NAG Library Routine Document

F01FDF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F01FDF computes the matrix exponential,

e^{A}

, of a complex Hermitian

n

n

matrix

A

2 Specification

SUBROUTINE F01FDF (

UPLO, N, A, LDA, IFAIL)

INTEGER	N, LDA, IFAIL
COMPLEX (KIND=nag_wp)	A(LDA,*)
CHARACTER(1)	UPLO

3 Description

e^{A}

is computed using a spectral factorization of

A

A = Q D Q^{H},

where

D

is the diagonal matrix whose diagonal elements,

d_{i}

, are the eigenvalues of

A

, and

Q

is a unitary matrix whose columns are the eigenvectors of

A

e^{A}

is then given by

e^{A} = Q e^{D} Q^{H},

where

e^{D}

is the diagonal matrix whose

i

th diagonal element is

e^{d_{i}}

. See for example Section 4.5 of Higham (2008).

4 References

Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl. 26(4) 1179–1193

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

Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Rev. 45 3–49

5 Parameters

1: $UPLO$ – CHARACTER(1)Input

On entry: if

UPLO ='U'

, the upper triangle of the matrix

A

is stored.

UPLO ='L'

, the lower triangle of the matrix

A

is stored.

Constraint:

UPLO ='U'

'L'

2: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: $A (LDA *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least

N

On entry: the

n

n

Hermitian matrix

A

If $UPLO ='U'$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $UPLO ='L'$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

IFAIL = 0

, the upper or lower triangular part of the

n

n

matrix exponential,

e^{A}

4: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F01FDF is called.

Constraint:

LDA \geq N

5: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ 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 parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL > 0$: The computation of the spectral factorization failed to converge.

$IFAIL = - 1$: On entry, UPLO was invalid.

$IFAIL = - 2$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 0$ .

$IFAIL = - 3$: An internal error occurred when computing the spectral factorization. Please contact NAG.

$IFAIL = - 4$: On entry, $LDA = 〈value〉$ and $N = 〈value〉$ .
Constraint: $LDA \geq N$ .

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

For an Hermitian matrix

A

, the matrix

e^{A}

, has the relative condition number

κ (A) = {‖A‖}_{2},

which is the minimal possible for the matrix exponential and so the computed matrix exponential is guaranteed to be close to the exact matrix. See Section 10.2 of Higham (2008) for details and further discussion.

8 Parallelism and Performance

F01FDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F01FDF 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.

9 Further Comments

The integer allocatable memory required is N, the real allocatable memory required is N and the complex allocatable memory required is approximately

(N + nb + 1) \times N

, where nb is the block size required by F08FNF (ZHEEV).

The cost of the algorithm is

O (n^{3})

As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003).

10 Example

This example finds the matrix exponential of the Hermitian matrix

A = (\begin{matrix} 1 & 2 + i & 3 + 2 i & 4 + 3 i \\ 2 - i & 1 & 2 + i & 3 + 2 i \\ 3 - 2 i & 2 - i & 1 & 2 + i \\ 4 - 3 i & 3 - 2 i & 2 - i & 1 \end{matrix}) .

NAG Library Routine DocumentF01FDF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

F01FDF