NAG Library Function Document
nag_matop_real_gen_matrix_fun_num (f01elc)
1
Purpose
nag_matop_real_gen_matrix_fun_num (f01elc) 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
#include <nag.h> 
#include <nagf01.h> 
void 
nag_matop_real_gen_matrix_fun_num (Integer n,
double a[],
Integer pda,
void 
(*f)(Integer *iflag,
Integer nz,
const Complex z[],
Complex fz[],
Nag_Comm *comm),


Nag_Comm *comm, Integer *iflag,
double *imnorm,
NagError *fail) 

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 function
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
Arguments
 1:
$\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
 2:
$\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension,
dim, of the array
a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
 3:
$\mathbf{pda}$ – IntegerInput

On entry: the stride separating matrix row elements in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.
 4:
$\mathbf{f}$ – function, supplied by the userExternal Function

The function
f evaluates
$f\left({z}_{i}\right)$ at a number of points
${z}_{i}$.
The specification of
f is:
void 
f (Integer *iflag,
Integer nz,
const Complex z[],
Complex fz[],
Nag_Comm *comm)


 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
$f\left({z}_{i}\right)$; for instance
$f\left({z}_{i}\right)$ may not be defined. If
iflag is returned as nonzero then
nag_matop_real_gen_matrix_fun_num (f01elc) will terminate the computation, with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 2:
$\mathbf{nz}$ – IntegerInput

On entry: ${n}_{z}$, the number of function values required.
 3:
$\mathbf{z}\left[{\mathbf{nz}}\right]$ – const ComplexInput

On entry: the ${n}_{z}$ points ${z}_{1},{z}_{2},\dots ,{z}_{{n}_{z}}$ at which the function $f$ is to be evaluated.
 4:
$\mathbf{fz}\left[{\mathbf{nz}}\right]$ – ComplexOutput

On exit: the ${n}_{z}$ function values.
${\mathbf{fz}}\left[\mathit{i}1\right]$ 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:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
f.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
nag_matop_real_gen_matrix_fun_num (f01elc) you may allocate memory and initialize these pointers with various quantities for use by
f when called from
nag_matop_real_gen_matrix_fun_num (f01elc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: f should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
nag_matop_real_gen_matrix_fun_num (f01elc). If your code inadvertently
does return any NaNs or infinities,
nag_matop_real_gen_matrix_fun_num (f01elc) is likely to produce unexpected results.
 5:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
 6:
$\mathbf{iflag}$ – Integer *Output

On exit:
${\mathbf{iflag}}=0$, unless
iflag has been set nonzero inside
f, in which case
iflag will be the value set and
fail will be set to
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 7:
$\mathbf{imnorm}$ – double *Output

On exit: if
$A$ has complex eigenvalues,
nag_matop_real_gen_matrix_fun_num (f01elc) 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,
nag_matop_real_gen_matrix_fun_num (f01elc) uses real arithmetic and
${\mathbf{imnorm}}=0$.
 8:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_CONVERGENCE

A Taylor series failed to converge after $40$ terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
An unexpected internal error occurred when ordering the eigenvalues of
$A$. Please contact
NAG.
The function was unable to compute the Schur decomposition of $A$.
Note: this failure should not occur and suggests that the function has been called incorrectly.
There was an error whilst reordering the Schur form of $A$.
Note: this failure should not occur and suggests that the function has been called incorrectly.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_USER_STOP

iflag has been set nonzero by the user.
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
nag_matop_real_gen_matrix_fun_num (f01elc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the usersupplied functions from within an OpenMP parallel region. Thus OpenMP pragmas 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. You must also ensure that you use the NAG communication argument
comm in a thread safe manner, which is best achieved by only using it to supply readonly data to the user functions.
nag_matop_real_gen_matrix_fun_num (f01elc) 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 function. 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 double 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
nag_matop_real_gen_matrix_fun_usd (f01emc) can be used to evaluate
$f\left(A\right)$ more accurately. If
$A$ is real symmetric then it is recommended that
nag_matop_real_symm_matrix_fun (f01efc) be used as it is more efficient and, in general, more accurate than
nag_matop_real_gen_matrix_fun_num (f01elc).
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
nag_matop_real_gen_matrix_cond_num (f01jbc) should be used.
nag_matop_complex_gen_matrix_fun_num (f01flc) 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 (f01elce.c)
10.2
Program Data
Program Data (f01elce.d)
10.3
Program Results
Program Results (f01elce.r)