NAG Library Routine Document

F01EFF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     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:     N – INTEGERInput

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

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

3:     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 $n$ 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:     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:     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, N, X, FX, IUSER, RUSER) INTEGER  IFLAG, N, IUSER(*) REAL (KIND=nag_wp)  X(N), FX(N), RUSER(*)

1:     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:     N – INTEGERInput

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

3:     X(N) – 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:     FX(N) – 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:     IUSER($*$) – INTEGER arrayUser Workspace

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

F is called with the parameters IUSER and RUSER as supplied to F01EFF. 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 F01EFF is called. Parameters denoted as Input must not be changed by this procedure.

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

7:     RUSER($*$) – 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 as an alternative to using COMMON global variables.

8:     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:     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

$\mathbf{IFAIL}<0\text{ and }\mathbf{IFAIL}\ne -999\text{ or }-6$

If $\mathbf{IFAIL}=-\mathit{i}$, the $i$th argument had an illegal value.

$\mathbf{IFAIL}=-6$

IFLAG has been set nonzero by the user.

$\mathbf{IFAIL}=-999$

Internal memory allocation failed.

The integer allocatable memory required is N, and the real allocatable memory required is approximately $\left(\mathbf{N}+\mathit{nb}+4\right)\times \mathbf{N}$, where nb is the block size required by F08FAF (DSYEV).

$\mathbf{IFAIL}=i\text{ and }\mathbf{IFAIL}>0$

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)).

Note:  this failure is unlikely to occur.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f01effe.f90)

9.2  Program Data

Program Data (f01effe.d)

9.3  Program Results

Program Results (f01effe.r)