NAG Library Function Document

nag_rand_bb (g05xbc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:     npaths – IntegerInput

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

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

3:     d – IntegerInput

On entry: the dimension of each Wiener sample path.

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

4:     start[d] – const doubleInput

On entry: the starting value of the Wiener process.

5:     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:     term[d] – const doubleInput

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

7:     z[$\mathit{dim}$] – doubleInput/Output

Note: the dimension, dim, of the array z must be at least

• $\mathbf{pdz}\times \mathbf{npaths}$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$;
• $\mathbf{pdz}\times (\mathbf{d}\times \left(\mathit{N}+1-\mathbf{a}\right))$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$.

The $\left(i,j\right)$th element of the matrix $Z$ is stored in

• $\mathbf{z}\left[\left(j-1\right)\times \mathbf{pdz}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{z}\left[\left(i-1\right)\times \mathbf{pdz}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

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

If quasi-random numbers are used, the $\mathbf{d}\times \left(\mathit{N}+1-\mathbf{a}\right)$-dimensional quasi-random points should be stored in successive rows of $Z$.

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

8:     pdz – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdz}\ge \mathbf{d}\times \left(\mathit{N}+1-\mathbf{a}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdz}\ge \mathbf{npaths}$.

9:     c[$\mathit{dim}$] – const doubleInput

Note: the dimension, dim, of the array c must be at least $\mathbf{pdc}\times \mathbf{d}$.

The $\left(i,j\right)$th element of the matrix $C$ is stored in $\mathbf{c}\left[\left(j-1\right)\times \mathbf{pdc}+i-1\right]$.

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:   pdc – IntegerInput

On entry: the stride separating matrix row elements in the array c.

Constraint: $\mathbf{pdc}\ge \mathbf{d}$.

11:   b[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array b must be at least

• $\mathbf{pdb}\times \mathbf{npaths}$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$;
• $\mathbf{pdb}\times (\mathbf{d}\times \left(\mathit{N}+1\right))$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$.

The $\left(i,j\right)$th element of the matrix $B$ is stored in

• $\mathbf{b}\left[\left(j-1\right)\times \mathbf{pdb}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{b}\left[\left(i-1\right)\times \mathbf{pdb}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

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 $1\le p\le \mathbf{npaths}$. The point $X_{p,i}^k$ is stored at $B\left(p,k+\left(i-1\right)\times \mathbf{d}\right)$. The starting value start is never stored, whereas the terminal value is always stored.

12:   pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdb}\ge \mathbf{d}\times \left(\mathit{N}+1\right)$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdb}\ge \mathbf{npaths}$.

13:   rcomm[$\mathit{dim}$] – const doubleCommunication Array

Note: the dimension, $\mathit{dim}$, 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 nag_rand_bb_init (g05xac) or nag_rand_bb (g05xbc).

On entry: communication array as returned by the last call to nag_rand_bb_init (g05xac) or nag_rand_bb (g05xbc). This array MUST NOT be directly modified.

14:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_ARRAY_SIZE

On entry, $\mathbf{pdb}=⟨\mathit{\text{value}}⟩$ and $\mathbf{d}\times \left(\mathbf{ntimes}+1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdb}\ge \mathbf{d}\times \left(\mathbf{ntimes}+1\right)$.

On entry, $\mathbf{pdb}=⟨\mathit{\text{value}}⟩$ and $\mathbf{npaths}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdb}\ge \mathbf{npaths}$.

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

On entry, $\mathbf{pdz}=⟨\mathit{\text{value}}⟩$ and $\mathbf{d}\times \left(\mathbf{ntimes}+1-\mathbf{a}\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdz}\ge \mathbf{d}\times \left(\mathbf{ntimes}+1-\mathbf{a}\right)$.

On entry, $\mathbf{pdz}=⟨\mathit{\text{value}}⟩$ and $\mathbf{npaths}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdz}\ge \mathbf{npaths}$.

NE_BAD_PARAM

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

NE_ILLEGAL_COMM

On entry, rcomm was not initialized or has been corrupted.

NE_INT

On entry, $\mathbf{a}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{a}=0\text{ or }1$.

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

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

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (g05xbce.c)

10.2  Program Data

10.3  Program Results

Program Results (g05xbce.r)