nag_rand_quasi_uniform (g05ym) generates a uniformly distributed low-discrepancy sequence as proposed by Sobol, Faure or Niederreiter. It must be preceded by a call to one of the initialization functions nag_rand_quasi_init (g05yl) or nag_rand_quasi_init_scrambled (g05yn).

Syntax

[quas, iref, ifail] = g05ym(n, iref, 'rcord', rcord)

[quas, iref, ifail] = nag_rand_quasi_uniform(n, iref, 'rcord', rcord)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 24:

rcord was made optional

Description

Low discrepancy (quasi-random) sequences are used in numerical integration, simulation and optimization. Like pseudorandom numbers they are uniformly distributed but they are not statistically independent, rather they are designed to give more even distribution in multidimensional space (uniformity). Therefore they are often more efficient than pseudorandom numbers in multidimensional Monte–Carlo methods.

nag_rand_quasi_uniform (g05ym) generates a set of points

x^{1}, x^{2}, \dots, x^{N}

with high uniformity in the

S

-dimensional unit cube

I^{S} = {[0, 1]}^{S}

Let

G

be a subset of

I^{S}

and define the counting function

S_{N} (G)

as the number of points

x^{i} \in G

. For each

x = (x_{1}, x_{2}, \dots, x_{S}) \in I^{S}

, let

G_{x}

be the rectangular

S

-dimensional region

G_{x} = [0, x_{1}) \times [0, x_{2}) \times \dots \times [0, x_{S})

with volume

x_{1}, x_{2}, \dots, x_{S}

. Then one measure of the uniformity of the points

x^{1}, x^{2}, \dots, x^{N}

is the discrepancy:

D_{N}^{*} (x^{1}, x^{2}, \dots, x^{N}) = \sup_{x \in I^{S}} |S_{N} (G_{x}) - N x_{1}, x_{2}, \dots, x_{S}| .

which has the form

D_{N}^{*} (x^{1}, x^{2}, \dots, x^{N}) \leq C_{S} {(\log N)}^{S} + O ({(\log N)}^{S - 1}) for all N \geq 2 .

The principal aim in the construction of low-discrepancy sequences is to find sequences of points in

I^{S}

with a bound of this form where the constant

C_{S}

is as small as possible.

The type of low-discrepancy sequence generated by nag_rand_quasi_uniform (g05ym) depends on the initialization function called and can include those proposed by Sobol, Faure or Niederreiter. If the initialization function nag_rand_quasi_init_scrambled (g05yn) was used then the sequence will be scrambled (see Description in nag_rand_quasi_init_scrambled (g05yn) for details).

References

Bratley P and Fox B L (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator ACM Trans. Math. Software 14(1) 88–100

Fox B L (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators ACM Trans. Math. Software 12(4) 362–376

Parameters

Note: the following variables are used in the parameter descriptions:

$idim = idim$ , the number of dimensions required, see nag_rand_quasi_init (g05yl) or nag_rand_quasi_init_scrambled (g05yn)
$liref = liref$ , the length of iref as supplied to the initialization function nag_rand_quasi_init (g05yl) or nag_rand_quasi_init_scrambled (g05yn)
$tdquas = n$ if $rcord = 1$ ; otherwise $tdquas = idim$

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: The number of quasi-random numbers required.

Constraint: $n \geq 0$ and $n + previous number of generated values \leq 2^{31} - 1$ .
2: $iref (liref)$ – int64int32nag_int array: Contains information on the current state of the sequence.

Optional Input Parameters

1: $rcord$ – int64int32nag_int scalar: Default: $1$
The order in which the generated values are returned.

Constraint: $rcord = 1$ or $2$ .

Output Parameters

1: $quas (ldquas, tdquas)$ – double array: Contains the n quasi-random numbers of dimension idim.
If $rcord = 1$ , $quas (i, j)$ holds the $j$ th value for the $i$ th dimension.

If $rcord = 2$ , $quas (i, j)$ holds the $i$ th value for the $j$ th dimension.
2: $iref (liref)$ – int64int32nag_int array: Contains updated information on the state of the sequence.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n \geq 0$ .

On entry, value of n would result in too many calls to the generator.

$ifail = 2$: Constraint: $rcord = 1$ or $2$ .

$ifail = 4$: Constraint: if $rcord = 1$ , $ldquas \geq idim$ .

Constraint: if $rcord = 2$ , $ldquas \geq n$ .

$ifail = 5$: On entry, iref has either not been initialized or has been corrupted.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

None.

Example

This example calls nag_rand_quasi_init (g05yl) and nag_rand_quasi_uniform (g05ym) to estimate the value of the integral

\int_{0}^{1} \dots \int_{0}^{1} \prod_{i = 1}^{s} |4 x_{i} - 2| d x_{1}, d x_{2}, \dots, d x_{s} = 1 .

In this example the number of dimensions

S

is set to

8

Open in the MATLAB editor: g05ym_example

function g05ym_example


fprintf('g05ym example results\n\n');

% Initialize the Sobol generator, skipping some variates
iskip = int64(1000);
idim  = int64(8);
genid = int64(1);
% Initialize the Sobol generator
[iref, ifail] = g05yl( ...
                       genid,idim,iskip);

% Number of variates
n = int64(200);

% Generate N quasi-random variates
[quas, iref, ifail] = g05ym( ...
                             n, iref);

% Evaluate the function, and sum
p(1:n) = prod(abs(4*quas(1:idim,:)-2));
fsum = sum(p);

% Convert sum to mean value
vsbl = fsum/double(n);
fprintf('Value of integral = %8.4f\n\n', vsbl);
fprintf('First 10 variates\n');
for i = 1:10
  fprintf(' %3d', i);
  fprintf(' %7.4f', quas(1:idim,i));
  fprintf('\n');
end

g05ym example results

Value of integral =   1.0410

First 10 variates
   1  0.7197  0.5967  0.0186  0.1768  0.7803  0.4072  0.5459  0.3994
   2  0.9697  0.3467  0.7686  0.9268  0.5303  0.1572  0.2959  0.1494
   3  0.4697  0.8467  0.2686  0.4268  0.0303  0.6572  0.7959  0.6494
   4  0.3447  0.4717  0.1436  0.3018  0.1553  0.7822  0.4209  0.0244
   5  0.8447  0.9717  0.6436  0.8018  0.6553  0.2822  0.9209  0.5244
   6  0.5947  0.2217  0.3936  0.0518  0.9053  0.0322  0.1709  0.7744
   7  0.0947  0.7217  0.8936  0.5518  0.4053  0.5322  0.6709  0.2744
   8  0.0635  0.1904  0.0498  0.4580  0.6240  0.2510  0.9521  0.8057
   9  0.5635  0.6904  0.5498  0.9580  0.1240  0.7510  0.4521  0.3057
  10  0.8135  0.4404  0.2998  0.2080  0.3740  0.5010  0.7021  0.0557

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