NAG FL Interface
f01hbf (complex_gen_matrix_actexp_rcomm)
1
Purpose
f01hbf computes the action of the matrix exponential ${e}^{tA}$, on the matrix $B$, where $A$ is a complex $n$ by $n$ matrix, $B$ is a complex $n$ by $m$ matrix and $t$ is a complex scalar. It uses reverse communication for evaluating matrix products, so that the matrix $A$ is not accessed explicitly.
2
Specification
Fortran Interface
Subroutine f01hbf ( 
irevcm, n, m, b, ldb, t, tr, b2, ldb2, x, ldx, y, ldy, p, r, z, ccomm, comm, icomm, ifail) 
Integer, Intent (In) 
:: 
n, m, ldb, ldb2, ldx, ldy 
Integer, Intent (Inout) 
:: 
irevcm, icomm(2*n+40), ifail 
Real (Kind=nag_wp), Intent (Inout) 
:: 
comm(3*n+14) 
Complex (Kind=nag_wp), Intent (In) 
:: 
t, tr 
Complex (Kind=nag_wp), Intent (Inout) 
:: 
b(ldb,*), b2(ldb2,*), x(ldx,*), y(ldy,*), p(n), r(n), z(n), ccomm(n*(m+2)) 

C Header Interface
#include <nag.h>
void 
f01hbf_ (Integer *irevcm, const Integer *n, const Integer *m, Complex b[], const Integer *ldb, const Complex *t, const Complex *tr, Complex b2[], const Integer *ldb2, Complex x[], const Integer *ldx, Complex y[], const Integer *ldy, Complex p[], Complex r[], Complex z[], Complex ccomm[], double comm[], Integer icomm[], Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f01hbf_ (Integer &irevcm, const Integer &n, const Integer &m, Complex b[], const Integer &ldb, const Complex &t, const Complex &tr, Complex b2[], const Integer &ldb2, Complex x[], const Integer &ldx, Complex y[], const Integer &ldy, Complex p[], Complex r[], Complex z[], Complex ccomm[], double comm[], Integer icomm[], Integer &ifail) 
}

The routine may be called by the names f01hbf or nagf_matop_complex_gen_matrix_actexp_rcomm.
3
Description
${e}^{tA}B$ is computed using the algorithm described in
Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the
${e}^{tA}B$ without explicitly forming
${e}^{tA}$.
The algorithm does not explicity need to access the elements of $A$; it only requires the result of matrix multiplications of the form $AX$ or ${A}^{\mathrm{H}}Y$. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.
4
References
Al–Mohy A H and Higham N J (2011) Computing the action of the matrix exponential, with an application to exponential integrators SIAM J. Sci. Statist. Comput. 33(2) 488511
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5
Arguments
Note: this routine uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the
argument irevcm. Between intermediate exits and reentries,
all arguments other than b2, x, y, p and r must remain unchanged.

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:

(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 $X={A}^{\mathrm{H}}Y$ and store the result in x;
if ${\mathbf{irevcm}}=4$: evaluate $p=Az$ and store the result in p;
if ${\mathbf{irevcm}}=5$: evaluate $r={A}^{\mathrm{H}}z$ and store the result in r.

(b)call f01hbf again with all other parameters unchanged.
On final exit: ${\mathbf{irevcm}}=0$.
Note: any values you return to f01hbf as part of the reverse communication procedure should not include floatingpoint NaN (Not a Number) or infinity values, since these are not handled by f01hbf. If your code does inadvertently return any NaNs or infinities, f01hbf 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:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) array
Input/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 reentry: must not be changed.
On final exit: the $n$ by $m$ matrix ${e}^{tA}B$.

5:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f01hbf is called.
Constraint:
${\mathbf{ldb}}\ge {\mathbf{n}}$.

6:
$\mathbf{t}$ – Complex (Kind=nag_wp)
Input

On entry: the scalar $t$.

7:
$\mathbf{tr}$ – Complex (Kind=nag_wp)
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:
$\mathbf{b2}\left({\mathbf{ldb2}},*\right)$ – Complex (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
b2
must be at least
${\mathbf{m}}$.
On initial entry: need not be set.
On intermediate reentry: if ${\mathbf{irevcm}}=1$, must contain $AB$.
On final exit: the array is undefined.

9:
$\mathbf{ldb2}$ – Integer
Input

On initial entry: the first dimension of the array
b2 as declared in the (sub)program from which
f01hbf is called.
Constraint:
${\mathbf{ldb2}}\ge {\mathbf{n}}$.

10:
$\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Complex (Kind=nag_wp) array
Input/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 reentry: if ${\mathbf{irevcm}}=3$, must contain ${A}^{\mathrm{H}}Y$.
On final exit: the array is undefined.

11:
$\mathbf{ldx}$ – Integer
Input

On entry: the first dimension of the array
x as declared in the (sub)program from which
f01hbf is called.
Constraint:
${\mathbf{ldx}}\ge {\mathbf{n}}$.

