NAG Library Function Document

nag_matop_complex_herm_matrix_exp (f01fdc)

+− 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

nag_matop_complex_herm_matrix_exp (f01fdc) computes the matrix exponential,

e^{A}

, of a complex Hermitian

n

n

matrix

A

2 Specification

#include <nag.h>

#include <nagf01.h>

void	nag_matop_complex_herm_matrix_exp (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, NagError *fail)

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 Arguments

1: order – Nag_OrderTypeInput: On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $order = Nag_RowMajor$ . See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $order = Nag_RowMajor$ or $Nag_ColMajor$ .
2: uplo – Nag_UploTypeInput: On entry: if $uplo = Nag_Upper$ , the upper triangle of the matrix $A$ is stored.
If $uplo = Nag_Lower$ , the lower triangle of the matrix $A$ is stored.

Constraint: $uplo = Nag_Upper$ or $Nag_Lower$ .
3: n – IntegerInput: On entry: $n$ , the order of the matrix $A$ .
Constraint: $n \geq 0$ .
4: a[ $\dim$ ] – ComplexInput/Output: Note: the dimension, dim, of the array a must be at least $pda \times n$ .
On entry: the $n$ by $n$ Hermitian matrix $A$ .
If $order = Nag_ColMajor$ , $A_{i j}$ is stored in $a [(j - 1) \times pda + i - 1]$ .
If $order = Nag_RowMajor$ , $A_{i j}$ is stored in $a [(i - 1) \times pda + j - 1]$ .
If $uplo = Nag_Upper$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo = Nag_Lower$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: the upper or lower triangular part of the $n$ by $n$ matrix exponential, $e^{A}$ .
5: pda – IntegerInput: On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.

Constraint: $pda \geq n$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: The computation of the spectral factorization failed to converge.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq n$ .
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.

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

nag_matop_complex_herm_matrix_exp (f01fdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_complex_herm_matrix_exp (f01fdc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

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 Function Documentnag_matop_complex_herm_matrix_exp (f01fdc)