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Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

The dimension of the arrays that must be passed as actual arguments to nag_quad_1d_gen_vec_multi_rcomm (d01ra) are dependent upon a number of factors. nag_quad_1d_gen_vec_multi_dimreq (d01rc) returns the correct size of these arrays enabling nag_quad_1d_gen_vec_multi_rcomm (d01ra) to be called successfully.

## Syntax

[lenxrq, ldfmrq, sdfmrq, licmin, licmax, lcmin, lcmax, ifail] = d01rc(ni, iopts, opts)
[lenxrq, ldfmrq, sdfmrq, licmin, licmax, lcmin, lcmax, ifail] = nag_quad_1d_gen_vec_multi_dimreq(ni, iopts, opts)

## Description

nag_quad_1d_gen_vec_multi_dimreq (d01rc) returns the minimum dimension of the arrays x ($\mathit{lenxrq}$), fm ($\mathit{ldfmrq}×\mathit{sdfmrq}$), icom ($\mathit{licmin}$) and com ($\mathit{lcmin}$) that must be passed to nag_quad_1d_gen_vec_multi_rcomm (d01ra) to enable the integration to commence given options currently set for the ni integrands. nag_quad_1d_gen_vec_multi_dimreq (d01rc) also returns the upper bounds $\mathit{licmax}$ and $\mathit{lcmax}$ for the dimension of the arrays icom and com, that could possibly be required with the chosen options.
All the minimum values $\mathit{lenxrq}$, $\mathit{ldfmrq}$, $\mathit{sdfmrq}$, $\mathit{licmin}$ and $\mathit{lcmin}$, and subsequently all the maximum values $\mathit{licmax}$ and $\mathit{lcmax}$ may be affected if different options are set, and hence nag_quad_1d_gen_vec_multi_dimreq (d01rc) should be called after any options are set, and before the first call to nag_quad_1d_gen_vec_multi_rcomm (d01ra).
A segment is here defined as a (possibly maximal) subset of the domain of integration. During subdivision, a segment is bisected into two new segments.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{ni}$int64int32nag_int scalar
${n}_{i}$, the number of integrals which will be approximated in the subsequent call to nag_quad_1d_gen_vec_multi_rcomm (d01ra).
Constraint: ${\mathbf{ni}}>0$.
2:     $\mathrm{iopts}\left(:\right)$int64int32nag_int array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument iopts in the previous call to nag_quad_opt_set (d01zk).
The integer option array as returned by nag_quad_opt_set (d01zk).
3:     $\mathrm{opts}\left(:\right)$ – double array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument opts in the previous call to nag_quad_opt_set (d01zk).
The real option array as returned by nag_quad_opt_set (d01zk).

None.

### Output Parameters

1:     $\mathrm{lenxrq}$int64int32nag_int scalar
$\mathit{lenxrq}$, the minimum dimension of the array x that can be used in a subsequent call to nag_quad_1d_gen_vec_multi_rcomm (d01ra).
2:     $\mathrm{ldfmrq}$int64int32nag_int scalar
$\mathit{ldfmrq}$, the minimum leading dimension of the array fm that can be used in a subsequent call to nag_quad_1d_gen_vec_multi_rcomm (d01ra).
3:     $\mathrm{sdfmrq}$int64int32nag_int scalar
$\mathit{sdfmrq}$, the minimum second dimension of the array fm that can be used in a subsequent call to nag_quad_1d_gen_vec_multi_rcomm (d01ra).
Note: the minimum dimension of the array fm is $\mathit{ldfmrq}×\mathit{sdfmrq}$.
4:     $\mathrm{licmin}$int64int32nag_int scalar
$\mathit{licmin}$, the minimum dimension of the array icom that must be passed to nag_quad_1d_gen_vec_multi_rcomm (d01ra) to enable it to calculate a single approximation to all the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
5:     $\mathrm{licmax}$int64int32nag_int scalar
$\mathit{licmax}$ the dimension of the array icom that must be passed to nag_quad_1d_gen_vec_multi_rcomm (d01ra) to enable it to exhaust the adaptive process controlled by the currently set options for the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
6:     $\mathrm{lcmin}$int64int32nag_int scalar
$\mathit{lcmin}$, the minimum dimension of the array com that must be passed to nag_quad_1d_gen_vec_multi_rcomm (d01ra) to enable it to calculate a single approximation to all the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
7:     $\mathrm{lcmax}$int64int32nag_int scalar
$\mathit{lcmax}$, the dimension of the array com that must be passed to nag_quad_1d_gen_vec_multi_rcomm (d01ra) to enable it to exhaust the adaptive process controlled by the currently set options for the ${n}_{i}$ integrals over the interval $\left[a,b\right]$ with ${s}_{\mathit{pri}}$ initial segments.
8:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=21$
Constraint: ${\mathbf{ni}}>0$.
${\mathbf{ifail}}=1001$
One of the option arrays iopts or opts has become corrupted. Re-initialize the arrays using nag_quad_opt_set (d01zk).
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

None.

