NAG CL Interface
f01gbc (real_​gen_​matrix_​actexp_​rcomm)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{irevcm}$ – Integer * Input/Output

On initial entry: must be set to $0$.

On intermediate exit: $\mathbf{irevcm}=1$, $2$, $3$, $4$ or $5$. The calling program must:

1. (a)if $\mathbf{irevcm}=1$: evaluate $B_2=AB$, where $B_2$ is an $n$ by $m$ matrix, and store the result in b2;
   if $\mathbf{irevcm}=2$: evaluate $Y=AX$, where $X$ and $Y$ are $n$ by $2$ matrices, and store the result in y;
   if $\mathbf{irevcm}=3$: evaluate $X=A^{\mathrm{T}}Y$ and store the result in x;
   if $\mathbf{irevcm}=4$: evaluate $p=Az$ and store the result in p;
   if $\mathbf{irevcm}=5$: evaluate $r=A^{\mathrm{T}}z$ and store the result in r.
2. (b)call f01gbc again with all other parameters unchanged.

On final exit: $\mathbf{irevcm}=0$.

Note: any values you return to f01gbc as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by f01gbc. If your code inadvertently does return any NaNs or infinities, f01gbc is likely to produce unexpected results.

2: $\mathbf{n}$ – Integer Input

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

3: $\mathbf{m}$ – Integer Input

On entry: the number of columns of the matrix $B$.

Constraint: $\mathbf{m}\ge 0$.

4: $\mathbf{b}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array b must be at least $\mathbf{pdb}\times \mathbf{m}$.

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

On initial entry: the $n$ by $m$ matrix $B$.

On intermediate exit: if $\mathbf{irevcm}=1$, contains the $n$ by $m$ matrix $B$.

On intermediate re-entry: must not be changed.

On final exit: the $n$ by $m$ matrix $e^{tA}B$.

5: $\mathbf{pdb}$ – Integer Input

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

Constraint: $\mathbf{pdb}\ge \mathbf{n}$.

6: $\mathbf{t}$ – double Input

On entry: the scalar $t$.

7: $\mathbf{tr}$ – double Input

On entry: the trace of $A$. If this is not available then any number can be supplied ($0.0$ is a reasonable default); however, in the trivial case, $n=1$, the result $e^{\mathbf{tr}t}B$ is immediately returned in the first row of $B$. See Section 9.

8: $\mathbf{b2}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array b2 must be at least $\mathbf{pdb2}\times \mathbf{m}$.

The $\left(i,j\right)$th element of the matrix is stored in $\mathbf{b2}\left[\left(j-1\right)\times \mathbf{pdb2}+i-1\right]$.

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}=1$, must contain $AB$.

On final exit: the array is undefined.

9: $\mathbf{pdb2}$ – Integer Input

On entry: the stride separating matrix row elements in the array b2.

Constraint: $\mathbf{pdb2}\ge \mathbf{n}$.

10: $\mathbf{x}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array x must be at least $\mathbf{pdx}\times 2$.

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

On initial entry: need not be set.

On intermediate exit: if $\mathbf{irevcm}=2$, contains the current $n$ by $2$ matrix $X$.

On intermediate re-entry: if $\mathbf{irevcm}=3$, must contain $A^{\mathrm{T}}Y$.

On final exit: the array is undefined.

11: $\mathbf{pdx}$ – Integer Input

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

Constraint: $\mathbf{pdx}\ge \mathbf{n}$.

12: $\mathbf{y}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array y must be at least $\mathbf{pdy}\times 2$.

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

On initial entry: need not be set.

On intermediate exit: if $\mathbf{irevcm}=3$, contains the current $n$ by $2$ matrix $Y$.

On intermediate re-entry: if $\mathbf{irevcm}=2$, must contain $AX$.

On final exit: the array is undefined.

13: $\mathbf{pdy}$ – Integer Input

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

Constraint: $\mathbf{pdy}\ge \mathbf{n}$.

14: $\mathbf{p}\left[\mathbf{n}\right]$ – double Input/Output

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}=4$, must contain $Az$.

On final exit: the array is undefined.

15: $\mathbf{r}\left[\mathbf{n}\right]$ – double Input/Output

On initial entry: need not be set.

On intermediate re-entry: if $\mathbf{irevcm}=5$, must contain $A^{\mathrm{T}}z$.

On final exit: the array is undefined.

16: $\mathbf{z}\left[\mathbf{n}\right]$ – double Input/Output

On initial entry: need not be set.

On intermediate exit: if $\mathbf{irevcm}=4$ or $5$, contains the vector $z$.

On intermediate re-entry: must not be changed.

On final exit: the array is undefined.

17: $\mathbf{comm}\left[\mathbf{n}\times \mathbf{m}+3\times \mathbf{n}+12\right]$ – double Communication Array

Note: the dimension, dim, of the array comm must be at least $\mathbf{n}\times \mathbf{m}+3\times \mathbf{n}+12$.

18: $\mathbf{icomm}\left[2\times \mathbf{n}+40\right]$ – Integer Communication Array

19: $\mathbf{fail}$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On initial entry, $\mathbf{irevcm}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{irevcm}=0$.

On intermediate re-entry, $\mathbf{irevcm}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{irevcm}=1$, $2$, $3$, $4$ or $5$.

NE_INT_2

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

On entry, $\mathbf{pdb2}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb2}\ge \mathbf{n}$.

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

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

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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

NW_SOME_PRECISION_LOSS

$e^{tA}B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Use of $Tr\left(A\right)$

9.2 When to use f01gbc

9.3 Use in Conjunction with NAG Library Functions

do {
f01gbc(&irevcm,n,m,b,tdb,t,tr,b2,tdb2,x,tdx,y,tdy,p,r,z,comm,icomm,&fail);
  if (irevcm == 1) {
    .. Code to compute B2=AB ..
  }
  else if (irevcm == 2){
    .. Code to compute Y=AX ..
  }
  else if (irevcm == 3){
    .. Code to compute X=A^T Y ..
  }
  else if (irevcm == 4){
    .. Code to compute P=AZ ..
  }
  else if (irevcm == 5){
    .. Code to compute R=A^T Z ..
  }
} (while irevcm !=0)

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results