# NAG Library Routine Document

## 1Purpose

f01kaf computes an estimate of the absolute condition number of a matrix function $f$ of a complex $n$ by $n$ matrix $A$ in the $1$-norm, where $f$ is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, $f\left(A\right)$, is also returned.

## 2Specification

Fortran Interface
 Subroutine f01kaf ( fun, n, a, lda,
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Out) :: conda, norma, normfa Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (*), Intent (In) :: fun
#include nagmk26.h
 void f01kaf_ (const char *fun, const Integer *n, Complex a[], const Integer *lda, double *conda, double *norma, double *normfa, Integer *ifail, const Charlen length_fun)

## 3Description

The absolute condition number of $f$ at $A$, ${\mathrm{cond}}_{\mathrm{abs}}\left(f,A\right)$ is given by the norm of the Fréchet derivative of $f$, $L\left(A\right)$, which is defined by
 $LX := maxE≠0 LX,E E ,$
where $L\left(X,E\right)$ is the Fréchet derivative in the direction $E$. $L\left(X,E\right)$ is linear in $E$ and can therefore be written as
 $vec LX,E = KX vecE ,$
where the $\mathrm{vec}$ operator stacks the columns of a matrix into one vector, so that $K\left(X\right)$ is ${n}^{2}×{n}^{2}$. f01kaf computes an estimate $\gamma$ such that $\gamma \le {‖K\left(X\right)‖}_{1}$, where ${‖K\left(X\right)‖}_{1}\in \left[{n}^{-1}{‖L\left(X\right)‖}_{1},n{‖L\left(X\right)‖}_{1}\right]$. The relative condition number can then be computed via
 $cond rel f,A = cond abs f,A A1 fA 1 .$
The algorithm used to find $\gamma$ is detailed in Section 3.4 of Higham (2008).
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 used.
${\mathbf{fun}}=\text{'EXP'}$
The matrix exponential, ${e}^{A}$, will be used.
${\mathbf{fun}}=\text{'SIN'}$
The matrix sine, $\mathrm{sin}\left(A\right)$, will be used.
${\mathbf{fun}}=\text{'COS'}$
The matrix cosine, $\mathrm{cos}\left(A\right)$, will be used.
${\mathbf{fun}}=\text{'SINH'}$
The hyperbolic matrix sine, $\mathrm{sinh}\left(A\right)$, will be used.
${\mathbf{fun}}=\text{'COSH'}$
The hyperbolic matrix cosine, $\mathrm{cosh}\left(A\right)$, will be used.
${\mathbf{fun}}=\text{'LOG'}$
The matrix logarithm, $\mathrm{log}\left(A\right)$, will be used.
Constraint: ${\mathbf{fun}}=\text{'EXP'}$, $\text{'SIN'}$, $\text{'COS'}$, $\text{'SINH'}$, $\text{'COSH'}$ or $\text{'LOG'}$.
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 f01kaf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
5:     $\mathbf{conda}$ – Real (Kind=nag_wp)Output
On exit: an estimate of the absolute condition number of $f$ at $A$.
6:     $\mathbf{norma}$ – Real (Kind=nag_wp)Output
On exit: the $1$-norm of $A$.
7:     $\mathbf{normfa}$ – Real (Kind=nag_wp)Output
On exit: the $1$-norm of $f\left(A\right)$.
8:     $\mathbf{ifail}$ – 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:
${\mathbf{ifail}}=1$
An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$. Please contact NAG.
${\mathbf{ifail}}=2$
An internal error occurred when evaluating the matrix function $f\left(A\right)$. You can investigate further by calling f01fcf, f01fjf or f01fkf with the matrix $A$.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{fun}}=〈\mathit{\text{value}}〉$ was an illegal value.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}<0$.
Input argument number $〈\mathit{\text{value}}〉$ is invalid.
${\mathbf{ifail}}=-4$
On entry, argument lda is invalid.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
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

f01kaf uses the norm estimation routine f04zdf to estimate a quantity $\gamma$, where $\gamma \le {‖K\left(X\right)‖}_{1}$ and ${‖K\left(X\right)‖}_{1}\in \left[{n}^{-1}{‖L\left(X\right)‖}_{1},n{‖L\left(X\right)‖}_{1}\right]$. For further details on the accuracy of norm estimation, see the documentation for f04zdf.

## 8Parallelism and Performance

f01kaf 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.
f01kaf 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.

Approximately $6{n}^{2}$ of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routines f01fcf, f01fjf or f01fkf.
f01kaf returns the matrix function $f\left(A\right)$. This is computed using f01fcf if ${\mathbf{fun}}=\text{'EXP'}$, f01fjf if ${\mathbf{fun}}=\text{'LOG'}$ and f01fkf otherwise. If only $f\left(A\right)$ is required, without an estimate of the condition number, then it is far more efficient to use f01fcf, f01fjf or f01fkf directly.
f01jaf can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.

## 10Example

This example estimates the absolute and relative condition numbers of the matrix sinh function for
 $A = 0.0+1.0i -1.0+0.0i 1.0+0.0i 2.0+0.0i 2.0+1.0i 0.0-1.0i 0.0+0.0i 1.0+0.0i 0.0+1.0i 0.0+0.0i 1.0+1.0i 0.0+2.0i 1.0+0.0i 2.0+0.0i -2.0+3.0i 0.0+1.0i .$

### 10.1Program Text

Program Text (f01kafe.f90)

### 10.2Program Data

Program Data (f01kafe.d)

### 10.3Program Results

Program Results (f01kafe.r)

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