F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08CUF (ZUNMQL)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08CUF (ZUNMQL) multiplies a general complex $m$ by $n$ matrix $C$ by the complex unitary matrix $Q$ from a $QL$ factorization computed by F08CSF (ZGEQLF).

## 2  Specification

 SUBROUTINE F08CUF ( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
 INTEGER M, N, K, LDA, LDC, LWORK, INFO COMPLEX (KIND=nag_wp) A(LDA,*), TAU(*), C(LDC,*), WORK(max(1,LWORK)) CHARACTER(1) SIDE, TRANS
The routine may be called by its LAPACK name zunmql.

## 3  Description

F08CUF (ZUNMQL) is intended to be used following a call to F08CSF (ZGEQLF), which performs a $QL$ factorization of a complex matrix $A$ and represents the unitary matrix $Q$ as a product of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QHC , CQ , CQH ,$
overwriting the result on $C$, which may be any complex rectangular $m$ by $n$ matrix.
A common application of this routine is in solving linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Section 9 in F08CSF (ZGEQLF).

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

## 5  Parameters

1:     SIDE – CHARACTER(1)Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{SIDE}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the left.
${\mathbf{SIDE}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{SIDE}}=\text{'L'}$ or $\text{'R'}$.
2:     TRANS – CHARACTER(1)Input
On entry: indicates whether $Q$ or ${Q}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{TRANS}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{TRANS}}=\text{'C'}$
${Q}^{\mathrm{H}}$ is applied to $C$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$ or $\text{'C'}$.
3:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{M}}\ge 0$.
4:     N – INTEGERInput
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{N}}\ge 0$.
5:     K – INTEGERInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraints:
• if ${\mathbf{SIDE}}=\text{'L'}$, ${\mathbf{M}}\ge {\mathbf{K}}\ge 0$;
• if ${\mathbf{SIDE}}=\text{'R'}$, ${\mathbf{N}}\ge {\mathbf{K}}\ge 0$.
6:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{K}}\right)$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08CSF (ZGEQLF).
On exit: is modified by F08CUF (ZUNMQL) but restored on exit.
7:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08CUF (ZUNMQL) is called.
Constraints:
• if ${\mathbf{SIDE}}=\text{'L'}$, ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$;
• if ${\mathbf{SIDE}}=\text{'R'}$, ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     TAU($*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{K}}\right)$.
On entry: further details of the elementary reflectors, as returned by F08CSF (ZGEQLF).
9:     C(LDC,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array C must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $m$ by $n$ matrix $C$.
On exit: C is overwritten by $QC$ or ${Q}^{\mathrm{H}}C$ or $CQ$ or $C{Q}^{\mathrm{H}}$ as specified by SIDE and TRANS.
10:   LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08CUF (ZUNMQL) is called.
Constraint: ${\mathbf{LDC}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
11:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
12:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08CUF (ZUNMQL) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, ${\mathbf{LWORK}}\ge {\mathbf{N}}×\mathit{nb}$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least ${\mathbf{M}}×\mathit{nb}$ if ${\mathbf{SIDE}}=\text{'R'}$, where $\mathit{nb}$ is the optimal block size.
Constraints:
• if ${\mathbf{SIDE}}=\text{'L'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ or ${\mathbf{LWORK}}=-1$;
• if ${\mathbf{SIDE}}=\text{'R'}$, ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ or ${\mathbf{LWORK}}=-1$.
13:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

The computed result differs from the exact result by a matrix $E$ such that
 $E2 = O⁡ε C2$
where $\epsilon$ is the machine precision.

The total number of floating point operations is approximately $8nk\left(2m-k\right)$ if ${\mathbf{SIDE}}=\text{'L'}$ and $8mk\left(2n-k\right)$ if ${\mathbf{SIDE}}=\text{'R'}$.