NAG CL Interface
f01hac (complex_​gen_​matrix_​actexp)

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1 Purpose

f01hac computes the action of the matrix exponential etA, on the matrix B, where A is a complex n×n matrix, B is a complex n×m matrix and t is a complex scalar.

2 Specification

#include <nag.h>
void  f01hac (Integer n, Integer m, Complex a[], Integer pda, Complex b[], Integer pdb, Complex t, NagError *fail)
The function may be called by the names: f01hac or nag_matop_complex_gen_matrix_actexp.

3 Description

etAB is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the product etAB without explicitly forming etA.

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

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
2: m Integer Input
On entry: m, the number of columns of the matrix B.
Constraint: m0.
3: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least pda×n.
The (i,j)th element of the matrix A is stored in a[(j-1)×pda+i-1].
On entry: the n×n matrix A.
On exit: A is overwritten during the computation.
4: pda Integer Input
On entry: the stride separating matrix row elements in the array a.
Constraint: pdan.
5: b[dim] Complex Input/Output
Note: the dimension, dim, of the array b must be at least pdb×m.
The (i,j)th element of the matrix B is stored in b[(j-1)×pdb+i-1].
On entry: the n×m matrix B.
On exit: the n×m matrix etAB.
6: pdb Integer Input
On entry: the stride separating matrix row elements in the array b.
Constraint: pdbn.
7: t Complex Input
On entry: the scalar t.
8: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value and n=value.
Constraint: pdan.
On entry, pdb=value and n=value.
Constraint: pdbn.
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.
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.
etAB 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 AH=A) the computed matrix etAB 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

Background information to multithreading can be found in the Multithreading documentation.
f01hac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 etAB could be computed by explicitly forming etA using f01fcc and multiplying B by the result. However, experiments show that it is usually both more accurate and quicker to use f01hac.
The cost of the algorithm is O(n2m). The precise cost depends on A since a combination of balancing, shifting and scaling is used prior to the Taylor series evaluation.
Approximately n2+ (2m+8) n of complex allocatable memory is required by f01hac.
f01gac can be used to compute etAB for real A, B, and t. f01hbc provides an implementation of the algorithm with a reverse communication interface, which returns control to the calling program when matrix multiplications are required. This should be used if A is large and sparse.

10 Example

This example computes etAB, 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 )  
t=-0.5+0.0i .  

10.1 Program Text

Program Text (f01hace.c)

10.2 Program Data

Program Data (f01hace.d)

10.3 Program Results

Program Results (f01hace.r)