NAG CL Interface
f01gac (real_gen_matrix_actexp)
1
Purpose
f01gac computes the action of the matrix exponential ${e}^{tA}$, on the matrix $B$, where $A$ is a real $n$ by $n$ matrix, $B$ is a real $n$ by $m$ matrix and $t$ is a real scalar.
2
Specification
void 
f01gac (Integer n,
Integer m,
double a[],
Integer pda,
double b[],
Integer pdb,
double t,
NagError *fail) 

The function may be called by the names: f01gac or nag_matop_real_gen_matrix_actexp.
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 product
${e}^{tA}B$ without explicitly forming
${e}^{tA}$.
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

1:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

2:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of columns of the matrix $B$.
Constraint:
${\mathbf{m}}\ge 0$.

3:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: $A$ is overwritten during the computation.

4:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.

5:
$\mathbf{b}\left[\mathit{dim}\right]$ – 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 ${\mathbf{b}}\left[\left(j1\right)\times {\mathbf{pdb}}+i1\right]$.
On entry: the $n$ by $m$ matrix $B$.
On exit: the $n$ by $m$ matrix ${e}^{tA}B$.

6:
$\mathbf{pdb}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
b.
Constraint:
${\mathbf{pdb}}\ge {\mathbf{n}}$.

7:
$\mathbf{t}$ – double
Input

On entry: the scalar $t$.

8:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdb}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
 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
For a symmetric matrix
$A$ (for which
${A}^{\mathrm{T}}=A$) 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 nonsymmetric matrices. See Section 4 of
Al–Mohy and Higham (2011) for details and further discussion.
8
Parallelism and Performance
f01gac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01gac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The matrix
${e}^{tA}B$ could be computed by explicitly forming
${e}^{tA}$ using
f01ecc and multiplying
$B$ by the result. However, experiments show that it is usually both more accurate and quicker to use
f01gac.
The cost of the algorithm is $\mathit{O}\left({n}^{2}m\right)$. The precise cost depends on $A$ since a combination of balancing, shifting and scaling is used prior to the Taylor series evaluation.
Approximately ${n}^{2}+\left(2m+8\right)n$ of real allocatable memory is required by f01gac.
f01hac can be used to compute
${e}^{tA}B$ for complex
$A$,
$B$, and
$t$.
f01gbc provides an implementation of the algorithm with a reverse communication interface, which returns control to the user when matrix multiplications are required. This should be used if
$A$ is large and sparse.
10
Example
This example computes
${e}^{tA}B$, where
and
10.1
Program Text
10.2
Program Data
10.3
Program Results