F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF01EKF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F01EKF computes the matrix exponential, sine, cosine, sinh or cosh, of a real $n$ by $n$ matrix $A$ using the Schur–Parlett algorithm.

## 2  Specification

 SUBROUTINE F01EKF ( FUN, N, A, LDA, IMNORM, IFAIL)
 INTEGER N, LDA, IFAIL REAL (KIND=nag_wp) A(LDA,*), IMNORM CHARACTER(*) FUN

## 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  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     A(LDA,$*$) – REAL (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:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01EKF is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     IMNORM – REAL (KIND=nag_wp)Output
On exit: if $A$ has complex eigenvalues, F01EKF 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, F01EKF uses real arithmetic and ${\mathbf{IMNORM}}=0$.
6:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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.
${\mathbf{IFAIL}}=2$
${\mathbf{IFAIL}}=3$
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.
${\mathbf{IFAIL}}=-1$
Input parameter number $⟨\mathit{\text{value}}⟩$ is invalid.
${\mathbf{IFAIL}}=-2$
Input parameter number $⟨\mathit{\text{value}}⟩$ is invalid.
${\mathbf{IFAIL}}=-4$
On entry, parameter LDA is invalid.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
${\mathbf{IFAIL}}=-999$
Allocation of memory failed.

## 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).

The integer allocatable memory required is $n$. If $A$ has real eigenvalues then up to $9{n}^{2}$ of real 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 ${n}^{4}/3$ floating point operations; see Algorithm 9.6 of Higham (2008).
If the matrix exponential is required then it is recommended that F01ECF be used. F01ECF uses an algorithm which is, in general, more accurate than the Schur–Parlett algorithm used by F01EKF.
If estimates of the condition number of the matrix function are required then F01JAF should be used.
F01FKF can be used to find the matrix exponential, sin, cos, sinh or cosh of a complex matrix.

## 9  Example

This example finds the matrix cosine of the matrix
 $A = 2 0 1 0 0 2 -2 1 0 2 3 1 1 4 0 0 .$

### 9.1  Program Text

Program Text (f01ekfe.f90)

### 9.2  Program Data

Program Data (f01ekfe.d)

### 9.3  Program Results

Program Results (f01ekfe.r)