# NAG Library Routine Document

## 1Purpose

f01fff computes the matrix function, $f\left(A\right)$, of a complex Hermitian $n$ by $n$ matrix $A$. $f\left(A\right)$ must also be a complex Hermitian matrix.

## 2Specification

Fortran Interface
 Subroutine f01fff ( uplo, n, a, lda, f,
 Integer, Intent (In) :: n, lda Integer, Intent (Inout) :: iuser(*), ifail Integer, Intent (Out) :: iflag Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo External :: f
C Header Interface
#include nagmk26.h
 void f01fff_ (const char *uplo, const Integer *n, Complex a[], const Integer *lda, void (NAG_CALL *f)(Integer *iflag, const Integer *n, const double x[], double fx[], Integer iuser[], double ruser[]),Integer iuser[], double ruser[], Integer *iflag, Integer *ifail, const Charlen length_uplo)

## 3Description

$f\left(A\right)$ is computed using a spectral factorization of $A$
 $A = Q D QH ,$
where $D$ is the real diagonal matrix whose diagonal elements, ${d}_{i}$, are the eigenvalues of $A$, $Q$ is a unitary matrix whose columns are the eigenvectors of $A$ and ${Q}^{\mathrm{H}}$ is the conjugate transpose of $Q$. $f\left(A\right)$ is then given by
 $fA = Q fD QH ,$
where $f\left(D\right)$ is the diagonal matrix whose $i$th diagonal element is $f\left({d}_{i}\right)$. See for example Section 4.5 of Higham (2008). $f\left({d}_{i}\right)$ is assumed to be real.

## 4References

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

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
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 $n$ Hermitian matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, the upper or lower triangular part of the $n$ by $n$ matrix function, $f\left(A\right)$.
4:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01fff is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5:     $\mathbf{f}$ – Subroutine, supplied by the user.External Procedure
The subroutine f evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.
The specification of f is:
Fortran Interface
 Subroutine f ( n, x, fx,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: iflag, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: fx(n)
C Header Interface
#include nagmk26.h
 void f (Integer *iflag, const Integer *n, const double x[], double fx[], Integer iuser[], double ruser[])
1:     $\mathbf{iflag}$ – 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 $f\left(x\right)$; for instance $f\left(x\right)$ may not be defined, or may be complex. If iflag is returned as nonzero then f01fff will terminate the computation, with ${\mathbf{ifail}}=-{\mathbf{6}}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of function values required.
3:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the $n$ points ${x}_{1},{x}_{2},\dots ,{x}_{n}$ at which the function $f$ is to be evaluated.
4:     $\mathbf{fx}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the $n$ function values. ${\mathbf{fx}}\left(\mathit{i}\right)$ should return the value $f\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$.
5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
f is called with the arguments iuser and ruser as supplied to f01fff. 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 f01fff 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 f01fff. If your code inadvertently does return any NaNs or infinities, f01fff is likely to produce unexpected results.
6:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
7:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by f01fff, but are passed directly to f and may be used to pass information to this routine.
8:     $\mathbf{iflag}$ – IntegerOutput
On exit: ${\mathbf{iflag}}=0$, unless you have set iflag nonzero inside f, in which case iflag will be the value you set and ifail will be set to ${\mathbf{ifail}}=-{\mathbf{6}}$.
9:     $\mathbf{ifail}$ – 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:
${\mathbf{ifail}}>0$
The computation of the spectral factorization failed to converge.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{uplo}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{uplo}}=\text{'L'}$ or $\text{'U'}$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
An internal error occurred when computing the spectral factorization. Please contact NAG.
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-6$
iflag was set to a nonzero value in f.
${\mathbf{ifail}}=-99$
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

Provided that $f\left(D\right)$ can be computed accurately then the computed matrix function will be close to the exact matrix function. See Section 10.2 of Higham (2008) for details and further discussion.

## 8Parallelism and Performance

f01fff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01fff 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.

## 9Further Comments

The integer allocatable memory required is n, the real allocatable memory required is $4×{\mathbf{n}}-2$ and the complex allocatable memory required is approximately $\left({\mathbf{n}}+\mathit{nb}+1\right)×{\mathbf{n}}$, where nb is the block size required by f08fnf (zheev).
The cost of the algorithm is $O\left({n}^{3}\right)$ plus the cost of evaluating $f\left(D\right)$. If ${\stackrel{^}{\lambda }}_{\mathit{i}}$ is the $\mathit{i}$th computed eigenvalue of $A$, then the user-supplied subroutine f will be asked to evaluate the function $f$ at $f\left({\stackrel{^}{\lambda }}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$.
For further information on matrix functions, see Higham (2008).
f01eff can be used to find the matrix function $f\left(A\right)$ for a real symmetric matrix $A$.

## 10Example

This example finds the matrix cosine, $\mathrm{cos}\left(A\right)$, of the Hermitian matrix
 $A= 1 2+i 3+2i 4+3i 2-i 1 2+i 3+2i 3-2i 2-i 1 2+i 4-3i 3-2i 2-i 1 .$

### 10.1Program Text

Program Text (f01fffe.f90)

### 10.2Program Data

Program Data (f01fffe.d)

### 10.3Program Results

Program Results (f01fffe.r)

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