12:
$\mathbf{y}\left({\mathbf{ldy}},*\right)$ – Complex (Kind=nag_wp) array
Input/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 reentry: if ${\mathbf{irevcm}}=2$, must contain $AX$.
On final exit: the array is undefined.

13:
$\mathbf{ldy}$ – Integer
Input

On entry: the first dimension of the array
y as declared in the (sub)program from which
f01hbf is called.
Constraint:
${\mathbf{ldy}}\ge {\mathbf{n}}$.

14:
$\mathbf{p}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) array
Input/Output

On initial entry: need not be set.
On intermediate reentry: if ${\mathbf{irevcm}}=4$, must contain $Az$.
On final exit: the array is undefined.

15:
$\mathbf{r}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) array
Input/Output

On initial entry: need not be set.
On intermediate reentry: if ${\mathbf{irevcm}}=5$, must contain ${A}^{\mathrm{H}}z$.
On final exit: the array is undefined.

16:
$\mathbf{z}\left({\mathbf{n}}\right)$ – Complex (Kind=nag_wp) array
Input/Output

On initial entry: need not be set.
On intermediate exit:
if ${\mathbf{irevcm}}=4$ or $5$, contains the vector $z$.
On intermediate reentry: must not be changed.
On final exit: the array is undefined.

17:
$\mathbf{ccomm}\left({\mathbf{n}}\times \left({\mathbf{m}}+2\right)\right)$ – Complex (Kind=nag_wp) array
Communication Array


18:
$\mathbf{comm}\left(3\times {\mathbf{n}}+14\right)$ – Real (Kind=nag_wp) array
Communication Array


19:
$\mathbf{icomm}\left(2\times {\mathbf{n}}+40\right)$ – Integer array
Communication Array


20:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface 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 argument, 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
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=2$

${e}^{tA}B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.
 ${\mathbf{ifail}}=1$

On initial entry, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate reentry, ${\mathbf{irevcm}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{irevcm}}=1$, $2$, $3$, $4$ or $5$.
 ${\mathbf{ifail}}=2$

On initial entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=3$

On initial entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 0$.
 ${\mathbf{ifail}}=5$

On initial entry, ${\mathbf{ldb}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{n}}$.
 ${\mathbf{ifail}}=9$

On initial entry, ${\mathbf{ldb2}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldb2}}\ge {\mathbf{n}}$.
 ${\mathbf{ifail}}=11$

On initial entry, ${\mathbf{ldx}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
 ${\mathbf{ifail}}=13$

On initial entry, ${\mathbf{ldy}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ldy}}\ge {\mathbf{n}}$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
For an Hermitian matrix
$A$ (for which
${A}^{\mathrm{H}}=A$) the computed matrix
${e}^{tA}B$ is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for nonHermitian matrices. See Section 4 of
Al–Mohy and Higham (2011) for details and further discussion.
8
Parallelism and Performance
f01hbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The elements of $A$ are not explicitly required by f01hbf. However, the trace of $A$ is used in the preprocessing phase of the algorithm. If $Tr\left(A\right)$ is not available to the calling subroutine then any number can be supplied ($0$ is recommended). This will not affect the stability of the algorithm, but it may reduce its efficiency.
f01hbf is designed to be used when $A$ is large and sparse. Whenever a matrix multiplication is required, the routine will return control to the calling program so that the multiplication can be done in the most efficient way possible. Note that ${e}^{tA}B$ will not, in general, be sparse even if $A$ is sparse.
If
$A$ is small and dense then
f01haf can be used to compute
${e}^{tA}B$ without the use of a reverse communication interface.
The real analog of
f01hbf is
f01gbf.
To compute
${e}^{tA}B$, the following skeleton code can normally be used:
revcm: Do
Call f01hbf(irevcm,n,m,b,ldb,t,tr,b2,ldb2,x,ldx,y,ldx,p,r,z, &
ccomm,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^h y ..
Else If (irevcm == 4) Then
.. Code to compute p=az ..
Else If (irevcm == 5) Then
.. Code to compute r=a^h z ..
End If
End Do revcm
The code used to compute the matrix products will vary depending on the way
$A$ is stored. If all the elements of
$A$ are stored explicitly, then
f06zaf can be used. If
$A$ is triangular then
f06zff should be used. If
$A$ is Hermitian, then
f06zcf should be used. If
$A$ is symmetric, then
f06ztf should be used. For sparse
$A$ stored in coordinate storage format
f11xnf and
f11xsf can be used. For sparse
$A$ stored in compressed column storage format (CCS) the program text of
Section 10 contains the routine matmul to perform matrix products.
10
Example
This example computes
${e}^{tA}B$ where
and
$A$ is stored in compressed column storage format (CCS) and matrix multiplications are performed using the routine matmul.
10.1
Program Text
10.2
Program Data
10.3
Program Results