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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_real_gen_matrix_actexp_rcomm (f01gb)

## Purpose

nag_matop_real_gen_matrix_actexp_rcomm (f01gb) computes the action of the matrix exponential etA${e}^{tA}$, on the matrix B$B$, where A$A$ is a real n$n$ by n$n$ matrix, B$B$ is a real n$n$ by m$m$ matrix and t$t$ is a real scalar. It uses reverse communication for evaluating matrix products, so that the matrix A$A$ is not accessed explicitly.

## Syntax

[irevcm, b, b2, x, y, p, r, z, comm, icomm, ifail] = f01gb(irevcm, m, b, t, b2, x, y, p, r, z, comm, icomm, 'n', n, 'tr', tr)
[irevcm, b, b2, x, y, p, r, z, comm, icomm, ifail] = nag_matop_real_gen_matrix_actexp_rcomm(irevcm, m, b, t, b2, x, y, p, r, z, comm, icomm, 'n', n, 'tr', tr)

## Description

etAB${e}^{tA}B$ is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the etAB${e}^{tA}B$ without explicitly forming etA${e}^{tA}$.
The algorithm does not explicity need to access the elements of A$A$; it only requires the result of matrix multiplications of the form AX$AX$ or ATY${A}^{\mathrm{T}}Y$. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.

## 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) 488-511
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## Parameters

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the parameter irevcm. Between intermediate exits and re-entries, all parameters other than b2, x, y, p and r must remain unchanged.

### Compulsory Input Parameters

1:     irevcm – int64int32nag_int scalar
On initial entry: must be set to 0$0$.
2:     m – int64int32nag_int scalar
The number of columns of the matrix B$B$.
Constraint: m0${\mathbf{m}}\ge 0$.
3:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least m${\mathbf{m}}$
On initial entry: the n$n$ by m$m$ matrix B$B$.
On intermediate re-entry: must not be changed.
4:     t – double scalar
The scalar t$t$.
5:     b2(ldb2, : $:$) – double array
The first dimension of the array b2 must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least m${\mathbf{m}}$
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 1${\mathbf{irevcm}}=1$, must contain AB$AB$.
6:     x(ldx, : $:$) – double array
The first dimension of the array x must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least 2$2$
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 3${\mathbf{irevcm}}=3$, must contain ATY${A}^{\mathrm{T}}Y$.
7:     y(ldy, : $:$) – double array
The first dimension of the array y must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least 2$2$
On initial entry: need not be set.
On intermediate re-entry: if irevcm = 2${\mathbf{irevcm}}=2$, must contain AX$AX$.
8:     p(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
On initial entry: need not be set.
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
On intermediate re-entry: if irevcm = 4${\mathbf{irevcm}}=4$, must contain Az$Az$.
9:     r(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
On initial entry: need not be set.
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
On intermediate re-entry: if irevcm = 5${\mathbf{irevcm}}=5$, must contain ATz${A}^{\mathrm{T}}z$.
10:   z(n) – double array
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
On initial entry: need not be set.
n, the dimension of the array, must satisfy the constraint n0${\mathbf{n}}\ge 0$.
On intermediate re-entry: must not be changed.
11:   comm(n × m + 3 × n + 12${\mathbf{n}}×{\mathbf{m}}+3×{\mathbf{n}}+12$) – double array
12:   icomm(2 × n + 40$2×{\mathbf{n}}+40$) – int64int32nag_int array

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays b, b2, x, y, comm and the dimension of the arrays icomm, p, r, z. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     tr – double scalar
The trace of A$A$. If this is not available then any number can be supplied (0$0$ is a reasonable default); however, in the trivial case, n = 1$n=1$, the result etrtB${e}^{{\mathbf{tr}}t}B$ is immediately returned in the first row of B$B$. See Section [Further Comments].
Default: 0$0$