## Example

See Example in nag_quad_1d_gen_vec_multi_rcomm (d01ra) for examples of the usage of nag_quad_1d_gen_vec_multi_dimreq (d01rc).
```function d01rc_example

fprintf('d01rc example results\n\n');

% Setup phase.

% set problem parameters
ni = int64(2);
nx = int64(0);
% lower (a) and upper (b) bounds
a = 0;
b = pi;
iopts = zeros(100, 1, 'int64');
opts  = zeros(100, 1);

% initialize option arrays
[iopts, opts, ifail] = d01zk('Initialize = d01ra', iopts, opts);

% set any non-default options required
[iopts, opts, ifail] = d01zk('Quadrature Rule = gk41', iopts, opts);
[iopts, opts, ifail] = d01zk('Absolute Tolerance = 1.0e-7', iopts, opts);
[iopts, opts, ifail] = d01zk('Relative Tolerance = 1.0e-7', iopts, opts);

% determine maximum required array lengths
[lenxrq, ldfmrq, sdfmrq, licmin, licmax, lcmin, lcmax, ifail] = ...
d01rc(ni, iopts, opts);

% allocate remaining arrays
needi  = zeros(ni, 1, 'int64');
comm   = zeros(lcmax, 1);
icomm  = zeros(licmax, 1, 'int64');
fm     = zeros(ldfmrq, sdfmrq);
dinest = zeros(ni, 1);
errest = zeros(ni, 1);
x      = zeros(1, lenxrq);

% Solve phase.

% Use d01ra to evaluate the definate integrals of:
%   f_1 = (x*sin(2*x))*cos(15*x)
%   f_2 = (x*sin(2*x))*(x*cos(50*x))

% set initial irevcm
irevcm = int64(1);

while irevcm ~= 0
[irevcm, sid, needi, x, nx, dinest, errest, icomm, comm, ifail] = ...
d01ra(irevcm, a, b, needi, x, nx, fm, dinest, errest, ...
iopts, opts, icomm, comm);

switch irevcm
case 11
% Initial returns.
% These will occur during the non-adaptive phase.
% All values must be supplied.
% dinest and errest do not contain approximations
% over the complete interval at this stage.

% Calculate x*sin(2*x), storing the result in fm(2,1:nx) for re-use.
fm(2, :) = x.*sin(2*x);

% Calculate f_1
fm(1, :) = fm(2, :).*cos(15*x);

% Calculate f_2
fm(2, :) = fm(2, :).*x.*cos(50*x);
case 12
% Intermediate returns.
% These will occur during the adaptive phase.
% All requested values must be supplied.
% dinest and errest do not contain approximations
% over the complete interval at this stage.

% Calculate x*sin(2*x).
fm(2, :) = x.*sin(2*x);

% Calculate f_1 if required
if needi(1) == 1
fm(1, :) = fm(2, :).*cos(15*x);
end

% Complete f_2 calculation if required.
if needi(2) == 1
fm(2, :) = fm(2, :).*x.*cos(50*x);
end
case 0
% Final return
end
end

% query some currently set options and statistics.
[ivalue, rvalue, cvalue, optype, ifail] = ...
[ivalue, rvalue, cvalue, optype, ifail] = ...
d01zl('Maximum Subdivisions', iopts, opts);
display_option('Maximum Subdivisions',optype,ivalue,rvalue,cvalue);
[ivalue, rvalue, cvalue, optype, ifail] = ...
d01zl('Extrapolation', iopts, opts);
display_option('Extrapolation',optype,ivalue,rvalue,cvalue);
[ivalue, rvalue, cvalue, optype, ifail] = ...
d01zl('Extrapolation Safeguard', iopts, opts);
display_option('Extrapolation Safeguard',optype,ivalue,rvalue,cvalue);

% print solution
fprintf('\nIntegral |  needi  |   dinest   |   errest   \n');
for j=1:ni
fprintf('%9d %9d %12.4e %12.4e\n', j, needi(j), dinest(j), errest(j));
end

function [dinest, errest, user] = monit(ni, ns, dinest, errest, fcount, ...
sinfoi, evals, ldi, sinfor, fs, ...
es, ldr, user)
% Display information on individual segments
fprintf('\nInformation on splitting and evaluations over subregions.\n');
for k=1:ns
sid = sinfoi(1,k);
parent = sinfoi(2,k);
child1 = sinfoi(3,k);
child2 = sinfoi(4,k);
level = sinfoi(5,k);
lbnd = sinfor(1,k);
ubnd = sinfor(2,k);
fprintf('\nSegment %3d Sid = %3d', k, sid);
fprintf(' Parent = %3d Level = %3d.\n', parent, level);
if (child1>0)
fprintf('Children = (%3d, %3d)\n', child1, child2);
end
fprintf('Bounds (%11.4e, %11.4e)\n', lbnd, ubnd);
for j = 1:ni
if (evals(j,k) ~= 0)
fprintf('Integral %2d approximation %11.4e\n', j, fs(j,k));
fprintf('Integral %2d error estimate %11.4e\n', j, es(j,k));
end
if (evals(j,k) ~= 1)
fprintf('Integral %2d evaluation', j);
fprintf(' has been superseded by descendants.\n');
end
end
end

function display_option(optstr,optype,ivalue,rvalue,cvalue)
% Query optype and print the appropriate option values

switch optype
case 1
fprintf('%30s: %13d\n', optstr, ivalue);
case 2
fprintf('%30s: %13.4e\n', optstr, rvalue);
case 3
fprintf('%30s: %16s\n', optstr, cvalue);
case 4
fprintf('%30s: %3d  %16s\n', optstr, ivalue, cvalue);
case 5
fprintf('%30s: %14.4e  %16s\n', optstr, rvalue, cvalue);
end
```
```d01rc example results

Maximum Subdivisions:            50
Extrapolation: ON
Extrapolation Safeguard:    1.0000e-12

Integral |  needi  |   dinest   |   errest
1         0  -2.8431e-02   1.1234e-14
2         0   7.9083e-03   2.6600e-09
```