NAG Library Routine Document

F04YAF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     JOB – INTEGERInput

On entry: specifies which elements of $C$ are required.

$\mathbf{JOB}=-1$

The upper triangular part of $C$ is required.

$\mathbf{JOB}=0$

The diagonal elements of $C$ are required.

$\mathbf{JOB}>0$

The elements of column JOB of $C$ are required.

Constraint: $-1\le \mathbf{JOB}\le \mathbf{P}$.

2:     P – INTEGERInput

On entry: $p$, the order of the variance-covariance matrix $C$.

Constraint: $\mathbf{P}\ge 1$.

3:     SIGMA – REAL (KIND=nag_wp)Input

On entry: $\sigma$, the standard error of the residual vector given by

$$\begin{array}{ll}\sigma =\sqrt{r^{\mathrm{T}}r/\left(n-k\right)}\text{,}& n>k\\ & \\ \sigma =0\text{,}& n=k\text{,}\end{array}$$

where $k$ is the rank of $X$.

Constraint: $\mathbf{SIGMA}\ge 0.0$.

4:     A(LDA,P) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if $\mathbf{SVD}=\mathrm{.FALSE.}$, A must contain the upper triangular matrix $U$ of the $QU$ factorization of $X$, or of the Cholesky factorization of $X^{\mathrm{T}}X$; elements of the array below the diagonal need not be set.

If $\mathbf{SVD}=\mathrm{.TRUE.}$, $A$ must contain the first $k$ rows of the matrix $V^{\mathrm{T}}$, where $k$ is the rank of $X$ and $V$ is the right-hand orthogonal matrix of the singular value decomposition of $X$. Thus the $i$th row must contain the $i$th right-hand singular vector of $X$.

On exit: if $\mathbf{JOB}\ge 0$, $A$ is unchanged.

If $\mathbf{JOB}=-1$, A contains the upper triangle of the symmetric matrix $C$.

If $\mathbf{SVD}=\mathrm{.TRUE.}$, elements of the array below the diagonal are used as workspace.

If $\mathbf{SVD}=\mathrm{.FALSE.}$, they are unchanged.

5:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F04YAF is called.

Constraints:

• if $\mathbf{SVD}=\mathrm{.FALSE.}$ or $\mathbf{JOB}=-1$, $\mathbf{LDA}\ge \mathbf{P}$;
• if $\mathbf{SVD}=\mathrm{.TRUE.}$ and $\mathbf{JOB}\ge 0$, $\mathbf{LDA}\ge \max \left(1,\mathbf{IRANK}\right)$.

6:     SVD – LOGICALInput

On entry: must be .TRUE. if the least squares solution was obtained from a singular value decomposition of $X$. SVD must be .FALSE. if the least squares solution was obtained from either a $QU$ factorization of $X$ or a Cholesky factorization of $X^{\mathrm{T}}X$. In the latter case the rank of $X$ is assumed to be $p$ and so is applicable only to full rank problems with $n\ge p$.

7:     IRANK – INTEGERInput

On entry: if $\mathbf{SVD}=\mathrm{.TRUE.}$, IRANK must specify the rank $k$ of the matrix $X$.

If $\mathbf{SVD}=\mathrm{.FALSE.}$, IRANK is not referenced and the rank of $X$ is assumed to be $p$.

Constraint: $0<\mathbf{IRANK}\le \min \left(n,\mathbf{P}\right)$.

8:     SV(P) – REAL (KIND=nag_wp) arrayInput

On entry: if $\mathbf{SVD}=\mathrm{.TRUE.}$, SV must contain the first IRANK singular values of $X$.

If $\mathbf{SVD}=\mathrm{.FALSE.}$, SV is not referenced.

9:     CJ(P) – REAL (KIND=nag_wp) arrayOutput

On exit: if $\mathbf{JOB}=0$, CJ returns the diagonal elements of $C$.

If $\mathbf{JOB}=j>0$, CJ returns the $j$th column of $C$.

If $\mathbf{JOB}=-1$, CJ is not referenced.

10:   WORK(P) – REAL (KIND=nag_wp) arrayWorkspace

If $\mathbf{JOB}>0$, WORK is not referenced.

11:   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

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{P}<1$,
or	$\mathbf{SIGMA}<0.0$,
or	$\mathbf{JOB}<-1$,
or	$\mathbf{JOB}>\mathbf{P}$,
or	$\mathbf{SVD}=\mathrm{.TRUE.}$ and ($\mathbf{IRANK}<0$ or $\mathbf{IRANK}>\mathbf{P}$) or ($\mathbf{JOB}\ge 0$ and $\mathbf{LDA}<\max \left(1,\mathbf{IRANK}\right)$) or ($\mathbf{JOB}=-1$ and $\mathbf{LDA}<\mathbf{P}$)),
or	$\mathbf{SVD}=\mathrm{.FALSE.}$ and $\mathbf{LDA}<\mathbf{P}$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{SVD}=\mathrm{.TRUE.}$ and $\mathbf{IRANK}=0$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{SVD}=\mathrm{.FALSE.}$ and overflow would occur in computing an element of $C$. The upper triangular matrix $U$ must be very nearly singular.

$\mathbf{IFAIL}=4$

On entry, $\mathbf{SVD}=\mathrm{.TRUE.}$ and one of the first IRANK singular values is zero. Either the first IRANK singular values or IRANK must be incorrect.

$\mathbf{\text{overflow}}$

If overflow occurs then either an element of $C$ is very large, or more likely, either the rank, or the upper triangular matrix, or the singular values or vectors have been incorrectly supplied.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f04yafe.f90)

9.2  Program Data

Program Data (f04yafe.d)

9.3  Program Results

Program Results (f04yafe.r)