerance

and

Relative Tolerance

respectively. Moreover, it returns errest, the entries of which in normal circumstances satisfy,

| F_{j} - dinest [j - 1] | \leq errest [j - 1] \leq tol .

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

d01rac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

d01rac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time required by d01rac is usually dominated by the time required to evaluate the values of the integrands

f_{j}

d01rac will be most efficient if any badly behaved integrands provided have irregularities over similar subsections of the domain. For example, evaluation of the integrals,

\int_{0}^{1} (\begin{matrix} \log (x) \\ x^{- \frac{1}{2}} \\ x^{2} \end{matrix}) d x

will be quite efficient, as the irregular behaviour of the first two integrands is at

x = 0

. On the contrary, the evaluation of the integrals,

\int_{0}^{1} (\begin{matrix} \log (x) \\ \log (1 - x) \end{matrix}) d x

will be less efficient, as the two integrands have singularities at opposite ends of the domain, which will result in subdivisions which are only of use to one integrand. In such cases, it will be more efficient to use two sets of calls to d01rac.

d01rac will flag extremely bad behaviour if a sub-interval

\bar{k}

with bounds

[a_{\bar{k}}, b_{\bar{k}}]

satisfying

| b_{\bar{k}} - a_{\bar{k}} | < \max (δ_{a}, δ_{r} \times | b - a |)

has a local error estimate greater than the requested tolerance for at least one integral. The values

δ_{a}

and

δ_{r}

can be set through the optional parameters

Absolute Interval Minimum

and

Relative Interval Minimum

respectively.

9.1 Details of the Computation

This section is recommended for expert users only. It describes the contents of the arrays com and icom upon exit from d01rac with

fail . code =

NE_NOERROR, NE_ACCURACY or NE_QUAD_BAD_SUBDIV_INT, and provided at least one iteration completed, failure due to insufficient licom or lcom.

The arrays icom and com contain details of the integration, including various scalars, one-dimensional arrays, and (effectively) two-dimensional arrays. The dimensions of these arrays vary depending on the arguments and options used and the progress of the algorithm. Here we describe some of these details, including how and where they are stored in icom and com.

Scalar quantities:

The indices in icom including the following scalars are available via query only options, see Section 11.2. For example,

I_{ldi}

is the integer value returned by the option

Index LDI

. Note the indices returned start at

1

and so the corresponding zero based indices require

1

to be subtracted as shown below. This is also true for the location of arrays within icom and com and consequently the indices of the elements of the one- and two-dimensional arrays must be modified as detailed below.

$ldi$	The leading dimension of the two-dimensional integer arrays stored in icom detailed below. $ldi = icom [I_{ldi} - 1]$ .
$ldr$	The leading dimension of the two-dimensional real arrays stored in com detailed below. $ldr = icom [I_{ldr} - 1]$ .
$nsdiv$	The number of segments that have been subdivided during the adaptive process. $nsdiv = icom [I_{nsdiv} - 1]$ .
$nseg$	The total number of segments formed. $nseg = 2 nsdiv + s_{pri}$ . $nseg = icom [I_{nseg} - 1]$ .
$dsp$	The reference of the first element of the array $ds$ stored in com. $dsp = icom [I_{dsp} - 1]$ .
$esp$	The reference of the first element of the array $es$ stored in com. $esp = icom [I_{esp} - 1]$ .
$evalsp$	The reference of the first element of the array $evals$ stored in icom. $evalsp = icom [I_{evalsp} - 1]$ .
$fcp$	The reference of the first element of the array $fcount$ stored in icom. $fcp = icom [I_{fcp} - 1]$ .
$sinforp$	The reference of the first element of the array $sinfor$ stored in com. $sinforp = icom [I_{sinforp} - 1]$ .
$sinfoip$	The reference of the first element of the array $sinfoi$ stored in icom. $sinfoip = icom [I_{sinfoip} - 1]$ .

One-dimensional arrays:

$fcount [n_{i}]$: $fcount [0] = icom [fcp - 1]$ .
$fcount [j - 1]$ contains the number of different approximations of integral $j$ calculated, for $j = 1, 2, \dots, n_{i}$ .

