# NAG Library Routine Document

## 1Purpose

f01haf 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.

## 2Specification

Fortran Interface
 Subroutine f01haf ( n, m, a, lda, b, ldb, t,
 Integer, Intent (In) :: n, m, lda, ldb Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (In) :: t Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*)
#include nagmk26.h
 void f01haf_ (const Integer *n, const Integer *m, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, const Complex *t, Integer *ifail)

## 3Description

${e}^{tA}B$ 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 ${e}^{tA}$.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of columns of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: $A$ is overwritten during the computation.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01haf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
5:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least ${\mathbf{m}}$.
On entry: the $n$ by $m$ matrix $B$.
On exit: the $n$ by $m$ matrix ${e}^{tA}B$.
6:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f01haf is called.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{n}}$.
7:     $\mathbf{t}$ – Complex (Kind=nag_wp)Input
On entry: the scalar $t$.
8:     $\mathbf{ifail}$ – IntegerInput/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 3.4 in How to Use the NAG Library and its Documentation 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).

## 6Error 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 entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-6$
On entry, ${\mathbf{ldb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For a 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 non-Hermitian matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

## 8Parallelism and Performance

f01haf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01haf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The matrix ${e}^{tA}B$ could be computed by explicitly forming ${e}^{tA}$ using f01fcf and multiplying $B$ by the result. However, experiments show that it is usually both more accurate and quicker to use f01haf.
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 complex allocatable memory is required by f01haf.
f01gaf can be used to compute ${e}^{tA}B$ for real $A$, $B$, and $t$. f01hbf 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.

## 10Example

This example computes ${e}^{tA}B$, where
 $A = 0.5+0.0i -0.2+0.0i 1.0+0.1i 0.0+0.4i 0.3+0.0i 0.5+1.2i 3.1+0.0i 1.0+0.2i 0.0+2.0i 0.1+0.0i 1.2+0.2i 0.5+0.0i 1.0+0.3i 0.0+0.2i 0.0+0.9i 0.5+0.0i ,$
 $B = 0.4+0.0i 1.2+0.0i 1.3+0.0i -0.2+0.1i 0.0+0.3i 2.1+0.0i 0.4+0.0i -0.9+0.0i$
and
 $t=-0.5+0.0i .$

### 10.1Program Text

Program Text (f01hafe.f90)

### 10.2Program Data

Program Data (f01hafe.d)

### 10.3Program Results

Program Results (f01hafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017