NAG Library Routine Document

D01RAF

1  Purpose

2  Specification

3  Description

liopts = 100
lopts = 100
allocate(iopts(liopts),opts(lopts))
call D01ZKF('initialize = d01raf',iopts,liopts,opts,lopts,
     ifail)
call D01ZKF('option = value',iopts,liopts,opts,lopts,
     ifail)
...
call D01RCF(ni,lenxrq,ldfmrq,sdfmrq,licmin,licmax,lcmin,lcmax,iopts,opts,
     ifail)
lenx = lenxrq
ldfm = ldfmrq
sdfm = sdfmrq
licomm = licmax
lcomm = lcmax
allocate(icomm(licomm),comm(lcomm),x(lenx),fm(ldfm,sdfm),needi(ni),
              dinest(ni),errest(ni))

irevcm = 1
while irevcm≠0
    call D01RAF(irevcm,ni,a,b,sid,needi,x,lenx,nx,fm,ldfm,   &
                dinest,errest,iopts,opts,icomm,licomm,comm, &
                lcomm,ifail)
    select case(irevcm)
    case(11)
          Initial solve phase 
          evaluate fm(1:ni,1:nx)
    case(12)
          Adaptive solve phase 
          evaluate fm(needi(1:ni)=1,1:nx)
    case(0)
          investigate ifail
    end select
end while

call D01ZLF('option',ivalue,rvalue,cvalue,optype,iopts,opts,ifail)
...
call D01RBF(monit,ni,dinest,errest,iopts,opts,icomm,licomm,comm,lcomm, &
                   iuser,ruser,ifail)

4  References

5  Parameters

1:     IREVCM – INTEGERInput/Output

On initial entry: $\mathbf{IREVCM}=1$.

$\mathbf{IREVCM}=1$

Set up data structures in ICOMM and COMM and start a new integration.

Constraint: $\mathbf{IREVCM}=1$ on initial entry.

On intermediate exit: $\mathbf{IREVCM}=11$ or $12$.

IREVCM requests the values of $f_j\left(x_i\right)$ be evaluated for all required $j\in 1,\dots ,n_i$ as indicated by NEEDI, and at all the points $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n_x$. Abscissae $x_i$ are provided in $\mathbf{X}\left(i\right)$ and $f_j\left(x_i\right)$ must be returned in $\mathbf{FM}\left(j,i\right)$.

During the initial solve phase:

$\mathbf{IREVCM}=11$

Function values are required to construct the inital estimates of the definite integrals.

If $\mathbf{NEEDI}\left(j\right)=1$, $f_j\left(x_i\right)$ must be supplied in $\mathbf{FM}\left(j,i\right)$. This will be the case unless you have abandoned the evaluation of specific integrals on a previous call.

If $\mathbf{NEEDI}\left(j\right)=0$, you have previously abandoned the evaluation of integral $j$, and hence should not supply the value of $f_j\left(x_i\right)$.

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, $\mathbf{NEEDI}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,n_i$, will be set to this negative value and $\mathbf{IFAIL}=-\mathbf{1}$ will be returned.

During the adaptive solve phase:

$\mathbf{IREVCM}=12$

Function values are required to improve the estimates of the definite integrals.

If $\mathbf{NEEDI}\left(j\right)=0$, any evaluation of $f_j\left(x_i\right)$ will be discarded, so there is no need to provide them.

If $\mathbf{NEEDI}\left(j\right)=1$, $f_j\left(x_i\right)$ must be returned.

If $\mathbf{NEEDI}\left(j\right)=2$, $3$ or $4$, the current error estimate of integral $j$ does not require integrand $j$ to be evaluated. Should you choose to, integrand $j$ can be evaluated in which case $\mathbf{NEEDI}\left(j\right)$ 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 either $\mathbf{IFAIL}=\mathbf{1}$, $\mathbf{2}$ or $\mathbf{3}$ will be returned and the elements of NEEDI will reflect the current state of of the adaptive process.

On intermediate re-entry: IREVCM should normally be left unchanged. However, if IREVCM is set to a negative value, D01RAF will terminate, (see $\mathbf{IREVCM}=11$ and $\mathbf{IREVCM}=12$ above).

On final exit: $\mathbf{IREVCM}=0$.

$\mathbf{IREVCM}=0$

Indicates the algorithm has completed.

