NAG Library Routine Document

f01eff (real_symm_matrix_fun)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{uplo}$ – Character(1)Input

On entry: if $\mathbf{uplo}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathbf{n}$ – IntegerInput

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

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

3:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – 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 $n$ symmetric matrix $A$.

• If $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{uplo}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if $\mathbf{ifail}=\mathbf{0}$, the upper or lower triangular part of the $n$ by $n$ matrix function, $f\left(A\right)$.

4:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f01eff is called.

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

5:     $\mathbf{f}$ – 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:

Fortran Interface

Subroutine f ( iflag, n, x, fx, iuser, ruser) Integer, Intent (In) :: n Integer, Intent (Inout) :: iflag, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: fx(n)

C Header Interface

#include <nagmk26.h> void  f (Integer *iflag, const Integer *n, const double x[], double fx[], Integer iuser[], double ruser[])

1:     $\mathbf{iflag}$ – 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 $f\left(x\right)$; for instance $f\left(x\right)$ may not be defined, or may be complex. If iflag is returned as nonzero then f01eff will terminate the computation, with $\mathbf{ifail}=-\mathbf{6}$.

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of function values required.

3:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the $n$ points $x_1,x_2,\dots ,x_n$ at which the function $f$ is to be evaluated.

4:     $\mathbf{fx}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the $n$ function values. $\mathbf{fx}\left(\mathit{i}\right)$ should return the value $f\left(x_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$.

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

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

f is called with the arguments iuser and ruser as supplied to f01eff. 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 f01eff 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 f01eff. If your code inadvertently does return any NaNs or infinities, f01eff is likely to produce unexpected results.

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

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

iuser and ruser are not used by f01eff, but are passed directly to f and may be used to pass information to this routine.

8:     $\mathbf{iflag}$ – IntegerOutput

On exit: $\mathbf{iflag}=0$, unless you have set iflag nonzero inside f, in which case iflag will be the value you set and ifail will be set to $\mathbf{ifail}=-\mathbf{6}$.

9:     $\mathbf{ifail}$ – 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

$\mathbf{ifail}>0$

The computation of the spectral factorization failed to converge.

If $\mathbf{ifail}=i$, the algorithm to compute the spectral factorization failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero (see f08faf (dsyev)).

$\mathbf{ifail}=-1$

On entry, $\mathbf{uplo}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{uplo}=\text{'L'}$ or $\text{'U'}$.

$\mathbf{ifail}=-2$

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

$\mathbf{ifail}=-3$

An internal error occurred when computing the spectral factorization. Please contact NAG.

$\mathbf{ifail}=-4$

On entry, $\mathbf{lda}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lda}\ge \mathbf{n}$.

$\mathbf{ifail}=-6$

iflag was set to a nonzero value in f.

$\mathbf{ifail}=-99$

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 (f01effe.f90)

10.2

Program Data

Program Data (f01effe.d)

10.3

Program Results

Program Results (f01effe.r)