NAG CL Interface
f01emc (real_gen_matrix_fun_usd)
1
Purpose
f01emc 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
void 
f01emc (Nag_OrderType order,
Integer n,
double a[],
Integer pda,
void 
(*f)(Integer m,
Integer *iflag,
Integer nz,
const Complex z[],
Complex fz[],
Nag_Comm *comm),


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

The function may be called by the names: f01emc or nag_matop_real_gen_matrix_fun_usd.
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 function
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.
f01emc 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
Arguments

1:
$\mathbf{order}$ – Nag_OrderType
Input

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
Input/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
 ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(i1\right)\times {\mathbf{pda}}+j1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.

4:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.

5:
$\mathbf{f}$ – function, supplied by the user
External Function

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



1:
$\mathbf{m}$ – Integer
Input

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:
$\mathbf{iflag}$ – 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
$f\left(z\right)$; for instance
$f\left({z}_{i}\right)$ may not be defined for a particular
${z}_{i}$. If
iflag is returned as nonzero then
f01emc will terminate the computation, with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

3:
$\mathbf{nz}$ – Integer
Input

On entry: ${n}_{z}$, the number of function or derivative values required.

4:
$\mathbf{z}\left[\mathit{dim}\right]$ – const Complex
Input

On entry: the ${n}_{z}$ points ${z}_{1},{z}_{2},\dots ,{z}_{{n}_{z}}$ at which the function $f$ is to be evaluated.

5:
$\mathbf{fz}\left[\mathit{dim}\right]$ – Complex
Output

On exit: the ${n}_{z}$ function or derivative values.
${\mathbf{fz}}\left[\mathit{i}1\right]$ 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:
$\mathbf{comm}$ – 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
f01emc you may allocate memory and initialize these pointers with various quantities for use by
f when called from
f01emc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: f should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
f01emc. If your code inadvertently
does return any NaNs or infinities,
f01emc is likely to produce unexpected results.

6:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

7:
$\mathbf{iflag}$ – 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.

8:
$\mathbf{imnorm}$ – double *
Output

On exit: if
$A$ has complex eigenvalues,
f01emc 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 function has given a reliable answer.
If $A$ has real eigenvalues, f01emc uses real arithmetic and ${\mathbf{imnorm}}=0$.

9:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
An unexpected internal error occurred when ordering the eigenvalues of
$A$. Please contact
NAG.
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.
The routine was unable to compute the Schur decomposition 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 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_USER_STOP

Termination requested in
f.
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.
8
Parallelism and Performance
f01emc 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.
f01emc 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.
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 evaluating
$f$ and its derivatives.
If the derivatives of
$f$ are not known analytically, then
f01elc can be used to evaluate
$f\left(A\right)$ using numerical differentiation.
If
$A$ is real symmetric then it is recommended that
f01efc be used as it is more efficient and, in general, more accurate than
f01emc.
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
f01jcc should be used.
f01fmc can be used to find the matrix function
$f\left(A\right)$ for a complex matrix
$A$.
10
Example
This example finds the
${e}^{2A}$ where
10.1
Program Text
10.2
Program Data
10.3
Program Results