2:     NI – INTEGERInput

On entry: $n_i$, number of integrands.

3:     A – REAL (KIND=nag_wp)Input

On entry: $a$, the lower bound of the domain.

4:     B – REAL (KIND=nag_wp)Input

On entry: $b$, the upper bound of the domain.

If $\left|b-a\right|<10\epsilon$, where $\epsilon$ is the machine precision (see X02AJF), then D01RAF will return $\mathbf{DINEST}\left(\mathit{j}\right)=\mathbf{ERREST}\left(\mathit{j}\right)=0.0$, for $\mathit{j}=1,2,\dots ,n_i$.

5:     SID – INTEGEROutput

For advanced users.

On intermediate exit: SID identifies a specific set of abscissae, $\mathbf{x}$, 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 $\mathbf{x}$, and reuse them should D01RAF return the same value of SID, i.e., the same set of abscissae was used.

6:     NEEDI(NI) – INTEGER arrayInput/Output

On initial entry: need not be set.

On intermediate exit: $\mathbf{NEEDI}\left(j\right)$ indicates what action must be taken for integral $j=1,2,\dots n_i$ (see IREVCM).

$\mathbf{NEEDI}\left(j\right)=0$

Do not provide $f_j\left(x_i\right)$. Any provided values will be ignored.

$\mathbf{NEEDI}\left(j\right)=1$

The values $f_j\left(x_{\mathit{i}}\right)$ must be provided in $\mathbf{FM}\left(j,\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n_x$.

$\mathbf{NEEDI}\left(j\right)=2$

The values $f_j\left(x_i\right)$ are not definitely 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 $\mathbf{NEEDI}\left(j\right)=1$, and supply $f_j\left(x_{\mathit{i}}\right)$ in $\mathbf{FM}\left(j,\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n_x$.

$\mathbf{NEEDI}\left(j\right)=3$

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 integral behaviour. However, providing the values $f_j\left(x_i\right)$ 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 $\mathbf{NEEDI}\left(j\right)=1$, and supply $f_j\left(x_{\mathit{i}}\right)$ in $\mathbf{FM}\left(j,\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n_x$.

$\mathbf{NEEDI}\left(j\right)=4$

The error estimate of integral $j$ is below the requested tolerance. If you believe this to be false, if for example the result in $\mathbf{DINEST}\left(j\right)$ 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 $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n_x$, and setting $\mathbf{NEEDI}\left(j\right)=1$. Integral and error estimation will be performed again during the next iteration.

On intermediate re-entry: $\mathbf{NEEDI}\left(j\right)$ may be used to indicate what action you have taken for integral $j$.

$\mathbf{NEEDI}\left(j\right)=1$

You have provided the values $f_j\left(x_{\mathit{i}}\right)$ in $\mathbf{FM}\left(j,\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n_x$.

$\mathbf{NEEDI}\left(j\right)<0$

You are abandoning the evaluation of integral $j$. The current values of $\mathbf{DINEST}\left(j\right)$ and $\mathbf{ERREST}\left(j\right)$ will be returned on final completion.

Otherwise you have not provided the value $f_j\left(x_i\right)$.

On final exit: $\mathbf{NEEDI}\left(j\right)$ indicates the final state of integral $j$.

$\mathbf{NEEDI}\left(j\right)=0$

The error estimate for $F_j$ is below the requested tolerance.

$\mathbf{NEEDI}\left(j\right)=1$

The error estimate for $F_j$ is below the requested tolerance after extrapolation.

$\mathbf{NEEDI}\left(j\right)=2$

The error estimate for $F_j$ is above the requested tolerance.

$\mathbf{NEEDI}\left(j\right)=3$

The error estimate for $F_j$ is above the requested tolerance, and extremely bad behaviour of integral $j$ has been detected.

$\mathbf{NEEDI}\left(j\right)<0$

You prohibited further evaluation of integral $j$.

7:     X(LENX) – REAL (KIND=nag_wp) arrayInput/Output

On initial entry: if $\mathbf{Primary\; Division\; Mode}=\mathrm{AUTOMATIC}$, X need not be set. This is the default behaviour.

