# NAG FL Interfacef01eff (real_​symm_​matrix_​fun)

## 1Purpose

f01eff computes the matrix function, $f\left(A\right)$, of a real symmetric $n$ by $n$ matrix $A$. $f\left(A\right)$ must also be a real symmetric matrix.

## 2Specification

Fortran Interface
 Subroutine f01eff ( 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) :: a(lda,*), ruser(*) Character (1), Intent (In) :: uplo External :: f
#include <nag.h>
 void f01eff_ (const char *uplo, const Integer *n, double 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)
The routine may be called by the names f01eff or nagf_matop_real_symm_matrix_fun.

## 3Description

$f\left(A\right)$ is computed using a spectral factorization of $A$
 $A = Q D QT ,$
where $D$ is the diagonal matrix whose diagonal elements, ${d}_{i}$, are the eigenvalues of $A$, and $Q$ is an orthogonal matrix whose columns are the eigenvectors of $A$. $f\left(A\right)$ is then given by
 $fA = Q fD QT ,$
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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least ${\mathbf{n}}$.
On entry: the $n$ by $n$ symmetric 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: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01eff 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)
 void f_ (Integer *iflag, const Integer *n, const double x[], double fx[], Integer iuser[], double ruser[])
1: $\mathbf{iflag}$Integer Input/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 f01eff will terminate the computation, with ${\mathbf{ifail}}=-{\mathbf{6}}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of function values required.
3: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
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) array Output
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 array User Workspace
6: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User Workspace
f is called with the arguments iuser and ruser as supplied to f01eff. 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 f01eff 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 f01eff. If your code inadvertently does return any NaNs or infinities, f01eff is likely to produce unexpected results.
6: $\mathbf{iuser}\left(*\right)$Integer array User Workspace
7: $\mathbf{ruser}\left(*\right)$Real (Kind=nag_wp) array User Workspace
iuser and ruser are not used by f01eff, but are passed directly to f and may be used to pass information to this routine.
8: $\mathbf{iflag}$Integer Output
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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.
The value of ifail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08faf).
${\mathbf{ifail}}=-1$
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: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-6$
Termination requested in f.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

f01eff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01eff 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 integer allocatable memory required is n, and the real allocatable memory required is approximately $\left({\mathbf{n}}+\mathit{nb}+4\right)×{\mathbf{n}}$, where nb is the block size required by f08faf.
The cost of the algorithm is $O\left({n}^{3}\right)$ plus the cost of evaluating $f\left(D\right)$. If ${\stackrel{^}{\lambda }}_{i}$ is the $i$th computed eigenvalue of $A$, then the user-supplied subroutine f will be asked to evaluate the function $f$ at $f\left({\stackrel{^}{\lambda }}_{i}\right)$, $i=1,2,\dots ,n$.
For further information on matrix functions, see Higham (2008).
f01fff can be used to find the matrix function $f\left(A\right)$ for a complex Hermitian matrix $A$.

## 10Example

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

### 10.1Program Text

Program Text (f01effe.f90)

### 10.2Program Data

Program Data (f01effe.d)

### 10.3Program Results

Program Results (f01effe.r)