ldb ldb2 ldx ldy

### Output Parameters

1:     irevcm – int64int32nag_int scalar
On intermediate exit: irevcm = 1${\mathbf{irevcm}}=1$, 2$2$, 3$3$, 4$4$ or 5$5$. The calling program must:
 (a) if irevcm = 1${\mathbf{irevcm}}=1$: evaluate B2 = AB${B}_{2}=AB$, where B2${B}_{2}$ is an n$n$ by m$m$ matrix, and store the result in b2; if irevcm = 2${\mathbf{irevcm}}=2$: evaluate Y = AX$Y=AX$, where X$X$ and Y$Y$ are n$n$ by 2$2$ matrices, and store the result in y; if irevcm = 3${\mathbf{irevcm}}=3$: evaluate X = ATY$X={A}^{\mathrm{T}}Y$ and store the result in x; if irevcm = 4${\mathbf{irevcm}}=4$: evaluate p = Az$p=Az$ and store the result in p; if irevcm = 5${\mathbf{irevcm}}=5$: evaluate r = ATz$r={A}^{\mathrm{T}}z$ and store the result in r. (b) call nag_matop_real_gen_matrix_actexp_rcomm (f01gb) again with all other parameters unchanged.
On final exit: irevcm = 0${\mathbf{irevcm}}=0$.
2:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be m${\mathbf{m}}$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On intermediate exit: if irevcm = 1${\mathbf{irevcm}}=1$, contains the n$n$ by m$m$ matrix B$B$.
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On final exit: the n$n$ by m$m$ matrix etAB${e}^{tA}B$.
3:     b2(ldb2, : $:$) – double array
The first dimension of the array b2 will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be m${\mathbf{m}}$
ldb2max (1,n)$\mathit{ldb2}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On final exit: the array is undefined.
4:     x(ldx, : $:$) – double array
The first dimension of the array x will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be 2$2$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On intermediate exit: if irevcm = 2${\mathbf{irevcm}}=2$, contains the current n$n$ by 2$2$ matrix X$X$.
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On final exit: the array is undefined.
5:     y(ldy, : $:$) – double array
The first dimension of the array y will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be 2$2$
ldymax (1,n)$\mathit{ldy}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On intermediate exit: if irevcm = 3${\mathbf{irevcm}}=3$, contains the current n$n$ by 2$2$ matrix Y$Y$.
ldymax (1,n)$\mathit{ldy}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On final exit: the array is undefined.
6:     p(n) – double array
On final exit: the array is undefined.
7:     r(n) – double array
On final exit: the array is undefined.
8:     z(n) – double array
On intermediate exit: if irevcm = 4${\mathbf{irevcm}}=4$ or 5$5$, contains the vector z$z$.
On final exit: the array is undefined.
9:     comm(n × m + 3 × n + 12${\mathbf{n}}×{\mathbf{m}}+3×{\mathbf{n}}+12$) – double array
10:   icomm(2 × n + 40$2×{\mathbf{n}}+40$) – int64int32nag_int array
11:   ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail = 1${\mathbf{ifail}}=1$
Note:  this failure should not occur, and suggests that the function has been called incorrectly. An unexpected internal error occurred when estimating a matrix norm.
W ifail = 2${\mathbf{ifail}}=2$
etAB${e}^{tA}B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.
ifail = 1${\mathbf{ifail}}=-1$
On initial entry.
Constraint: irevcm = 0${\mathbf{irevcm}}=0$.
On intermediate re-entry.
Constraint: irevcm = 1${\mathbf{irevcm}}=1$, 2$2$, 3$3$, 4$4$ or 5$5$.
ifail = 2${\mathbf{ifail}}=-2$
On initial entry.
Constraint: n0${\mathbf{n}}\ge 0$.
ifail = 3${\mathbf{ifail}}=-3$
On initial entry.
Constraint: m0${\mathbf{m}}\ge 0$.
ifail = 5${\mathbf{ifail}}=-5$
On initial entry, ldb = _$\mathit{ldb}=_$ and .
Constraint: ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
ifail = 9${\mathbf{ifail}}=-9$
On initial entry, ldb2 = _$\mathit{ldb2}=_$ and .
Constraint: ldb2max (1,n)$\mathit{ldb2}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
ifail = 11${\mathbf{ifail}}=-11$
On initial entry, ldx = _$\mathit{ldx}=_$ and .
Constraint: ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
ifail = 13${\mathbf{ifail}}=-13$
On initial entry, ldy = _$\mathit{ldy}=_$ and .
Constraint: ldymax (1,n)$\mathit{ldy}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

## Accuracy

For a symmetric matrix A$A$ (for which AT = A${A}^{\mathrm{T}}=A$) the computed matrix etAB${e}^{tA}B$ is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-symmetric matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

### Use of TrA

The elements of A$A$ are not explicitly required by nag_matop_real_gen_matrix_actexp_rcomm (f01gb). However, the trace of A$A$ is used in the preprocessing phase of the algorithm. If Tr(A)$Tr\left(A\right)$ is not available to the calling function then any number can be supplied (0$0$ is recommended). This will not affect the stability of the algorithm, but it may reduce its efficiency.

