NAG Library Routine Document

G05ZTF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{NS}$ – INTEGERInput

On entry: the number of steps (points) to be generated in realizations of the increments of the fractional Brownian motion. This must be the same value as supplied to G05ZNF when calculating the eigenvalues of the embedding matrix.

Note: in the context of fractional Brownian motion, NS represents the number of steps from a zero starting state. Realizations returned in Z include this starting state and so $\mathbf{NS}+1$ values are returned for each realization..

Constraint: $\mathbf{NS}\ge 1$.

2:     $\mathrm{S}$ – INTEGERInput

On entry: $S$, the number of realizations of the fractional Brownian motion to simulate.

Constraint: $\mathbf{S}\ge 1$.

3:     $\mathrm{M}$ – INTEGERInput

On entry: the size, $M$, of the embedding matrix, as returned by G05ZMF or G05ZNF.

Constraint: $\mathbf{M}\ge \max \left(1,2\left(\mathbf{NS}-1\right)\right)$.

4:     $\mathrm{XMAX}$ – REAL (KIND=nag_wp)Input

On entry: the upper bound for the interval over which the fractional Brownian motion is to be simulated, as input to G05ZMF or G05ZNF.

Constraint: $\mathbf{XMAX}>0.0$.

5:     $\mathrm{H}$ – REAL (KIND=nag_wp)Input

On entry: the Hurst parameter, $H$, for the fractional Brownian motion. This must be the same value as supplied to G05ZNF in $\mathbf{PARAMS}\left(1\right)$, when the eigenvalues of the embedding matrix were calculated.

Constraint: $0.0<\mathbf{H}<1.0$.

6:     $\mathrm{LAM}\left(\mathbf{M}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by G05ZMF or G05ZNF.

Constraint: $\mathbf{LAM}\left(\mathit{i}\right)\ge 0$, for $\mathit{i}=1,2,\dots ,\mathbf{M}$.

7:     $\mathrm{RHO}$ – REAL (KIND=nag_wp)Input

On entry: indicates the scaling of the covariance matrix, as returned by G05ZMF or G05ZNF.

Constraint: $0.0<\mathbf{RHO}\le 1.0$.

8:     $\mathrm{STATE}\left(*\right)$ – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

9:     $\mathrm{Z}\left(\mathbf{NS}+1,\mathbf{S}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: contains the realizations of the fractional Brownian motion, $Z$. The $\mathit{j}$th realization, for the $\mathit{i}$th point $\mathbf{XX}\left(\mathit{i}\right)$, is stored in $\mathbf{Z}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,\mathbf{S}$ and $\mathit{i}=1,2,\dots ,\mathbf{NS}+1$.

10:   $\mathrm{XX}\left(\mathbf{NS}+1\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the points at which values of the fractional Brownian motion are output. The first point is always zero, and the subsequent NS points represent the equispaced steps towards the last point, XMAX. Note that in G05ZMF and G05ZNF, the returned NS sample points are the mid-points of the grid returned in XX here.

11:   $\mathrm{IFAIL}$ – INTEGERInput/Output

On 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, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On 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

$\mathbf{IFAIL}=1$

On entry, $\mathbf{NS}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{NS}\ge 1$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{S}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{S}\ge 1$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{M}=〈\mathit{\text{value}}〉$, and $\mathbf{NS}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{M}\ge \max \left(1,2\left(\mathbf{NS}-1\right)\right)$.

$\mathbf{IFAIL}=4$

On entry, $\mathbf{XMAX}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{XMAX}>0.0$.

$\mathbf{IFAIL}=5$

On entry, $\mathbf{H}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{H}<1.0$.

$\mathbf{IFAIL}=6$

On entry, at least one element of LAM was negative.
Constraint: all elements of LAM must be non-negative.

$\mathbf{IFAIL}=7$

On entry, $\mathbf{RHO}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{RHO}\le 1.0$.

$\mathbf{IFAIL}=8$

On entry, STATE vector has been corrupted or not initialized.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g05ztfe.f90)

10.2  Program Data

Program Data (g05ztfe.d)

10.3  Program Results

Program Results (g05ztfe.r)