# NAG Library Routine Document

## 1Purpose

f08bkf (dormrz) multiplies a general real $m$ by $n$ matrix $C$ by the real orthogonal matrix $Z$ from an $RZ$ factorization computed by f08bhf (dtzrzf).

## 2Specification

Fortran Interface
 Subroutine f08bkf ( side, m, n, k, l, a, lda, tau, c, ldc, work, info)
 Integer, Intent (In) :: m, n, k, l, lda, ldc, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: tau(*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: side, trans
#include <nagmk26.h>
 void f08bkf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, const Integer *l, double a[], const Integer *lda, const double tau[], double c[], const Integer *ldc, double work[], const Integer *lwork, Integer *info, const Charlen length_side, const Charlen length_trans)
The routine may be called by its LAPACK name dormrz.

## 3Description

f08bkf (dormrz) is intended to be used following a call to f08bhf (dtzrzf), which performs an $RZ$ factorization of a real upper trapezoidal matrix $A$ and represents the orthogonal matrix $Z$ as a product of elementary reflectors.
This routine may be used to form one of the matrix products
 $ZC , ZTC , CZ , CZT ,$
overwriting the result on $C$, which may be any real rectangular $m$ by $n$ matrix.

## 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 http://www.netlib.org/lapack/lug

## 5Arguments

1:     $\mathbf{side}$ – Character(1)Input
On entry: indicates how $Z$ or ${Z}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Z$ or ${Z}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Z$ or ${Z}^{\mathrm{T}}$ 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 $Z$ or ${Z}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Z$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Z}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
3:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Z$.
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:     $\mathbf{l}$ – IntegerInput
On entry: $l$, the number of columns of the matrix $A$ containing the meaningful part of the Householder reflectors.
Constraints:
• if ${\mathbf{side}}=\text{'L'}$, ${\mathbf{m}}\ge {\mathbf{l}}\ge 0$;
• if ${\mathbf{side}}=\text{'R'}$, ${\mathbf{n}}\ge {\mathbf{l}}\ge 0$.
7:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'R'}$.
On entry: the $\mathit{i}$th row of a must contain the vector which defines the elementary reflector ${H}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$, as returned by f08bhf (dtzrzf).
8:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bkf (dormrz) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
9:     $\mathbf{tau}\left(*\right)$ – Real (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: ${\mathbf{tau}}\left(i\right)$ must contain the scalar factor of the elementary reflector ${H}_{i}$, as returned by f08bhf (dtzrzf).
10:   $\mathbf{c}\left({\mathbf{ldc}},*\right)$ – Real (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 $ZC$ or ${Z}^{\mathrm{T}}C$ or $CZ$ or ${Z}^{\mathrm{T}}C$ as specified by side and trans.
11:   $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08bkf (dormrz) is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
12:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ 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 f08bkf (dormrz) 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

$-999<{\mathbf{info}}<0$
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 3.7 in How to Use the NAG Library and its Documentation 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
 $E2 = O⁡ε C2$
where $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08bkf (dormrz) 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 floating-point operations is approximately $4nlk$ if ${\mathbf{side}}=\text{'L'}$ and $4mlk$ if ${\mathbf{side}}=\text{'R'}$.
The complex analogue of this routine is f08bxf (zunmrz).

## 10Example

See Section 10 in f08bhf (dtzrzf).