If $\mathbf{Primary\; Division\; Mode}=\mathrm{MANUAL}$, X may be used to supply a set of initial ‘break points’ inside the domain of integration. Specifically, $\mathbf{X}\left(i\right)$ must contain a break point $x_{\mathit{i}}^0$, for $\mathit{i}=1,2,\dots ,\left(s_{\mathit{pri}}-1\right)$, where $s_{\mathit{pri}}$ is the number of Primary Divisions.

Constraint: if break points are supplied, $x_{\mathit{i}}^0\in \left(a,b\right)$, $\left|x_{\mathit{i}}^0-a\right|>10.0\epsilon$, $\left|x_{\mathit{i}}^0-b\right|>10.0\epsilon$, for $\mathit{i}=1,2,\dots ,\left(s_{\mathit{pri}}-1\right)$.

On intermediate exit: $\mathbf{X}\left(\mathit{i}\right)$ is the abscissa $x_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n_x$, at which the appropriate integrals must be evaluated.

8:     LENX – INTEGERInput

On entry: the dimension of the array X as declared in the (sub)program from which D01RAF is called. Currently $\mathbf{LENX}=\max \left(122,s_{\mathit{pri}}-1\right)$ will be sufficient for all cases.

Constraint: $\mathbf{LENX}\ge \mathit{lenxrq}$, where $\mathit{lenxrq}$ is dependent upon the options currently set (see Section 10). $\mathit{lenxrq}$ is returned as LENXRQ from D01RCF.

9:     NX – INTEGERInput/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(LDFM,$*$) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array FM must be at least $\mathit{sdfmrq}$, where $\mathit{sdfmrq}$ is dependent upon $n_i$ and the options currently set. $\mathit{sdfmrq}$ is returned as SDFMRQ from D01RCF. If default options are chosen, $\mathit{sdfmrq}=\mathit{lenxrq}$.

On initial entry: need not be set.

On intermediate re-entry: if indicated by $\mathbf{NEEDI}\left(j\right)$ you must supply the values $f_j\left(x_i\right)$ in $\mathbf{FM}\left(\mathit{j},\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n_x$ and $\mathit{j}=1,2,\dots ,n_i$.

11:   LDFM – INTEGERInput

On entry: the first dimension of the array FM as declared in the (sub)program from which D01RAF is called.

Constraint: $\mathbf{LDFM}\ge \mathit{ldfmrq}$, where $\mathit{ldfmrq}$ is dependent upon $n_i$ and the options currently set. $\mathit{ldfmrq}$ is returned as LDFMRQ from D01RCF. If default options are chosen, $\mathit{ldfmrq}=n_i$, implying $\mathbf{LDFM}\ge \mathbf{NI}$.

12:   DINEST(NI) – REAL (KIND=nag_wp) arrayInput/Output

$\mathbf{DINEST}\left(j\right)$ 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 $\mathbf{IREVCM}=0$, this will be the final solution.

13:   ERREST(NI) – REAL (KIND=nag_wp) arrayInput/Output

$\mathbf{ERREST}\left(j\right)$ 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 $\mathbf{IREVCM}=0$, ERREST contains the final error estimates of the NI integrals.

14:   IOPTS($100$) – INTEGER arrayInput

On entry: integer option array generated by D01ZKF.

Constraint: IOPTS must not be altered between calls to D01RAF, D01RCF, D01ZKF and D01ZLF.

15:   OPTS($100$) – REAL (KIND=nag_wp) arrayInput

On entry: real option array generated by D01ZKF.

Constraint: OPTS must not be altered between calls to D01RAF, D01RCF, D01ZKF and D01ZLF.

16:   ICOMM(LICOMM) – INTEGER arrayCommunication Array

The contents of this array must not be changed directly once the integration process has started. ICOMM 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. You are free to examine the contents of ICOMM using the diagnostic routine D01RBF at any time during, or after, the integration process.

17:   LICOMM – INTEGERInput

On entry: the dimension of the array ICOMM.

Constraint: $\mathbf{LICOMM}\ge \mathit{licmin}$, where $\mathit{licmin}$ is dependent upon NI and the current options set. $\mathit{licmin}$ is returned as LICMIN from D01RCF. If the default options are set, then $\mathit{licmin}=55+6\times \mathbf{NI}$. Larger values than $\mathit{licmin}$ are recommended if you anticipate that any integrals will require the domain to be further subdivided.