### When to use nag_matop_real_gen_matrix_actexp_rcomm (f01gb)

nag_matop_real_gen_matrix_actexp_rcomm (f01gb) is designed to be used when A$A$ is large and sparse. Whenever a matrix multiplication is required, the function will return control to the calling program so that the multiplication can be done in the most efficient way possible. Note that etAB${e}^{tA}B$ will not, in general, be sparse even if A$A$ is sparse.
If A$A$ is small and dense then nag_matop_real_gen_matrix_actexp (f01ga) can be used to compute etAB${e}^{tA}B$ without the use of a reverse communication interface.
The complex analog of nag_matop_real_gen_matrix_actexp_rcomm (f01gb) is nag_matop_complex_gen_matrix_actexp_rcomm (f01hb).

### Use in Conjunction with NAG Toolbox Functions

To compute etAB${e}^{tA}B$, the following skeleton code can normally be used:
```while irevcm ~= 0
switch irevcm
case {1}
% Compute ab and store in b2
case {2}
% Compute ax and store in y
case {3}
% Compute a'y and store in x
case {4}
% compute az and store in p
otherwise
% compute a'z and store in r
end
```

## Example

```function nag_matop_real_gen_matrix_actexp_rcomm_example
a = [0.4,  -0.2,   1.3,   0.6;
0.3,   0.8,   1.0,   1.0;
3.0,   4.8,   0.2,   0.7;
0.5,   0.0,  -5.0,   0.7];
b = [0.1,  1.1;
1.7, -0.2;
0.5,  1.0;
0.4, -0.2];
n = 4;
m = int64(2);
t = -0.2;
b2 = zeros(n, m);
x = zeros(n, 2);
y = zeros(n, 2);
p = zeros(n, 1);
r = zeros(n, 1);
z = zeros(n, 1);
comm =  zeros(n*m+3*n+12, 1);
icomm = zeros(2*n+40, 1, 'int64');
% Compute the trace of a
tr = 0;
for i=1:n
tr = tr + a(i,i);
end

% Compute exp(ta)b
irevcm = int64(0);
[irevcm, b, b2, x, y, p, r, z, comm, icomm, ifail] = ...
nag_matop_real_gen_matrix_actexp_rcomm(irevcm, m, b, t, b2, x, y, ...
p, r, z, comm, icomm, 'tr', tr);

while irevcm ~= 0
switch irevcm
case {1}
% Compute ab and store in b2
b2 = a*b;
case {2}
% Compute ax and store in y
y = a*x;
case {3}
% Compute a'y and store in x
x = a'*y;
case {4}
% compute az and store in p
p = a*z;
otherwise
% compute a'z and store in r
r = a'*z;
end

[irevcm, b, b2, x, y, p, r, z, comm, icomm, ifail] = ...
nag_matop_real_gen_matrix_actexp_rcomm(irevcm, m, b, t, b2, x, y, ...
p, r, z, comm, icomm, 'tr', tr);
end

fprintf('\n exp(tA)B\n');
disp(b);
```
```

exp(tA)B
0.1933    0.7812
1.4423   -0.4055
-1.0756    0.6686
0.0276    0.4900

```
```function f01gb_example
a = [0.4,  -0.2,   1.3,   0.6;
0.3,   0.8,   1.0,   1.0;
3.0,   4.8,   0.2,   0.7;
0.5,   0.0,  -5.0,   0.7];
b = [0.1,  1.1;
1.7, -0.2;
0.5,  1.0;
0.4, -0.2];
n = 4;
m = int64(2);
t = -0.2;
b2 = zeros(n, m);
x = zeros(n, 2);
y = zeros(n, 2);
p = zeros(n, 1);
r = zeros(n, 1);
z = zeros(n, 1);
comm =  zeros(n*m+3*n+12, 1);
icomm = zeros(2*n+40, 1, 'int64');
% Compute the trace of a
tr = 0;
for i=1:n
tr = tr + a(i,i);
end

% Compute exp(ta)b
irevcm = int64(0);
[irevcm, b, b2, x, y, p, r, z, comm, icomm, ifail] = ...
f01gb(irevcm, m, b, t, b2, x, y, p, r, z, comm, icomm, 'tr', tr);

while irevcm ~= 0
switch irevcm
case {1}
% Compute ab and store in b2
b2 = a*b;
case {2}
% Compute ax and store in y
y = a*x;
case {3}
% Compute a'y and store in x
x = a'*y;
case {4}
% compute az and store in p
p = a*z;
otherwise
% compute a'z and store in r
r = a'*z;
end

[irevcm, b, b2, x, y, p, r, z, comm, icomm, ifail] = ...
f01gb(irevcm, m, b, t, b2, x, y, p, r, z, comm, icomm, 'tr', tr);
end

fprintf('\n exp(tA)B\n');
disp(b);
```
```

exp(tA)B
0.1933    0.7812
1.4423   -0.4055
-1.0756    0.6686
0.0276    0.4900

```