Two-dimensional arrays:

$sinfoi [5 \times nseg]$: $sinfoi [0] = icom [sinfoip - 1]$ .
$sinfoi$ contains information about the hierarchy of splitting.
$sinfoi [(k - 1) \times ldi]$ contains the split identifier for segment $k$ , for $k = 1, 2, \dots, nseg$ .

$sinfoi [(k - 1) \times ldi + 1]$ contains the parent segment number of segment $k$ (i.e., the segment was split to create segment $k$ ), for $k = 1, 2, \dots, nseg$ .

$sinfoi [(k - 1) \times ldi + 2]$ and $sinfoi [(k - 1) \times ldi + 3]$ contain the segment numbers of the two child segments formed from segment $k$ , if segment $k$ has been split. If segment $k$ has not been split, these will be negative.

$sinfoi [(k - 1) \times ldi + 4]$ contains the level at which the segment exists, corresponding to $n_{a} + 1$ , where $n_{a}$ is the number of ancestor segments of segment $k$ , for $k = 1, 2, \dots, nseg$ . A negative level indicates that segment $k$ will not be split further, the level is then given by the absolute value of $sinfoi [(k - 1) \times ldi + 4]$ .
$sinfor [2 \times nseg]$: $sinfor [0] = com [sinforp - 1]$ .
$sinfor$ contains the bounds of each segment.
$sinfor [(k - 1) \times ldr]$ contains the lower bound of segment $k$ , for $k = 1, 2, \dots, nseg$ .

$sinfor [(k - 1) \times ldr + 1]$ contains the upper bound of segment $k$ , for $k = 1, 2, \dots, nseg$ .
$evals [n_{i} \times nseg]$: $evals [0] = icom [evalsp - 1]$ .
$evals$ contains information to indicate whether an estimate of the integral $j$ has been obtained over segment $k$ , and if so whether this evaluation still contributes to the direct estimate of $F_{j}$ , for $j = 1, 2, \dots, n_{i}$ and $k = 1, 2, \dots, nseg$ .

$evals [(k - 1) \times ldi + j - 1] = 0$ indicates that integral $j$ has not been evaluated over segment $k$ .

$evals [(k - 1) \times ldi + j - 1] = 1$ indicates that integral $j$ has been evaluated over segment $k$ , and that this evaluation contributes to the direct estimate of $F_{j}$ .

$evals [(k - 1) \times ldi + j - 1] = 2$ indicates that integral $j$ has been evaluated over segment $k$ , that this evaluation contributes to the direct estimate of $F_{j}$ , and that you have requested no further evaluation of this integral at this segment by setting $needi [j - 1] < 0$ .

$evals [(k - 1) \times ldi + j - 1] = 3$ indicates that integral $j$ has been evaluated over segment $k$ , and this evaluation no longer contributes to the direct estimate of $F_{j}$ .

$evals [(k - 1) \times ldi + j - 1] = 4$ indicates that integral $j$ has been evaluated over segment $k$ , that this evaluation contributes to the direct estimate of $F_{j}$ , and that this segment is too small for any further splitting to be performed. Integral $j$ also has a local error estimate over this segment above the requested tolerance. Such segments cause d01rac to return $fail . code =$ NE_QUAD_BAD_SUBDIV_INT, indicating extremely bad behaviour.

$evals [(k - 1) \times ldi + j - 1] = 5$ indicates that integral $j$ has been evaluated over segment $k$ , that this evaluation contributes to the direct estimate of $F_{j}$ , and that this segment is too small for any further splitting to be performed. The local error estimate is however below the requested tolerance.
$ds [n_{i} \times nseg]$: $ds [0] = com [dsp - 1]$ .
$ds [(k - 1) \times ldr + j - 1]$ contains the definite integral estimate of the $j$ th integral over the $k$ th segment, ${ds}_{j, k}$ , provided it has been evaluated, for $j = 1, 2, \dots, n_{i}$ and $k = 1, 2, \dots, nseg$ .
$es [n_{i} \times nseg]$: $es [0] = com [esp - 1]$ .
$es [(k - 1) \times ldr + j - 1]$ contains the definite integral error estimate of the $j$ th integral over the $k$ th segment, ${es}_{j, k}$ , provided it has been evaluated, for $j = 1, 2, \dots, n_{i}$ and $k = 1, 2, \dots, nseg$ .

