F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF01FFF

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

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.

## 2  Specification

 SUBROUTINE F01FFF ( UPLO, N, A, LDA, F, IUSER, RUSER, IFLAG, IFAIL)
 INTEGER N, LDA, IUSER(*), IFLAG, IFAIL REAL (KIND=nag_wp) RUSER(*) COMPLEX (KIND=nag_wp) A(LDA,*) CHARACTER(1) UPLO EXTERNAL F

## 3  Description

$f\left(A\right)$ 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.

## 4  References

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

## 5  Parameters

1:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     A(LDA,$*$) – 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:     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:     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:
 SUBROUTINE F ( IFLAG, N, X, FX, IUSER, RUSER)
 INTEGER IFLAG, N, IUSER(*) REAL (KIND=nag_wp) X(N), FX(N), RUSER(*)
1:     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:     N – INTEGERInput
On entry: $n$, the number of function values required.
3:     X(N) – 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:     FX(N) – 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:     IUSER($*$) – INTEGER arrayUser Workspace
6:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
F is called with the parameters IUSER and RUSER as supplied to F01FFF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.
F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F01FFF is called. Parameters denoted as Input must not be changed by this procedure.
6:     IUSER($*$) – INTEGER arrayUser Workspace
7:     RUSER($*$) – 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 as an alternative to using COMMON global variables.
8:     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:     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:
If ${\mathbf{IFAIL}}=-{\mathbit{i}}$, the $i$th argument had an illegal value.
${\mathbf{IFAIL}}=-6$
IFLAG has been set nonzero by the user.
${\mathbf{IFAIL}}=-999$
Internal memory allocation failed.
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 algorithm to compute the spectral factorization failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see F08FNF (ZHEEV)).
Note:  this failure is unlikely to occur.

## 7  Accuracy

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.

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 $A$.

## 9  Example

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

### 9.1  Program Text

Program Text (f01fffe.f90)

### 9.2  Program Data

Program Data (f01fffe.d)

### 9.3  Program Results

Program Results (f01fffe.r)