# NAG FL Interfacef08kgf (dormbr)

## 1Purpose

f08kgf multiplies an arbitrary real $m$ by $n$ matrix $C$ by one of the real orthogonal matrices $Q$ or $P$ which were determined by f08kef when reducing a real matrix to bidiagonal form.

## 2Specification

Fortran Interface
 Subroutine f08kgf ( 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 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) :: vect, side, trans
#include <nag.h>
 void f08kgf_ (const char *vect, const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double c[], const Integer *ldc, double work[], const Integer *lwork, Integer *info, const Charlen length_vect, const Charlen length_side, const Charlen length_trans)
The routine may be called by the names f08kgf, nagf_lapackeig_dormbr or its LAPACK name dormbr.

## 3Description

f08kgf is intended to be used after a call to f08kef, which reduces a real rectangular matrix $A$ to bidiagonal form $B$ by an orthogonal transformation: $A=QB{P}^{\mathrm{T}}$. f08kef represents the matrices $Q$ and ${P}^{\mathrm{T}}$ as products of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QTC , CQ , CQT , PC , PTC , CP ​ or ​ CPT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).

## 4References

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 ${P}^{\mathrm{T}}$: 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 ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{vect}}=\text{'Q'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$.
${\mathbf{vect}}=\text{'P'}$
$P$ or ${P}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{vect}}=\text{'Q'}$ or $\text{'P'}$.
2: $\mathbf{side}$Character(1) Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ or $P$ or ${P}^{\mathrm{T}}$ 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 ${Q}^{\mathrm{T}}$ or ${P}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ or $P$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Q}^{\mathrm{T}}$ or ${P}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
4: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $C$.
Constraint: ${\mathbf{m}}\ge 0$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $C$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{k}$Integer Input
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: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$ 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 f08kef.
8: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08kgf 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: $\mathbf{tau}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{r},{\mathbf{k}}\right)\right)$.
On entry: further details of the elementary reflectors, as returned by f08kef in its argument tauq if ${\mathbf{vect}}=\text{'Q'}$, or in its argument taup if ${\mathbf{vect}}=\text{'P'}$.
10: $\mathbf{c}\left({\mathbf{ldc}},*\right)$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 matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or ${C}^{\mathrm{T}}Q$ or $PC$ or ${P}^{\mathrm{T}}C$ or $CP$ or ${C}^{\mathrm{T}}P$ as specified by vect, 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 f08kgf 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 f08kgf 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

${\mathbf{info}}<0$
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

f08kgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kgf 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
• if ${\mathbf{side}}=\text{'L'}$ and $m\ge k$, $2nk\left(2m-k\right)$;
• if ${\mathbf{side}}=\text{'R'}$ and $n\ge k$, $2mk\left(2n-k\right)$;
• if ${\mathbf{side}}=\text{'L'}$ and $m, $2{m}^{2}n$;
• if ${\mathbf{side}}=\text{'R'}$ and $n, $2m{n}^{2}$,
where $k$ is the value of the argument k.
The complex analogue of this routine is f08kuf.

## 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
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 .$
The routine first performs a $QR$ factorization of $A$ as $A={Q}_{a}R$ and then reduces the factor $R$ to bidiagonal form $B$: $R={Q}_{b}B{P}^{\mathrm{T}}$. Finally it forms ${Q}_{a}$ and calls f08kgf to form $Q={Q}_{a}{Q}_{b}$.
In the second example, $m, and
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 .$
The routine first performs an $LQ$ factorization of $A$ as $A=L{P}_{a}^{\mathrm{T}}$ and then reduces the factor $L$ to bidiagonal form $B$: $L=QB{P}_{b}^{\mathrm{T}}$. Finally it forms ${P}_{b}^{\mathrm{T}}$ and calls f08kgf to form ${P}^{\mathrm{T}}={P}_{b}^{\mathrm{T}}{P}_{a}^{\mathrm{T}}$.

### 10.1Program Text

Program Text (f08kgfe.f90)

### 10.2Program Data

Program Data (f08kgfe.d)

### 10.3Program Results

Program Results (f08kgfe.r)