For each integral

j

, the direct approximation

D_{j}

F_{j}

, and its error estimate

E_{j}

, may be constructed as,

\begin{matrix} F_{j} \approx D_{j} = \sum_{K_{j}} {ds}_{j, k}, \\ | F_{j} - D_{j} | \approx E_{j} = \sum_{K_{j}} {es}_{j, k}, \end{matrix}

where

K_{j}

is the set of all contributing segments,

K_{j} = {k ∣ evals [(k - 1) \times ldi + j - 1] = 1, ​ 2, ​ 4 ​ or ​ 5, ​ 1 \leq k \leq nseg}

D_{j}

will have been returned in

dinest [j - 1]

, unless extrapolation was successful, as indicated by

needi [j - 1]

Similarly,

E_{j}

will have been returned in

errest [j - 1]

unless extrapolation was successful, in which case the error estimate from the extrapolation will have been returned. If for a given integral

j

one or more contributing segments have unacceptable error estimates, it may be possible to improve the direct approximation by replacing the contributions from these segments with more accurate estimates should these be calculable by some means. Indeed for any segment

\bar{k} \in k

, with lower bound

a_{\bar{k}} = sinfor (1, \bar{k})

and upper bound

b_{\bar{k}} = sinfor (2, \bar{k})

, one may alter the direct approximation

D_{j}

by the following,

\begin{matrix} {ds}_{j, \bar{k}}^{new} \approx \int_{a_{\bar{k}}}^{b_{\bar{k}}} f_{j} (x) ​ ​ dx \\ D_{j} = \sum_{K_{j}} {ds}_{j, k} - {ds}_{j, \bar{k}} + {ds}_{j, \bar{k}}^{new} . \end{matrix}

The error estimate

E_{j}

may be altered similarly.

10 Example

This example integrates

F = \int_{0}^{π} (\begin{matrix} x \sin (2 x) \cos (15 x) \\ x^{2} \sin (2 x) \cos (50 x) \end{matrix}) d x .

10.1 Program Text

Program Text (d01race.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (d01race.r)

11 Optional Parameters

This section can be skipped if you wish to use the default values for all optional parameters, otherwise, the following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 11.1.

Absolute Interval Minimum
Absolute Tolerance
Extrapolation
Extrapolation Safeguard
Maximum Subdivisions
Primary Division Mode
Primary Divisions
Prioritize Error
Quadrature Rule
Relative Interval Minimum
Relative Tolerance

The following optional parameters, see Section 11.2, may be utilized by expert users in conjunction with the information provided in Section 9.1.

$Index LDI$
$Index LDR$
$Index NSDIV$
$Index NSEG$
$Index FCP$
$Index EVALSP$
$Index DSP$
$Index ESP$
$Index SINFOIP$
$Index SINFORP$

11.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

the keywords, where the minimum abbreviation of each keyword is underlined;
a parameter value, where the letters $a$ , $i$ and $r$ denote options that take character, integer and real values respectively;
the default value.

The following symbol represents various machine constants:

$ε$ represents the machine precision (see X02AJC).

All options accept the value ‘DEFAULT’ in order to return single options to their default states.

Keywords and character values are case insensitive, however they must be separated by at least one space.

Unsetable options will return the appropriate value when calling d01zlc. They will have no effect if passed to d01zkc.

For d01rac the maximum length of the argument cvalue used by d01zlc is

15

Absolute Interval Minimum

r

Default

= 128.0 ε

r = δ_{a}

, the absolute lower limit for a segment to be considered for subdivision. See also

Relative Interval Minimum

and Section 9.

Constraint:

r \geq 128 ε

Absolute Tolerance

r

Default

= 1024 ε

r = ε_{a}

, the absolute tolerance required. See also

Relative Tolerance

and Section 3.

Constraint:

r \geq 0.0

Extrapolation

a

Default

= ON

Activate or deactivate the use of the

ε

algorithm (Wynn (1956)).

Extrapolation

often reduces the number of iterations required to achieve the desired solution, but it can occasionally lead to premature convergence towards an incorrect answer.

$ON$: Use extrapolation.
$OFF$: Disable extrapolation.

