NAG Library Routine Document

F01GBF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     IREVCM – INTEGERInput/Output

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

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

(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.

(b)

call F01GBF again with all other parameters unchanged.

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

2:     N – INTEGERInput

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

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

3:     M – INTEGERInput

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

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

4:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output

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

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:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F01GBF is called.

Constraint: $\mathbf{LDB}\ge \max \left(1,\mathbf{N}\right)$.

6:     T – REAL (KIND=nag_wp)Input

On entry: the scalar $t$.

7:     TR – REAL (KIND=nag_wp)Input

On entry: the trace of $A$. If this is not available then any number can be supplied ($0$ 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 8.

8:     B2(LDB2,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B2 must be at least $\mathbf{M}$.

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:     LDB2 – INTEGERInput

On initial entry: the first dimension of the array B2 as declared in the (sub)program from which F01GBF is called.

Constraint: $\mathbf{LDB2}\ge \max \left(1,\mathbf{N}\right)$.

10:   X(LDX,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array X must be at least $2$.

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:   LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F01GBF is called.

Constraint: $\mathbf{LDX}\ge \max \left(1,\mathbf{N}\right)$.

12:   Y(LDY,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Y must be at least $2$.

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:   LDY – INTEGERInput

On entry: the first dimension of the array Y as declared in the (sub)program from which F01GBF is called.

Constraint: $\mathbf{LDY}\ge \max \left(1,\mathbf{N}\right)$.

14:   P(N) – REAL (KIND=nag_wp) arrayInput/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:   R(N) – REAL (KIND=nag_wp) arrayInput/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:   Z(N) – REAL (KIND=nag_wp) arrayInput/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:   COMM($\mathbf{N}\times \mathbf{M}+3\times \mathbf{N}+12$) – REAL (KIND=nag_wp) arrayCommunication Array

18:   ICOMM($2\times \mathbf{N}+40$) – INTEGER arrayCommunication Array

19:   IFAIL – INTEGERInput/Output

On initial 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, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On final 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

$\mathbf{IFAIL}=1$

Note:  this failure should not occur, and suggests that the routine has been called incorrectly. An unexpected internal error occurred when estimating a matrix norm.

$\mathbf{IFAIL}=2$

An unexpected internal error has occurred. Please contact NAG.

$\mathbf{IFAIL}=-1$

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 $5$.

$\mathbf{IFAIL}=-2$

On initial entry, $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{N}\ge 0$.

$\mathbf{IFAIL}=-3$

On initial entry, $\mathbf{M}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{M}\ge 0$.

$\mathbf{IFAIL}=-5$

On initial entry, $\mathbf{LDB}=⟨\mathit{\text{value}}⟩$ and $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LDB}\ge \max \left(1,\mathbf{N}\right)$.

$\mathbf{IFAIL}=-9$

On initial entry, $\mathbf{LDB2}=⟨\mathit{\text{value}}⟩$ and $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LDB2}\ge \max \left(1,\mathbf{N}\right)$.

$\mathbf{IFAIL}=-11$

On initial entry, $\mathbf{LDX}=⟨\mathit{\text{value}}⟩$ and $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LDX}\ge \max \left(1,\mathbf{N}\right)$.

$\mathbf{IFAIL}=-13$

On initial entry, $\mathbf{LDY}=⟨\mathit{\text{value}}⟩$ and $\mathbf{N}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{LDY}\ge \max \left(1,\mathbf{N}\right)$.

7  Accuracy

8  Further Comments

8.1  Use of $Tr\left(A\right)$

8.2  When to use F01GBF

8.3  Use in Conjunction with NAG Library Routines

revcm: Do 
  Call F01GBF(IREVCM,N,M,B,LDB,T,TR,B2,LDB2,X,LDX,Y,LDY,P,R,Z, &
              COMM,ICOMM,IFAIL)
  If (IREVCM == 0) Then 
      Exit revcm 
  Else If (IREVCM == 1) Then
      .. Code to compute B2=AB ..
  Else If (IREVCM == 2) Then
      .. Code to compute Y=AX ..
  Else If (IREVCM == 3) Then
      .. Code to compute X=A^T Y ..
  Else If (IREVCM == 4) Then
      .. Code to compute P=AZ ..
  Else If (IREVCM == 5) Then 
      .. Code to compute R=A^T Z ..
  End If
End Do revcm

9  Example

9.1  Program Text

Program Text (f01gbfe.f90)

9.2  Program Data

Program Data (f01gbfe.d)

9.3  Program Results

Program Results (f01gbfe.r)