NAG Library Function Document
nag_matop_complex_gen_matrix_fun_std (f01fkc)
1
Purpose
nag_matop_complex_gen_matrix_fun_std (f01fkc) computes the matrix exponential, sine, cosine, sinh or cosh, of a complex $n$ by $n$ matrix $A$ using the Schur–Parlett algorithm.
2
Specification
#include <nag.h> 
#include <nagf01.h> 
void 
nag_matop_complex_gen_matrix_fun_std (Nag_OrderType order,
Nag_MatFunType fun,
Integer n,
Complex a[],
Integer pda,
NagError *fail) 

3
Description
$f\left(A\right)$, where
$f$ is either the exponential, sine, cosine, sinh or cosh, is computed using the Schur–Parlett algorithm described in
Higham (2008) and
Davies and Higham (2003).
4
References
Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5
Arguments
 1:
$\mathbf{order}$ – Nag_OrderTypeInput

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.
 2:
$\mathbf{fun}$ – Nag_MatFunTypeInput

On entry: indicates which matrix function will be computed.
 ${\mathbf{fun}}=\mathrm{Nag\_Exp}$
 The matrix exponential, ${e}^{A}$, will be computed.
 ${\mathbf{fun}}=\mathrm{Nag\_Sin}$
 The matrix sine, $\mathrm{sin}\left(A\right)$, will be computed.
 ${\mathbf{fun}}=\mathrm{Nag\_Cos}$
 The matrix cosine, $\mathrm{cos}\left(A\right)$, will be computed.
 ${\mathbf{fun}}=\mathrm{Nag\_Sinh}$
 The hyperbolic matrix sine, $\mathrm{sinh}\left(A\right)$, will be computed.
 ${\mathbf{fun}}=\mathrm{Nag\_Cosh}$
 The hyperbolic matrix cosine, $\mathrm{cosh}\left(A\right)$, will be computed.
Constraint:
${\mathbf{fun}}=\mathrm{Nag\_Exp}$, $\mathrm{Nag\_Sin}$, $\mathrm{Nag\_Cos}$, $\mathrm{Nag\_Sinh}$ or $\mathrm{Nag\_Cosh}$.
 3:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
 4:
$\mathbf{a}\left[\mathit{dim}\right]$ – ComplexInput/Output

Note: the dimension,
dim, of the array
a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The
$\left(i,j\right)$th element of the matrix
$A$ is stored in
 ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(i1\right)\times {\mathbf{pda}}+j1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
 5:
$\mathbf{pda}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.
 6:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_CONVERGENCE

A Taylor series failed to converge.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
 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.
An unexpected internal error occurred when evaluating the function at a point. Please contact
NAG.
An unexpected internal error occurred when ordering the eigenvalues of
$A$. Please contact
NAG.
The function was unable to compute the Schur decomposition of $A$.
Note: this failure should not occur and suggests that the function has been called incorrectly.
There was an error whilst reordering the Schur form of $A$.
Note: this failure should not occur and suggests that the function has been called incorrectly.
 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.
 NE_SINGULAR

The linear equations to be solved are nearly singular and the Padé approximant used to compute the exponential may have no correct figures.
Note: this failure should not occur and suggests that the function has been called incorrectly.
7
Accuracy
For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$), the Schur decomposition is diagonal and the algorithm reduces to evaluating $f$ at the eigenvalues of $A$ and then constructing $f\left(A\right)$ using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm.
For further discussion of the Schur–Parlett algorithm see Section 9.4 of
Higham (2008).
8
Parallelism and Performance
nag_matop_complex_gen_matrix_fun_std (f01fkc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the usersupplied functions from within an OpenMP parallel region. Thus OpenMP pragmas within the user functions can only be used if you are compiling the usersupplied function and linking the executable in accordance with the instructions in the
Users' Note for your implementation.
nag_matop_complex_gen_matrix_fun_std (f01fkc) 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 implementationspecific information.
The Integer allocatable memory required is $n$, and the Complex allocatable memory required is approximately $9{n}^{2}$.
The cost of the Schur–Parlett algorithm depends on the spectrum of
$A$, but is roughly between
$28{n}^{3}$ and
${n}^{4}/3$ floatingpoint operations; see Algorithm 9.6 of
Higham (2008).
If the matrix exponential is required then it is recommended that
nag_matop_complex_gen_matrix_exp (f01fcc) be used.
nag_matop_complex_gen_matrix_exp (f01fcc) uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by
nag_matop_complex_gen_matrix_fun_std (f01fkc).
If estimates of the condition number of the matrix function are required then
nag_matop_complex_gen_matrix_cond_std (f01kac) should be used.
nag_matop_real_gen_matrix_fun_std (f01ekc) can be used to find the matrix exponential, sin, cos, sinh or cosh of a real matrix
$A$.
10
Example
This example finds the matrix sinh of the matrix
10.1
Program Text
Program Text (f01fkce.c)
10.2
Program Data
Program Data (f01fkce.d)
10.3
Program Results
Program Results (f01fkce.r)