NAG Library Routine Document

F02WUF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     N – INTEGERInput

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

If $\mathbf{N}=0$, an immediate return is effected.

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 $\max \left(1,\mathbf{N}\right)$.

On entry: the leading $n$ by $n$ upper triangular part of the array A must contain the upper triangular matrix $R$.

On exit: if $\mathbf{WANTP}=\mathrm{.TRUE.}$, the $n$ by $n$ part of A will contain the $n$ by $n$ orthogonal matrix $P^{\mathrm{T}}$, otherwise the $n$ by $n$ upper triangular part of A is used as internal workspace, but the strictly lower triangular part of A is not referenced.

3:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F02WUF is called.

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

4:     NCOLB – INTEGERInput

On entry: $\mathit{ncolb}$, the number of columns of the matrix $B$.

If $\mathbf{NCOLB}=0$, the array B is not referenced.

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

5:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least $\max \left(1,\mathbf{NCOLB}\right)$.

On entry: with $\mathbf{NCOLB}>0$, the leading $n$ by $\mathit{ncolb}$ part of the array B must contain the matrix to be transformed.

On exit: the leading $n$ by $\mathit{ncolb}$ part of the array B is overwritten by the matrix $Q^{\mathrm{T}}B$.

6:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F02WUF is called.

Constraints:

• if $\mathbf{NCOLB}>0$, $\mathbf{LDB}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDB}\ge 1$.

7:     WANTQ – LOGICALInput

On entry: must be .TRUE. if the matrix $Q$ is required.

If $\mathbf{WANTQ}=\mathrm{.FALSE.}$, the array Q is not referenced.

8:     Q(LDQ,$*$) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Q must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, and at least $1$ otherwise.

On exit: with $\mathbf{WANTQ}=\mathrm{.TRUE.}$, the leading $n$ by $n$ part of the array Q will contain the orthogonal matrix $Q$. Otherwise the array Q is not referenced.

9:     LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F02WUF is called.

Constraints:

• if $\mathbf{WANTQ}=\mathrm{.TRUE.}$, $\mathbf{LDQ}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDQ}\ge 1$.

10:   SV(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the array SV will contain the $n$ diagonal elements of the matrix $S$.

11:   WANTP – LOGICALInput

On entry: must be .TRUE. if the matrix $P^{\mathrm{T}}$ is required, in which case $P^{\mathrm{T}}$ is overwritten on the array A, otherwise WANTP must be .FALSE..

12:   WORK($*$) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array WORK must be at least $\max \left(1,2\times \left(\mathbf{N}-1\right)\right)$ if $\mathbf{NCOLB}=0$ and $\mathbf{WANTQ}=\mathrm{.FALSE.}$ and $\mathbf{WANTP}=\mathrm{.FALSE.}$, $\max \left(1,3\times \left(\mathbf{N}-1\right)\right)$ if ($\mathbf{NCOLB}=0$ and $\mathbf{WANTQ}=\mathrm{.FALSE.}$ and $\mathbf{WANTP}=\mathrm{.TRUE.}$) or ($\mathbf{WANTP}=\mathrm{.FALSE.}$ and ($\mathbf{NCOLB}>0$ or $\mathbf{WANTQ}=\mathrm{.TRUE.}$)), and at least $\max \left(1,5\times \left(\mathbf{N}-1\right)\right)$ otherwise.

On exit: $\mathbf{WORK}\left(\mathbf{N}\right)$ contains the total number of iterations taken by the $QR$ algorithm.

The rest of the array is used as internal workspace.

13:   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,	$\mathbf{N}<0$,
or	$\mathbf{LDA}<\mathbf{N}$,
or	$\mathbf{NCOLB}<0$,
or	$\mathbf{LDB}<\mathbf{N}$ and $\mathbf{NCOLB}>0$,
or	$\mathbf{LDQ}<\mathbf{N}$ and $\mathbf{WANTQ}=\mathrm{.TRUE.}$.

$\mathbf{IFAIL}>0$

The $QR$ algorithm has failed to converge in $50\times \mathbf{N}$ iterations. In this case $\mathbf{SV}\left(1\right),\mathbf{SV}\left(2\right),\dots ,\mathbf{SV}\left(\mathbf{IFAIL}\right)$ may not have been found correctly and the remaining singular values may not be the smallest. The matrix $R$ will nevertheless have been factorized as $R=QE{P}^{\mathrm{T}}$, where $E$ is a bidiagonal matrix with $\mathbf{SV}\left(1\right),\mathbf{SV}\left(2\right),\dots ,\mathbf{SV}\left(n\right)$ as the diagonal elements and $\mathbf{WORK}\left(1\right),\mathbf{WORK}\left(2\right),\dots ,\mathbf{WORK}\left(n-1\right)$ as the superdiagonal elements.

This failure is not likely to occur.

7  Accuracy

8  Further Comments

IRANK = F06KLF(N,SV,1,TOL)

9  Example

9.1  Program Text

Program Text (f02wufe.f90)

9.2  Program Data

Program Data (f02wufe.d)

9.3  Program Results

Program Results (f02wufe.r)