# NAG FL Interfacef08cuf (zunmql)

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## 1Purpose

f08cuf multiplies a general complex $m×n$ matrix $C$ by the complex unitary matrix $Q$ from a $QL$ factorization computed by f08csf.

## 2Specification

Fortran Interface
 Subroutine f08cuf ( side, m, n, k, a, lda, tau, c, ldc, work, info)
 Integer, Intent (In) :: m, n, k, lda, ldc, lwork Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: tau(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: side, trans
#include <nag.h>
 void f08cuf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, Complex a[], const Integer *lda, const Complex tau[], Complex c[], const Integer *ldc, Complex work[], const Integer *lwork, Integer *info, const Charlen length_side, const Charlen length_trans)
The routine may be called by the names f08cuf, nagf_lapackeig_zunmql or its LAPACK name zunmql.

## 3Description

f08cuf is intended to be used following a call to f08csf, 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 $m×n$ 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 10 in f08csf.

## 4References

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 https://www.netlib.org/lapack/lug

## 5Arguments

1: $\mathbf{side}$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: $\mathbf{trans}$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: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{k}$Integer Input
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: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input
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.
On exit: is modified by f08cuf but restored on exit.
7: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08cuf 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: $\mathbf{tau}\left(*\right)$Complex (Kind=nag_wp) array Input
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.
9: $\mathbf{c}\left({\mathbf{ldc}},*\right)$Complex (Kind=nag_wp) array Input/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 $m×n$ 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: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08cuf is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
11: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
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: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08cuf 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: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

$-999<{\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The computed result differs from the exact result by a matrix $E$ such that
 $‖E‖2 = O⁡ε ‖C‖2$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

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 ${\mathbf{side}}=\text{'R'}$.