NAG C Library Function Document

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

5

Arguments

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

Constraint: $n \geq 0$ .
2: $m$ – IntegerInput: On entry: $m$ , the number of columns of the matrix $B$ .

Constraint: $m \geq 0$ .
3: $a [\dim]$ – ComplexInput/Output: Note: the dimension, dim, of the array a must be at least $pda \times n$ .

The $(i, j)$ th element of the matrix $A$ is stored in $a [(j - 1) \times pda + i - 1]$ .

On entry: the $n$ by $n$ matrix $A$ .

On exit: $A$ is overwritten during the computation.
4: $pda$ – IntegerInput: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
5: $b [\dim]$ – ComplexInput/Output: Note: the dimension, dim, of the array b must be at least $pdb \times m$ .

The $(i, j)$ th element of the matrix $B$ is stored in $b [(j - 1) \times pdb + i - 1]$ .

On entry: the $n$ by $m$ matrix $B$ .

On exit: the $n$ by $m$ matrix $e^{t A} B$ .
6: $pdb$ – IntegerInput: On entry: the stride separating matrix row elements in the array b.

Constraint: $pdb \geq n$ .
7: $t$ – ComplexInput: On entry: the scalar $t$ .
8: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq n$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq n$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_SOME_PRECISION_LOSS: $e^{t A} B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.

7

Accuracy

For a Hermitian matrix

A

(for which

A^{H} = A

) the computed matrix

e^{t A} 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.

8

Parallelism and Performance

nag_matop_complex_gen_matrix_actexp (f01hac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_complex_gen_matrix_actexp (f01hac) 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 implementation-specific information.

9

Further Comments

The matrix

e^{t A} B

could be computed by explicitly forming

e^{t A}

using nag_matop_complex_gen_matrix_exp (f01fcc) and multiplying

B

by the result. However, experiments show that it is usually both more accurate and quicker to use nag_matop_complex_gen_matrix_actexp (f01hac).

The cost of the algorithm is

O (n^{2} m)

. 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} + (2 m + 8) n

of complex allocatable memory is required by nag_matop_complex_gen_matrix_actexp (f01hac).

nag_matop_real_gen_matrix_actexp (f01gac) can be used to compute

e^{t A} B

for real

A

B

, and

t

. nag_matop_complex_gen_matrix_actexp_rcomm (f01hbc) 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^{t A} B

, where

A = (\begin{array}{r} 0.5 + 0.0 i & - 0.2 + 0.0 i & 1.0 + 0.1 i & 0.0 + 0.4 i \\ 0.3 + 0.0 i & 0.5 + 1.2 i & 3.1 + 0.0 i & 1.0 + 0.2 i \\ 0.0 + 2.0 i & 0.1 + 0.0 i & 1.2 + 0.2 i & 0.5 + 0.0 i \\ 1.0 + 0.3 i & 0.0 + 0.2 i & 0.0 + 0.9 i & 0.5 + 0.0 i \end{array}),

B = (\begin{array}{r} 0.4 + 0.0 i & 1.2 + 0.0 i \\ 1.3 + 0.0 i & - 0.2 + 0.1 i \\ 0.0 + 0.3 i & 2.1 + 0.0 i \\ 0.4 + 0.0 i & - 0.9 + 0.0 i \end{array})

and

t = - 0.5 + 0.0 i .

NAG C Library Function Document

nag_matop_complex_gen_matrix_actexp (f01hac)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results