NAG Library Routine Document
F01ELF
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
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: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 2: $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – 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: $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array
A as declared in the (sub)program from which F01ELF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 4: $\mathrm{F}$ – 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:
INTEGER 
IFLAG, NZ, IUSER(*) 
REAL (KIND=nag_wp) 
RUSER(*) 
COMPLEX (KIND=nag_wp) 
Z(NZ), FZ(NZ) 

 1: $\mathrm{IFLAG}$ – 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: $\mathrm{NZ}$ – INTEGERInput

On entry: ${n}_{z}$, the number of function values required.
 3: $\mathrm{Z}\left({\mathbf{NZ}}\right)$ – 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: $\mathrm{FZ}\left({\mathbf{NZ}}\right)$ – 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: $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace
 6: $\mathrm{RUSER}\left(*\right)$ – 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: $\mathrm{IUSER}\left(*\right)$ – INTEGER arrayUser Workspace
 6: $\mathrm{RUSER}\left(*\right)$ – 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: $\mathrm{IFLAG}$ – 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: $\mathrm{IMNORM}$ – 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: $\mathrm{IFAIL}$ – 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 $\u2329\mathit{\text{value}}\u232a$ is invalid.
 ${\mathbf{IFAIL}}=3$

On entry, parameter
LDA is invalid.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
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 a discussion of numerical differentiation.
8 Parallelism and Performance
F01ELF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this routine may make calls to the usersupplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions can only be used if you are compiling the usersupplied function and linking the executable in accordance with the instructions in the
Users' Note for your implementation. The user workspace arrays
IUSER and
RUSER are classified as OpenMP shared memory and use of
IUSER and
RUSER has to take account of this in order to preserve thread safety whenever information is written back to either of these arrays. If at all possible, it is recommended that these arrays are only used to supply readonly data to the user functions when a multithreaded implementation is being used.
F01ELF 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 implementationspecific information.
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
${n}^{4}/3$ floatingpoint 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 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
F01JBF should be used.
F01FLF can be used to find the matrix function
$f\left(A\right)$ for a complex matrix
$A$.
10 Example
This example finds
$\mathrm{cos}2A$ where
10.1 Program Text
Program Text (f01elfe.f90)
10.2 Program Data
Program Data (f01elfe.d)
10.3 Program Results
Program Results (f01elfe.r)