# NAG Library Routine Document

## 1Purpose

f08kuf (zunmbr) multiplies an arbitrary complex $m$ by $n$ matrix $C$ by one of the complex unitary matrices $Q$ or $P$ which were determined by f08ksf (zgebrd) when reducing a complex matrix to bidiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08kuf ( vect, 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) :: vect, side, trans
#include nagmk26.h
 void f08kuf_ (const char *vect, 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_vect, const Charlen length_side, const Charlen length_trans)
The routine may be called by its LAPACK name zunmbr.

## 3Description

f08kuf (zunmbr) is intended to be used after a call to f08ksf (zgebrd), which reduces a complex rectangular matrix $A$ to real bidiagonal form $B$ by a unitary transformation: $A=QB{P}^{\mathrm{H}}$. f08ksf (zgebrd) represents the matrices $Q$ and ${P}^{\mathrm{H}}$ as products of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QHC , CQ , CQH , PC , PHC , CP ​ or ​ CPH ,$
overwriting the result on $C$ (which may be any complex rectangular matrix).
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

Note: in the descriptions below, $\mathit{r}$ denotes the order of $Q$ or ${P}^{\mathrm{H}}$: if ${\mathbf{side}}=\text{'L'}$, $\mathit{r}={\mathbf{m}}$ and if ${\mathbf{side}}=\text{'R'}$, $\mathit{r}={\mathbf{n}}$.
1:     $\mathbf{vect}$ – Character(1)Input
On entry: indicates whether $Q$ or ${Q}^{\mathrm{H}}$ or $P$ or ${P}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{vect}}=\text{'Q'}$
$Q$ or ${Q}^{\mathrm{H}}$ is applied to $C$.
${\mathbf{vect}}=\text{'P'}$
$P$ or ${P}^{\mathrm{H}}$ is applied to $C$.
Constraint: ${\mathbf{vect}}=\text{'Q'}$ or $\text{'P'}$.
2:     $\mathbf{side}$ – Character(1)Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{H}}$ or $P$ or ${P}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{H}}$ or $P$ or ${P}^{\mathrm{H}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{H}}$ or $P$ or ${P}^{\mathrm{H}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
3:     $\mathbf{trans}$ – Character(1)Input
On entry: indicates whether $Q$ or $P$ or ${Q}^{\mathrm{H}}$ or ${P}^{\mathrm{H}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ or $P$ is applied to $C$.
${\mathbf{trans}}=\text{'C'}$
${Q}^{\mathrm{H}}$ or ${P}^{\mathrm{H}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'C'}$.
4:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
5:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     $\mathbf{k}$ – IntegerInput
On entry: if ${\mathbf{vect}}=\text{'Q'}$, the number of columns in the original matrix $A$.
If ${\mathbf{vect}}=\text{'P'}$, the number of rows in the original matrix $A$.
Constraint: ${\mathbf{k}}\ge 0$.
7:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$ if ${\mathbf{vect}}=\text{'Q'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}\right)$ if ${\mathbf{vect}}=\text{'P'}$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08ksf (zgebrd).
8:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08kuf (zunmbr) is called.
Constraints:
• if ${\mathbf{vect}}=\text{'Q'}$, ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{r}\right)$;
• if ${\mathbf{vect}}=\text{'P'}$, ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$.
9:     $\mathbf{tau}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$.
On entry: further details of the elementary reflectors, as returned by f08ksf (zgebrd) in its argument tauq if ${\mathbf{vect}}=\text{'Q'}$, or in its argument taup if ${\mathbf{vect}}=\text{'P'}$.
10:   $\mathbf{c}\left({\mathbf{ldc}},*\right)$ – 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 matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{H}}C$ or $CQ$ or ${C}^{\mathrm{H}}Q$ or $PC$ or ${P}^{\mathrm{H}}C$ or $CP$ or ${C}^{\mathrm{H}}P$ as specified by vect, side and trans.
11:   $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08kuf (zunmbr) is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
12:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – 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.
13:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kuf (zunmbr) 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 ${\mathbf{lwork}}=-1$.
14:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

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

## 8Parallelism and Performance

f08kuf (zunmbr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kuf (zunmbr) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately
• if ${\mathbf{side}}=\text{'L'}$ and $m\ge k$, $8nk\left(2m-k\right)$;
• if ${\mathbf{side}}=\text{'R'}$ and $n\ge k$, $8mk\left(2n-k\right)$;
• if ${\mathbf{side}}=\text{'L'}$ and $m, $8{m}^{2}n$;
• if ${\mathbf{side}}=\text{'R'}$ and $n, $8m{n}^{2}$,
where $k$ is the value of the argument k.
The real analogue of this routine is f08kgf (dormbr).

## 10Example

For this routine two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix $A$ may be preceded by a $QR$ or $LQ$ factorization of $A$.
In the first example, $m>n$, and
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$
The routine first performs a $QR$ factorization of $A$ as $A={Q}_{a}R$ and then reduces the factor $R$ to bidiagonal form $B$: $R={Q}_{b}B{P}^{\mathrm{H}}$. Finally it forms ${Q}_{a}$ and calls f08kuf (zunmbr) to form $Q={Q}_{a}{Q}_{b}$.
In the second example, $m, and
 $A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i .$
The routine first performs an $LQ$ factorization of $A$ as $A=L{P}_{a}^{\mathrm{H}}$ and then reduces the factor $L$ to bidiagonal form $B$: $L=QB{P}_{b}^{\mathrm{H}}$. Finally it forms ${P}_{b}^{\mathrm{H}}$ and calls f08kuf (zunmbr) to form ${P}^{\mathrm{H}}={P}_{b}^{\mathrm{H}}{P}_{a}^{\mathrm{H}}$.

### 10.1Program Text

Program Text (f08kufe.f90)

### 10.2Program Data

Program Data (f08kufe.d)

### 10.3Program Results

Program Results (f08kufe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017