NAG Library Routine Document

F01EMF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     N – INTEGERInput

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

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

2:     A(LDA,$*$) – 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:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F01EMF is called.

Constraint: $\mathbf{LDA}\ge \max \left(1,\mathbf{N}\right)$.

4:     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:

SUBROUTINE F ( M, IFLAG, NZ, Z, FZ, IUSER, RUSER) INTEGER  M, IFLAG, NZ, IUSER(*) REAL (KIND=nag_wp)  RUSER(*) COMPLEX (KIND=nag_wp)  Z(NZ), FZ(NZ)

1:     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:     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 $f\left(z_i\right)$ 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:     NZ – INTEGERInput

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

4:     Z(NZ) – 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:     FZ(NZ) – 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:     IUSER($*$) – INTEGER arrayUser Workspace

7:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser Workspace

F is called with the parameters IUSER and RUSER as supplied to F01EMF. 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 F01EMF is called. Parameters denoted as Input must not be changed by this procedure.

5:     IUSER($*$) – INTEGER arrayUser Workspace

6:     RUSER($*$) – REAL (KIND=nag_wp) arrayUser 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 as an alternative to using COMMON global variables.

7:     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 $\mathbf{IFAIL}=\mathbf{2}$.

8:     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:     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

$\mathbf{IFAIL}=1$

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, parameter LDA is invalid.
Constraint: $\mathbf{LDA}\ge \mathbf{N}$.

$\mathbf{IFAIL}=-999$

Allocation of memory failed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f01emfe.f90)

9.2  Program Data

Program Data (f01emfe.d)

9.3  Program Results

Program Results (f01emfe.r)