The maximum value that may be required, $\mathit{licmax}$, is returned as LICMAX from D01RCF. If default options are chosen, except for possibly increasing the value of $s_{\mathit{pri}}$, then $\mathit{licmax}=50+5\times \mathbf{NI}+\left(s_{\mathit{pri}}+100\right)\times \left(5+\mathbf{NI}\right)$.

18:   COMM(LCOMM) – REAL (KIND=nag_wp) arrayCommunication Array

The contents of this array must not be changed directly once the integration process has started. COMM 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. You are free to examine the contents of COMM using the diagnostic routine D01RBF at any time during, or after, the solution phase.

19:   LCOMM – INTEGERInput

On entry: the dimension of the array COMM.

Constraint: $\mathbf{LCOMM}>\mathit{lcmin}$, where $\mathit{lcmin}$ is dependent upon NI, $s_{\mathit{pri}}$ and the current options set. $\mathit{lcmin}$ is returned as LCMIN from D01RCF. If default options are set, then $\mathit{lcmin}=96+12\times \mathbf{NI}$. Larger values are recommended if you anticipate that any integrals will require the domain to be further subdivided.

Given the current options and parameters, the maximum value, $\mathit{lcmax}$, of LCOMM that may be required, is returned as LCMAX from D01RCF. If default options are chosen, $\mathit{lcmax}=94+9\times \mathbf{NI}+⌈\mathbf{NI}/2⌉+\left(s_{\mathit{pri}}+100\right)\times \left(2+⌈\mathbf{NI}/2⌉+2\times \mathbf{NI}\right)$.

20:   IFAIL – INTEGERInput/Output

On initial entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On final exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

At least one error estimate exceeded the requested tolerances.

$\mathbf{IFAIL}=2$

Extremely bad behaviour was detected for at least one integral.

$\mathbf{IFAIL}=3$

Extremely bad behaviour was detected for at least one integral. At least one other integral error estimate was above the requested tolerance.

$\mathbf{IFAIL}=11$

IREVCM had an illegal value.
On entry, $\mathbf{IREVCM}=⟨\mathit{\text{value}}⟩$.

$\mathbf{IFAIL}=21$

On entry, $\mathbf{NI}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{NI}\ge 1$.

$\mathbf{IFAIL}=71$

On entry, $\mathbf{Primary\; Division\; Mode}=\mathrm{MANUAL}$ and at least one supplied breakpoint in X is outside of the domain of integration.

$\mathbf{IFAIL}=81$

LENX is insufficient for the chosen options.
On entry, $\mathbf{LENX}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LENX}\ge ⟨\mathit{\text{value}}⟩$.

$\mathbf{IFAIL}=111$

$\mathbf{LDFM}<\mathit{ldfmrq}$. If default options are chosen, this implies $\mathbf{LDFM}<\mathbf{NI}$.
On entry, $\mathbf{LDFM}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LDFM}\ge ⟨\mathit{\text{value}}⟩$.

$\mathbf{IFAIL}=171$

The length of LICOMM is insufficient for additional subdivision.
On entry, $\mathbf{LICOMM}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LICOMM}\ge ⟨\mathit{\text{value}}⟩$.

$\mathbf{IFAIL}=191$

The length of LCOMM is insufficient for additional subdivision.
On entry, $\mathbf{LCOMM}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LCOMM}\ge ⟨\mathit{\text{value}}⟩$.

$\mathbf{IFAIL}=1001$

Either the option arrays IOPTS and OPTS have not been initialized for D01RAF, or they have become corrupted.

$\mathbf{IFAIL}=1101$

On entry, one of ICOMM and COMM has become corrupted.

$\mathbf{IFAIL}=-1$

Evaluation of all integrals has been stopped during the initial phase.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d01rafe.f90)

9.2  Program Data

9.3  Program Results

Program Results (d01rafe.r)

10  Optional Parameters

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

10.1  Description of the Optional Parameters

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

AUTOMATIC

D01RAF will automatically generate $s_{\mathit{pri}}$ segments of equal size covering the interval $\left[a,b\right]$.

MANUAL

D01RAF will use the break points $x_{\mathit{i}}^0$, for $\mathit{i}=1,2,\dots ,s_{\mathit{pri}}-1$, supplied in X on initial entry to generate the initial segments covering $\left[a,b\right]$. 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.