NAG Library Routine Document

F01JBF

1  Purpose

2  Specification

3  Description

4  References

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 F01JBF 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:

SUBROUTINE F ( IFLAG, NZ, Z, FZ, IUSER, RUSER) 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 $f\left(z\right)$; for instance $f\left(z\right)$ may not be defined. If IFLAG is returned as nonzero then F01JBF will terminate the computation, with $\mathbf{IFAIL}=\mathbf{3}$.

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 F01JBF. 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 F01JBF 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 F01JBF, 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 $\mathbf{IFAIL}=\mathbf{3}$.

8:     $\mathrm{CONDA}$ – REAL (KIND=nag_wp)Output

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

9:     $\mathrm{NORMA}$ – REAL (KIND=nag_wp)Output

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

10:   $\mathrm{NORMFA}$ – REAL (KIND=nag_wp)Output

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

11:   $\mathrm{IFAIL}$ – 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

$\mathbf{IFAIL}=1$

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 F01ELF 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, 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

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f01jbfe.f90)

10.2  Program Data

Program Data (f01jbfe.d)

10.3  Program Results

Program Results (f01jbfe.r)