# NAG Library Routine Document

## 1Purpose

f01jcf computes an estimate of the absolute condition number of a matrix function $f$ at a real $n$ by $n$ matrix $A$ in the $1$-norm, using analytical derivatives of $f$ you have supplied.

## 2Specification

Fortran Interface
 Subroutine f01jcf ( n, a, lda, f,
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: iuser(*), ifail Integer, Intent (Out) :: iflag Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), ruser(*) Real (Kind=nag_wp), Intent (Out) :: conda, norma, normfa External :: f
#include nagmk26.h
 void f01jcf_ (const Integer *n, double a[], const Integer *lda, void (NAG_CALL *f)(const Integer *m, Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[]),Integer iuser[], double ruser[], Integer *iflag, double *conda, double *norma, double *normfa, Integer *ifail)

## 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 ${n}^{2}×{n}^{2}$. f01jcf 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).
The function $f$, and the derivatives of $f$, are returned by subroutine f which, given an integer $m$, evaluates ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$ at a number of (generally complex) points ${z}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. For any $z$ on the real line, $f\left(z\right)$ must also be real. f01jcf is therefore appropriate for routines that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.

## 4References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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, $f\left(A\right)$.
3:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01jcf is called.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
4:     $\mathbf{f}$ – Subroutine, supplied by the user.External Procedure
Given an integer $m$, the subroutine f evaluates ${f}^{\left(m\right)}\left({z}_{i}\right)$ at a number of points ${z}_{i}$.
The specification of f is:
Fortran Interface
 Subroutine f ( m, nz, z, fz,
 Integer, Intent (In) :: m, nz Integer, Intent (Inout) :: iflag, iuser(*) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Complex (Kind=nag_wp), Intent (In) :: z(nz) Complex (Kind=nag_wp), Intent (Out) :: fz(nz)
#include nagmk26.h
 void f (const Integer *m, Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[])
1:     $\mathbf{m}$ – IntegerInput
On entry: the order, $m$, of the derivative required.
If ${\mathbf{m}}=0$, $f\left({z}_{i}\right)$ should be returned. For ${\mathbf{m}}>0$, ${f}^{\left(m\right)}\left({z}_{i}\right)$ should be returned.
2:     $\mathbf{iflag}$ – IntegerInput/Output
On entry: iflag will be zero.
On exit: iflag should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function $f\left(z\right)$; for instance $f\left(z\right)$ may not be defined. If iflag is returned as nonzero then f01jcf will terminate the computation, with ${\mathbf{ifail}}={\mathbf{3}}$.
3:     $\mathbf{nz}$ – IntegerInput
On entry: ${n}_{z}$, the number of function or derivative values required.
4:     $\mathbf{z}\left({\mathbf{nz}}\right)$ – Complex (Kind=nag_wp) arrayInput
On entry: the ${n}_{z}$ points ${z}_{1},{z}_{2},\dots ,{z}_{{n}_{z}}$ at which the function $f$ is to be evaluated.
5:     $\mathbf{fz}\left({\mathbf{nz}}\right)$ – Complex (Kind=nag_wp) arrayOutput
On exit: the ${n}_{z}$ function or derivative values. ${\mathbf{fz}}\left(\mathit{i}\right)$ should return the value ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must ${f}^{\left(m\right)}\left({z}_{i}\right)$.
6:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
7:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
f is called with the arguments iuser and ruser as supplied to f01jcf. You should use the arrays iuser and ruser to supply information to f.
f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f01jcf is called. Arguments denoted as Input must not be changed by this procedure.
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01jcf. If your code inadvertently does return any NaNs or infinities, f01jcf is likely to produce unexpected results.
5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by f01jcf, but are passed directly to f and may be used to pass information to this routine.
7:     $\mathbf{iflag}$ – IntegerOutput
On exit: ${\mathbf{iflag}}=0$, unless iflag has been set nonzero inside f, in which case iflag will be the value set and ifail will be set to ${\mathbf{ifail}}={\mathbf{3}}$.
8:     $\mathbf{conda}$ – Real (Kind=nag_wp)Output
On exit: an estimate of the absolute condition number of $f$ at $A$.
9:     $\mathbf{norma}$ – Real (Kind=nag_wp)Output
On exit: the $1$-norm of $A$.
10:   $\mathbf{normfa}$ – Real (Kind=nag_wp)Output
On exit: the $1$-norm of $f\left(A\right)$.
11:   $\mathbf{ifail}$ – 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 f01emf with the matrix $A$ and the function $f$.
${\mathbf{ifail}}=3$
iflag has been set nonzero by the user-supplied subroutine.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}<0$.
Input argument number $〈\mathit{\text{value}}〉$ is invalid.
${\mathbf{ifail}}=-3$
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

f01jcf uses the norm estimation routine f04ydf 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 f04ydf.

## 8Parallelism and Performance

f01jcf 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. The user workspace arrays iuser and ruser are classified as OpenMP shared memory and use of iuser and ruser has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays. If at all possible, it is recommended that these arrays are only used to supply read-only data to the user functions when a multithreaded implementation is being used.
f01jcf 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 matrix function is computed using the underlying matrix function routine f01emf. Approximately $6{n}^{2}$ of real allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine.
If only $f\left(A\right)$ is required, without an estimate of the condition number, then it is far more efficient to use the underlying matrix function routine directly.
The complex analogue of this routine is f01kcf.

## 10Example

This example estimates the absolute and relative condition numbers of the matrix function ${e}^{2A}$ where
 $A= 0 -1 -1 1 -2 0 1 -1 2 -1 2 -2 -1 -2 0 -1 .$

### 10.1Program Text

Program Text (f01jcfe.f90)

### 10.2Program Data

Program Data (f01jcfe.d)

### 10.3Program Results

Program Results (f01jcfe.r)

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