NAG FL Interface
f01emf (real_​gen_​matrix_​fun_​usd)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{n}$ – Integer Input

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

2: $\mathbf{a}\left(\mathbf{lda},*\right)$ – Real (Kind=nag_wp) array Input/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: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f01emf is called.

Constraint: $\mathbf{lda}\ge \mathbf{n}$.

4: $\mathbf{f}$ – 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:

Fortran Interface

Subroutine f ( m, iflag, nz, z, fz, iuser, ruser) Integer, Intent (In) :: m, nz Integer, Intent (Inout) :: iflag, iuser(*) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Complex (Kind=nag_wp), Intent (In) :: z(nz) Complex (Kind=nag_wp), Intent (Out) :: fz(nz)

C Header Interface

void  f_ (const Integer *m, Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[])

C++ Header Interface

#include <nag.h> extern "C" { void  f_ (const Integer &m, Integer &iflag, const Integer &nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[]) }

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 f01emf will terminate the computation, with $\mathbf{ifail}=\mathbf{2}$.

3: $\mathbf{nz}$ – Integer Input

On entry: $n_z$, the number of function or derivative values required.

4: $\mathbf{z}\left(\mathbf{nz}\right)$ – Complex (Kind=nag_wp) array 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: $\mathbf{fz}\left(\mathbf{nz}\right)$ – Complex (Kind=nag_wp) array Output

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: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

7: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User Workspace

f is called with the arguments iuser and ruser as supplied to f01emf. You should use the arrays iuser and ruser to supply information to f.

f must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f01emf is called. Arguments denoted as Input must not be changed by this procedure.

Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01emf. If your code inadvertently does return any NaNs or infinities, f01emf is likely to produce unexpected results.

5: $\mathbf{iuser}\left(*\right)$ – Integer array User Workspace

6: $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) array User 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.

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 ifail will be set to $\mathbf{ifail}=\mathbf{2}$.

8: $\mathbf{imnorm}$ – 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: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface 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 argument, 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

$\mathbf{ifail}=1$

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.

$\mathbf{ifail}=5$

An unexpected internal error occurred. Please contact NAG.

$\mathbf{ifail}=-1$

Input argument number $〈\mathit{\text{value}}〉$ is invalid.

$\mathbf{ifail}=-3$

On entry, argument 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 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results