NAG Library Routine Document
f01fdf (complex_herm_matrix_exp)
1
Purpose
f01fdf computes the matrix exponential, ${e}^{A}$, of a complex Hermitian $n$ by $n$ matrix $A$.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n, lda  Integer, Intent (Inout)  ::  ifail  Complex (Kind=nag_wp), Intent (Inout)  ::  a(lda,*)  Character (1), Intent (In)  ::  uplo 

C Header Interface
#include <nagmk26.h>
void 
f01fdf_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, Integer *ifail, const Charlen length_uplo) 

3
Description
${e}^{A}$ is computed using a spectral factorization of
$A$
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
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, twentyfive years later SIAM Rev. 45 3–49
5
Arguments
 1: $\mathbf{uplo}$ – Character(1)Input

On entry: if
${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix
$A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint:
${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
 2: $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
 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
$n$ Hermitian matrix
$A$.
 If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
 If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, the upper or lower triangular part of the $n$ by $n$ matrix exponential, ${e}^{A}$.
 4: $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array
a as declared in the (sub)program from which
f01fdf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{n}}$.
 5: $\mathbf{ifail}$ – 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).
6
Error 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:
 ${\mathbf{ifail}}>0$

The computation of the spectral factorization failed to converge.
If
${\mathbf{ifail}}=i$, the algorithm to compute the spectral factorization failed to converge;
$i$ offdiagonal elements of an intermediate tridiagonal form did not converge to zero (see
f08fnf (zheev)).
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{uplo}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{uplo}}=\text{'L'}$ or $\text{'U'}$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 ${\mathbf{ifail}}=3$

An internal error occurred when computing the spectral factorization. Please contact
NAG.
 ${\mathbf{ifail}}=4$

On entry, ${\mathbf{lda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
7
Accuracy
For an Hermitian matrix
$A$, the matrix
${e}^{A}$, has the relative condition number
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 implementationspecific information.
The integer allocatable memory required is
n, the real allocatable memory required is
n and the
complex
allocatable memory required is approximately
$\left({\mathbf{n}}+\mathit{nb}+1\right)\times {\mathbf{n}}$, where
nb is the block size required by
f08fnf (zheev).
The cost of the algorithm is $O\left({n}^{3}\right)$.
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
10.1
Program Text
Program Text (f01fdfe.f90)
10.2
Program Data
Program Data (f01fdfe.d)
10.3
Program Results
Program Results (f01fdfe.r)