Extrapolation Safeguard

r

Default

= 1.0e−12

r = ε_{safe}

. If

ε_{q}

is the estimated error from the quadrature evaluation alone, and

ε_{ex}

is the error estimate determined using extrapolation, then the extrapolated solution will only be accepted if

ε_{safe} ε_{q} \leq ε_{ex}

Maximum Subdivisions

i

Default

= 50

i = \max_{sdiv}

, the maximum number of subdivisions the algorithm may use in the adaptive phase, forming at most an additional

(2 \times \max_{sdiv})

segments.

Primary Divisions

i

Default

= 1

i = s_{pri}

, the number of initial segments of the domain

[a, b]

. By default the initial segment is the entire domain.

Constraint:

0 < i < 1000000

Primary Division Mode

a

Default

= AUTOMATIC

Determines how the initial set of

s_{pri}

segments will be generated.

AUTOMATIC: d01rac will automatically generate $s_{pri}$ segments of equal size covering the interval $[a, b]$ .
MANUAL: d01rac will use the break-points $x_{i}^{0}$ , for $i = 1, 2, \dots, s_{pri} - 1$ , supplied in x on initial entry to generate the initial segments covering $[a, b]$ . These may be supplied in any order, however it will be more efficient to supply them in ascending (or descending if $a > b$ ) order. Repeated break-points are allowed, although this will generate fewer initial segments.

Note: an absolute bound on the size of an initial segment of

10.0 ε

is automatically applied in all cases, and will result in fewer initial subdivisions being generated if automatically generated or supplied break-points result in segments smaller than this.

Prioritize Error

a

Default

= LEVEL

Indicates how new subdivisions of segments sustaining unacceptable local errors for integrals should be prioritized.

$LEVEL$: Segments with lower level with unsatisfactory error estimates will be chosen over segments with greater error on higher levels. This will probably lead to more integrals being improved in earlier iterations of the algorithm, and hence will probably lead to fewer repeated returns (see argument sid), and to more integrals being satisfactorily estimated if computational exhaustion occurs.
$MAXERR$: The segment with the worst overall error will be split, regardless of level. This will more rapidly improve the worst integral estimates, although it will probably result in the fewest integrals being improved in earlier iterations, and may hence lead to more repeated returns (see argument sid), and potentially fewer integrals satisfying the requested tolerances if computational exhaustion occurs.

Quadrature Rule

a

Default

= GK15

The basic quadrature rule to be used during the integration. Currently

6

Gauss–Kronrod rules are available, all identifiable by the letters GK followed by the number of points required by the Kronrod rule. Higher order rules generally provide higher accuracy with fewer subdivisons. However, for integrands with sharp singularities, lower order rules may be more efficient, particularly if the integrand away from the singularity is well behaved. With higher order rules, you may need to increase the

Absolute Interval Minimum

and the

Relative Interval Minimum

to maintain numerical difference between the abscissae and the segment bounds.

GK15: The Gauss–Kronrod rule based on $7$ Gauss points and $15$ Kronrod points.
GK21: The Gauss–Kronrod rule based on $10$ Gauss points and $21$ Kronrod points. This is the rule used by d01rjc.
GK31: The Gauss–Kronrod rule based on $15$ Gauss points and $31$ Kronrod points.
GK41: The Gauss–Kronrod rule based on $20$ Gauss points and $41$ Kronrod points.
GK51: The Gauss–Kronrod rule based on $25$ Gauss points and $51$ Kronrod points.
GK61: The Gauss–Kronrod rule based on $30$ Gauss points and $61$ Kronrod points. This is the highest order rule, most suitable for highly oscilliatory integrals.

Relative Interval Minimum

r

Default

= 1.0e−6

r = δ_{r}

, the relative factor in the lower limit,

δ_{r} | b - a |

, for a segment to be considered for subdivision. See also

Absolute Interval Minimum

and Section 9.

Constraint:

r \geq 0.0

Relative Tolerance

r

Default

= \sqrt{ε}

r = ε_{r}

, the required relative tolerance. See also

Absolute Tolerance

and Section 3.

Constraint:

r \geq 0.0

Note: setting both

ε_{r} = ε_{a} = 0.0

is possible, although it will most likely result in an excessive amount of computational effort.

11.2 Diagnostic Options

These options are provided for expert users who wish to examine and modify the precise details of the computation. They should only be used after d01rac returns, as opposed to the options listed in Section 11.1 which must be used before the first call to d01rac.

Index LDI