NAG CL Interface
f01ekc (real_gen_matrix_fun_std)
1
Purpose
f01ekc computes the matrix exponential, sine, cosine, sinh or cosh, of a real $n$ by $n$ matrix $A$ using the Schur–Parlett algorithm.
2
Specification
void 
f01ekc (Nag_OrderType order,
Nag_MatFunType fun,
Integer n,
double a[],
Integer pda,
double *imnorm,
NagError *fail) 

The function may be called by the names: f01ekc or nag_matop_real_gen_matrix_fun_std.
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_OrderType
Input

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 3.1.3 in the Introduction to the NAG Library CL Interface 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_MatFunType
Input

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:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

4:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
Input/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:
$\mathbf{pda}$ – Integer
Input

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:
$\mathbf{imnorm}$ – double *
Output

On exit: if
$A$ has complex eigenvalues,
f01ekc will use complex arithmetic to compute the matrix function. The imaginary part is discarded at the end of the computation, because it will theoretically vanish.
imnorm contains the
$1$norm of the imaginary part, which should be used to check that the routine has given a reliable answer.
If $A$ has real eigenvalues, f01ekc uses real arithmetic and ${\mathbf{imnorm}}=0$.

7:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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 ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), 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
f01ekc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01ekc 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 $n$. If $A$ has real eigenvalues then up to $9{n}^{2}$ of double allocatable memory may be required. If $A$ has complex eigenvalues then up to $9{n}^{2}$ of Complex allocatable memory may be required.
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
f01ecc be used.
f01ecc uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by
f01ekc.
If estimates of the condition number of the matrix function are required then
f01jac should be used.
f01fkc can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.
10
Example
This example finds the matrix cosine of the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results