NAG Library Routine Document

G05XBF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{NPATHS}$ – INTEGERInput

On entry: the number of Wiener sample paths to create.

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

2:     $\mathrm{RCORD}$ – INTEGERInput

On entry: the order in which Normal random numbers are stored in Z and in which the generated values are returned in B.

Constraint: $\mathbf{RCORD}=1$ or $2$.

3:     $\mathrm{D}$ – INTEGERInput

On entry: the dimension of each Wiener sample path.

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

4:     $\mathrm{START}\left(\mathbf{D}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the starting value of the Wiener process.

5:     $\mathrm{A}$ – INTEGERInput

On entry: if $\mathbf{A}=0$, a free Wiener process is created beginning at START and TERM is ignored.

If $\mathbf{A}=1$, a non-free Wiener process is created beginning at START and ending at TERM.

Constraint: $\mathbf{A}=0$ or $1$.

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

On entry: the terminal value at which the non-free Wiener process should end. If $\mathbf{A}=0$, TERM is ignored.

7:     $\mathrm{Z}\left(\mathbf{LDZ},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Z must be at least $\mathbf{NPATHS}$ if $\mathbf{RCORD}=1$ and at least $\mathbf{D}\times \left(\mathit{N}+1-\mathbf{A}\right)$ if $\mathbf{RCORD}=2$.

On entry: the Normal random numbers used to construct the sample paths.

If $\mathbf{RCORD}=1$ and quasi-random numbers are used, the $\mathbf{D}\times \left(\mathit{N}+1-\mathbf{A}\right)$, where $\mathit{N}=\mathrm{nint}$ $\mathbf{RCOMM}\left(2\right)$-dimensional quasi-random points should be stored in successive columns of Z.

If $\mathbf{RCORD}=2$ and quasi-random numbers are used, the $\mathbf{D}\times \left(\mathit{N}+1-\mathbf{A}\right)$, where $\mathit{N}=\mathrm{nint}$ $\mathbf{RCOMM}\left(2\right)$-dimensional quasi-random points should be stored in successive rows of Z.

On exit: the Normal random numbers premultiplied by $C$.

8:     $\mathrm{LDZ}$ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which G05XBF is called.

Constraints:

• if $\mathbf{RCORD}=1$, $\mathbf{LDZ}\ge \mathbf{D}\times \left(\mathit{N}+1-\mathbf{A}\right)$;
• if $\mathbf{RCORD}=2$, $\mathbf{LDZ}\ge \mathbf{NPATHS}$.

9:     $\mathrm{C}\left(\mathbf{LDC},*\right)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array C must be at least $\mathbf{D}$.

On entry: the lower triangular Cholesky factorization $C$ such that $C{C}^{\mathrm{T}}$ gives the covariance matrix of the Wiener process. Elements of $C$ above the diagonal are not referenced.

10:   $\mathrm{LDC}$ – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which G05XBF is called.

Constraint: $\mathbf{LDC}\ge \mathbf{D}$.

11:   $\mathrm{B}\left(\mathbf{LDB},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array B must be at least $\mathbf{NPATHS}$ if $\mathbf{RCORD}=1$ and at least $\mathbf{D}\times \left(\mathit{N}+1\right)$ if $\mathbf{RCORD}=2$.

On exit: the values of the Wiener sample paths.

Let $X_{p,i}^k$ denote the $k$th dimension of the $i$th point of the $p$th sample path where $1\le k\le \mathbf{D}$, $1\le i\le \mathit{N}+1$ and $1\le p\le \mathbf{NPATHS}$.

If $\mathbf{RCORD}=1$, the point $X_{p,i}^k$ will be stored at $\mathbf{B}\left(k+\left(i-1\right)\times \mathbf{D},p\right)$.

If $\mathbf{RCORD}=2$, the point $X_{p,i}^k$ will be stored at $\mathbf{B}\left(p,k+\left(i-1\right)\times \mathbf{D}\right)$.

The starting value START is never stored, whereas the terminal value is always stored.

12:   $\mathrm{LDB}$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which G05XBF is called.

Constraints:

• if $\mathbf{RCORD}=1$, $\mathbf{LDB}\ge \mathbf{D}\times \left(\mathit{N}+1\right)$;
• if $\mathbf{RCORD}=2$, $\mathbf{LDB}\ge \mathbf{NPATHS}$.

13:   $\mathrm{RCOMM}\left(*\right)$ – REAL (KIND=nag_wp) arrayCommunication 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 RCOMM in the previous call to G05XAF or G05XBF.

On entry: communication array as returned by the last call to G05XAF or G05XBF. This array must not be directly modified.

14:   $\mathrm{IFAIL}$ – 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, RCOMM was not initialized or has been corrupted.

$\mathbf{IFAIL}=2$

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

$\mathbf{IFAIL}=3$

On entry, $\mathbf{RCORD}=〈\mathit{\text{value}}〉$ was an illegal value.

$\mathbf{IFAIL}=4$

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

$\mathbf{IFAIL}=5$

On entry, $\mathbf{A}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{A}=0\text{ or }1$.

$\mathbf{IFAIL}=6$

On entry, $\mathbf{LDZ}=〈\mathit{\text{value}}〉$ and $\mathbf{D}\times \left(\mathbf{NTIMES}+1-\mathbf{A}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{LDZ}\ge \mathbf{D}\times \left(\mathbf{NTIMES}+1-\mathbf{A}\right)$.

On entry, $\mathbf{LDZ}=〈\mathit{\text{value}}〉$ and $\mathbf{NPATHS}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{LDZ}\ge \mathbf{NPATHS}$.

$\mathbf{IFAIL}=7$

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

$\mathbf{IFAIL}=8$

On entry, $\mathbf{LDB}=〈\mathit{\text{value}}〉$ and $\mathbf{D}\times \left(\mathbf{NTIMES}+1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{LDB}\ge \mathbf{D}\times \left(\mathbf{NTIMES}+1\right)$.

On entry, $\mathbf{LDB}=〈\mathit{\text{value}}〉$ and $\mathbf{NPATHS}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{LDB}\ge \mathbf{NPATHS}$.

$\mathbf{IFAIL}=-99$

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 (g05xbfe.f90)

10.2  Program Data

10.3  Program Results

Program Results (g05xbfe.r)