F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF01EMF

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

F01EMF computes the matrix function, $f\left(A\right)$, of a real $n$ by $n$ matrix $A$, using analytical derivatives of $f$ you have supplied.

## 2  Specification

 SUBROUTINE F01EMF ( N, A, LDA, F, IUSER, RUSER, IFLAG, IMNORM, IFAIL)
 INTEGER N, LDA, IUSER(*), IFLAG, IFAIL REAL (KIND=nag_wp) A(LDA,*), RUSER(*), IMNORM EXTERNAL F

## 3  Description

$f\left(A\right)$ is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).
The scalar function $f$, and the derivatives of $f$, are returned by the subroutine F which, given an integer $m$, should evaluate ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$ 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. F01EMF is therefore appropriate for functions that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.

## 4  References

Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## 5  Parameters

1:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     A(LDA,$*$) – 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:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01EMF is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
4:     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:
 SUBROUTINE F ( M, IFLAG, NZ, Z, FZ, IUSER, RUSER)
 INTEGER M, IFLAG, NZ, IUSER(*) REAL (KIND=nag_wp) RUSER(*) COMPLEX (KIND=nag_wp) Z(NZ), FZ(NZ)
1:     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:     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 $f\left({z}_{i}\right)$ may not be defined for a particular ${z}_{i}$. If IFLAG is returned as nonzero then F01EMF will terminate the computation, with ${\mathbf{IFAIL}}={\mathbf{2}}$.
3:     NZ – INTEGERInput
On entry: ${n}_{z}$, the number of function or derivative values required.
4:     Z(NZ) – 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:     FZ(NZ) – 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:     IUSER($*$) – INTEGER arrayUser Workspace
7:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
F is called with the parameters IUSER and RUSER as supplied to F01EMF. 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 F01EMF is called. Parameters denoted as Input must not be changed by this procedure.
5:     IUSER($*$) – INTEGER arrayUser Workspace
6:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace
IUSER and RUSER are not used by F01EMF, but are passed directly to F and may be used to pass information to this routine as an alternative to using COMMON global variables.
7:     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 ${\mathbf{IFAIL}}={\mathbf{2}}$.
8:     IMNORM – REAL (KIND=nag_wp)Output
On exit: if $A$ has complex eigenvalues, F01EMF will use complex arithmetic to compute $f\left(A\right)$. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. IMNORM contains the $1$-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.
If $A$ has real eigenvalues, F01EMF uses real arithmetic and ${\mathbf{IMNORM}}=0$.
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:
${\mathbf{IFAIL}}=1$
A Taylor series failed to converge.
${\mathbf{IFAIL}}=2$
IFLAG has been set nonzero by the user.
${\mathbf{IFAIL}}=3$
There was an error whilst reordering the Schur form of $A$.
Note:  this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{IFAIL}}=4$
The routine was unable to compute the Schur decomposition of $A$.
Note:  this failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{IFAIL}}=5$
${\mathbf{IFAIL}}=-1$
Input argument number $⟨\mathit{\text{value}}⟩$ is invalid.
${\mathbf{IFAIL}}=-3$
On entry, parameter LDA is invalid.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
${\mathbf{IFAIL}}=-999$
Allocation of memory failed.

## 7  Accuracy

For a normal matrix $A$ (for which ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$), the Schur decomposition is diagonal and the algorithm reduces to evaluating $f$ at the eigenvalues of $A$ and then constructing $f\left(A\right)$ using the Schur vectors. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm.

If $A$ has real eigenvalues then up to $6{n}^{2}$ of real allocatable memory may be required. If $A$ has complex eigenvalues then up to $6{n}^{2}$ of complex allocatable memory may be required.
The cost of the Schur–Parlett algorithm depends on the spectrum of $A$, but is roughly between $28{n}^{3}$ and ${n}^{4}/3$ floating point operations. There is an additional cost in evaluating $f$ and its derivatives. If the derivatives of $f$ are not known analytically, then F01ELF can be used to evaluate $f\left(A\right)$ using numerical differentiation. If $A$ is real symmetric then it is recommended that F01EFF be used as it is more efficient and, in general, more accurate than F01EMF.
For any $z$ on the real line, $f\left(z\right)$ must be real. $f$ must also be complex analytic on the spectrum of $A$. These conditions ensure that $f\left(A\right)$ is real for real $A$.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then F01JCF should be used.
F01FMF can be used to find the matrix function $f\left(A\right)$ for a complex matrix $A$.

## 9  Example

This example finds the ${e}^{2A}$ where
 $A= 1 0 -2 1 -1 2 0 1 2 0 1 0 1 0 -1 2 .$

### 9.1  Program Text

Program Text (f01emfe.f90)

### 9.2  Program Data

Program Data (f01emfe.d)

### 9.3  Program Results

Program Results (f01emfe.r)