# NAG FL Interfacef08bkf (dormrz)

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## 1Purpose

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

## 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 <nag.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 the names f08bkf, nagf_lapackeig_dormrz or its LAPACK name dormrz.

## 3Description

f08bkf is intended to be used following a call to f08bhf, 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 $m×n$ 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 https://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: $\mathbf{m}$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 $Z$.
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: $\mathbf{l}$Integer Input
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) array Input
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.
8: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08bkf 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) 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: ${\mathbf{tau}}\left(i\right)$ must contain the scalar factor of the elementary reflector ${H}_{i}$, as returned by f08bhf.
10: $\mathbf{c}\left({\mathbf{ldc}},*\right)$Real (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 $ZC$ or ${Z}^{\mathrm{T}}C$ or $CZ$ or ${Z}^{\mathrm{T}}C$ as specified by side and trans.
11: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08bkf 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) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
13: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08bkf 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: $\mathbf{info}$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 $4nlk$ if ${\mathbf{side}}=\text{'L'}$ and $4mlk$ if ${\mathbf{side}}=\text{'R'}$.