NAG Library Routine Document

f01jcf (real_gen_matrix_cond_usd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{n}$ – IntegerInput

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) 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:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f01jcf 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

#include <nagmk26.h> void  f (const Integer *m, Integer *iflag, const Integer *nz, const Complex z[], Complex fz[], Integer iuser[], double ruser[])

1:     $\mathbf{m}$ – IntegerInput

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:     $\mathbf{iflag}$ – 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 $f\left(z\right)$; for instance $f\left(z\right)$ may not be defined. If iflag is returned as nonzero then f01jcf will terminate the computation, with $\mathbf{ifail}=\mathbf{3}$.

3:     $\mathbf{nz}$ – IntegerInput

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

4:     $\mathbf{z}\left(\mathbf{nz}\right)$ – 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.

5:     $\mathbf{fz}\left(\mathbf{nz}\right)$ – Complex (Kind=nag_wp) arrayOutput

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 arrayUser Workspace

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

f is called with the arguments iuser and ruser as supplied to f01jcf. 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 f01jcf 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 f01jcf. If your code inadvertently does return any NaNs or infinities, f01jcf is likely to produce unexpected results.

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

6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by f01jcf, but are passed directly to f and may be used to pass information to this routine.

7:     $\mathbf{iflag}$ – 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 $\mathbf{ifail}=\mathbf{3}$.

8:     $\mathbf{conda}$ – Real (Kind=nag_wp)Output

On exit: an estimate of the absolute condition number of $f$ at $A$.

9:     $\mathbf{norma}$ – Real (Kind=nag_wp)Output

On exit: the $1$-norm of $A$.

10:   $\mathbf{normfa}$ – Real (Kind=nag_wp)Output

On exit: the $1$-norm of $f\left(A\right)$.

11:   $\mathbf{ifail}$ – IntegerInput/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 3.4 in How to Use the NAG Library and its Documentation 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

$\mathbf{ifail}=1$

An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$. Please contact NAG.

$\mathbf{ifail}=2$

An internal error occurred when evaluating the matrix function $f\left(A\right)$. You can investigate further by calling f01emf with the matrix $A$ and the function $f$.

$\mathbf{ifail}=3$

iflag has been set nonzero by the user-supplied subroutine.

$\mathbf{ifail}=-1$

On entry, $\mathbf{n}<0$.
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 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f01jcfe.f90)

10.2

Program Data

Program Data (f01jcfe.d)

10.3

Program Results

Program Results (f01jcfe.r)