NAG FL Interface
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: $\mathbf{n}$ – Integer Input

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) array Input/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: $\mathbf{lda}$ – Integer Input

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

void  f_ (Integer *iflag, const Integer *n, const double x[], double fx[], Integer iuser[], double ruser[])

C++ Header Interface

#include <nag.h> extern "C" { void  f_ (Integer &iflag, const Integer &n, const double x[], double fx[], Integer iuser[], double ruser[]) }

1: $\mathbf{iflag}$ – 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 $f\left(x\right)$; 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: $\mathbf{n}$ – Integer Input

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

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

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) array Output

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 array User Workspace

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

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

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

On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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.

The value of ifail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08faf).

$\mathbf{ifail}=-1$

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

$\mathbf{ifail}=-6$

Termination requested in f.

$\mathbf{ifail}=-99$

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