# NAG Library Routine Document

## 1Purpose

f01fkf computes the matrix exponential, sine, cosine, sinh or cosh, of a complex $n$ by $n$ matrix $A$ using the Schur–Parlett algorithm.

## 2Specification

Fortran Interface
 Subroutine f01fkf ( fun, n, a, lda,
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (*), Intent (In) :: fun
#include nagmk26.h
 void f01fkf_ (const char *fun, const Integer *n, Complex a[], const Integer *lda, Integer *ifail, const Charlen length_fun)

## 3Description

$f\left(A\right)$, 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).

## 4References

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

## 5Arguments

1:     $\mathbf{fun}$ – Character(*)Input
On entry: indicates which matrix function will be computed.
${\mathbf{fun}}=\text{'EXP'}$
The matrix exponential, ${e}^{A}$, will be computed.
${\mathbf{fun}}=\text{'SIN'}$
The matrix sine, $\mathrm{sin}\left(A\right)$, will be computed.
${\mathbf{fun}}=\text{'COS'}$
The matrix cosine, $\mathrm{cos}\left(A\right)$, will be computed.
${\mathbf{fun}}=\text{'SINH'}$
The hyperbolic matrix sine, $\mathrm{sinh}\left(A\right)$, will be computed.
${\mathbf{fun}}=\text{'COSH'}$
The hyperbolic matrix cosine, $\mathrm{cosh}\left(A\right)$, will be computed.
Constraint: ${\mathbf{fun}}=\text{'EXP'}$, $\text{'SIN'}$, $\text{'COS'}$, $\text{'SINH'}$ or $\text{'COSH'}$.
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 $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01fkf 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).

## 6Error 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:
${\mathbf{ifail}}=1$
A Taylor series failed to converge.
${\mathbf{ifail}}=2$
An unexpected internal error occurred when evaluating the function at a point. Please contact NAG.
${\mathbf{ifail}}=3$
There was an error whilst reordering the Schur form of $A$.
Note:  this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{ifail}}=4$
The routine was unable to compute the Schur decomposition of $A$.
Note:  this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{ifail}}=5$
${\mathbf{ifail}}=6$
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 routine has been called incorrectly.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{fun}}=〈\mathit{\text{value}}〉$ was an illegal value.
${\mathbf{ifail}}=-2$
Input argument number $〈\mathit{\text{value}}〉$ is invalid.
${\mathbf{ifail}}=-4$
On entry, argument lda is invalid.
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.

## 7Accuracy

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).

## 8Parallelism and Performance

f01fkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this routine may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation.
f01fkf 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.

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 ${n}^{4}/3$ floating-point operations; see Algorithm 9.6 of Higham (2008).
If the matrix exponential is required then it is recommended that f01fcf be used. f01fcf uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by f01fkf.
If estimates of the condition number of the matrix function are required then f01kaf should be used.
f01ekf can be used to find the matrix exponential, sin, cos, sinh or cosh of a real matrix $A$.

## 10Example

This example finds the matrix sinh of the matrix
 $A = 1.0+1.0i 0.0+0.0i 1.0+3.0i 0.0+0.0i 0.0+0.0i 2.0+0.0i 0.0+0.0i 1.0+2.0i 3.0+1.0i 0.0+4.0i 1.0+1.0i 0.0+0.0i 1.0+1.0i 0.0+2.0i 0.0+0.0i 1.0+0.0i .$

### 10.1Program Text

Program Text (f01fkfe.f90)

### 10.2Program Data

Program Data (f01fkfe.d)

### 10.3Program Results

Program Results (f01fkfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017