F01ELF (PDF version)
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F01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF01ELF

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

F01ELF computes the matrix function, $f\left(A\right)$, of a real $n$ by $n$ matrix $A$. Numerical differentiation is used to evaluate the derivatives of $f$ when they are required.

## 2  Specification

 SUBROUTINE F01ELF ( 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 coefficients of the Taylor series used in the algorithm are evaluated using the numerical differentiation algorithm of Lyness and Moler (1967).
The scalar function $f$ is supplied via subroutine F which evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.

## 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
Lyness J N and Moler C B (1967) Numerical differentiation of analytic functions SIAM J. Numer. Anal. 4(2) 202–210

## 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 F01ELF 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
The subroutine F evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.
The specification of F is:
 SUBROUTINE F ( IFLAG, NZ, Z, FZ, IUSER, RUSER)
 INTEGER IFLAG, NZ, IUSER(*) REAL (KIND=nag_wp) RUSER(*) COMPLEX (KIND=nag_wp) Z(NZ), FZ(NZ)
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 $f\left({z}_{i}\right)$; for instance $f\left({z}_{i}\right)$ may not be defined. If IFLAG is returned as nonzero then F01ELF will terminate the computation, with ${\mathbf{IFAIL}}={\mathbf{2}}$.
2:     NZ – INTEGERInput
On entry: ${n}_{z}$, the number of function values required.
3:     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.
4:     FZ(NZ) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the ${n}_{z}$ function values. ${\mathbf{FZ}}\left(\mathit{i}\right)$ should return the value $f\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must $f\left({z}_{i}\right)$.
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 F01ELF. 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 F01ELF 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 F01ELF, 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, F01ELF 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, F01ELF 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 after $40$ terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
${\mathbf{IFAIL}}=2$
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.
${\mathbf{IFAIL}}=5$
An unexpected internal error occurred. Please contact NAG.
${\mathbf{IFAIL}}=-1$
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. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm, and Lyness and Moler (1967) for discussion of the numerical differentiation subroutine.

## 8  Further Comments

The integer allocatable memory required is $n$. 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 numerically differentiating $f$, in order to obtain the Taylor series coefficients. If the derivatives of $f$ are known analytically, then F01EMF can be used to evaluate $f\left(A\right)$ more accurately. If $A$ is real symmetric then it is recommended that F01EFF be used as it is more efficient and, in general, more accurate than F01ELF.
For any $z$ on the real line, $f\left(z\right)$ must be real. $f$ must also be complex analytic ont he 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 F01JBF should be used.
F01FLF can be used to find the matrix function $f\left(A\right)$ for a complex matrix $A$.

## 9  Example

This example finds $\mathrm{cos}2A$ where
 $A= 3 0 1 2 -1 1 3 1 0 2 2 1 2 1 -1 1 .$

### 9.1  Program Text

Program Text (f01elfe.f90)

### 9.2  Program Data

Program Data (f01elfe.d)

### 9.3  Program Results

Program Results (f01elfe.r)

F01ELF (PDF version)
F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual