F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08GGF (DOPMTR)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08GGF (DOPMTR) multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ which was determined by F08GEF (DSPTRD) when reducing a real symmetric matrix to tridiagonal form.

## 2  Specification

 SUBROUTINE F08GGF ( SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
 INTEGER M, N, LDC, INFO REAL (KIND=nag_wp) AP(*), TAU(*), C(LDC,*), WORK(*) CHARACTER(1) SIDE, UPLO, TRANS
The routine may be called by its LAPACK name dopmtr.

## 3  Description

F08GGF (DOPMTR) is intended to be used after a call to F08GEF (DSPTRD), which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. F08GEF (DSPTRD) represents the orthogonal matrix $Q$ as a product of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real rectangular matrix).
A common application of this routine is to transform a matrix $Z$ of eigenvectors of $T$ to the matrix $QZ$ of eigenvectors of $A$.

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     $\mathrm{SIDE}$ – CHARACTER(1)Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{SIDE}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{SIDE}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{SIDE}}=\text{'L'}$ or $\text{'R'}$.
2:     $\mathrm{UPLO}$ – CHARACTER(1)Input
On entry: this must be the same parameter UPLO as supplied to F08GEF (DSPTRD).
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{TRANS}$ – CHARACTER(1)Input
On entry: indicates whether $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{TRANS}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{TRANS}}=\text{'T'}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$ or $\text{'T'}$.
4:     $\mathrm{M}$ – INTEGERInput
On entry: $m$, the number of rows of the matrix $C$; $m$ is also the order of $Q$ if ${\mathbf{SIDE}}=\text{'L'}$.
Constraint: ${\mathbf{M}}\ge 0$.
5:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the number of columns of the matrix $C$; $n$ is also the order of $Q$ if ${\mathbf{SIDE}}=\text{'R'}$.
Constraint: ${\mathbf{N}}\ge 0$.
6:     $\mathrm{AP}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}×\left({\mathbf{M}}+1\right)/2\right)$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$ if ${\mathbf{SIDE}}=\text{'R'}$.
On entry: details of the vectors which define the elementary reflectors, as returned by F08GEF (DSPTRD).
On exit: is used as internal workspace prior to being restored and hence is unchanged.
7:     $\mathrm{TAU}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}-1\right)$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$ if ${\mathbf{SIDE}}=\text{'R'}$.
On entry: further details of the elementary reflectors, as returned by F08GEF (DSPTRD).
8:     $\mathrm{C}\left({\mathbf{LDC}},*\right)$ – 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 $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by SIDE and TRANS.
9:     $\mathrm{LDC}$ – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08GGF (DOPMTR) is called.
Constraint: ${\mathbf{LDC}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
10:   $\mathrm{WORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{SIDE}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ if ${\mathbf{SIDE}}=\text{'R'}$.
11:   $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.

## 7  Accuracy

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

## 8  Parallelism and Performance

F08GGF (DOPMTR) is not threaded by NAG in any implementation.
F08GGF (DOPMTR) 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 $2{m}^{2}n$ if ${\mathbf{SIDE}}=\text{'L'}$ and $2m{n}^{2}$ if ${\mathbf{SIDE}}=\text{'R'}$.
The complex analogue of this routine is F08GUF (ZUPMTR).

## 10  Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix $A$, where
 $A = 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ,$
using packed storage. Here $A$ is symmetric and must first be reduced to tridiagonal form $T$ by F08GEF (DSPTRD). The program then calls F08JJF (DSTEBZ) to compute the requested eigenvalues and F08JKF (DSTEIN) to compute the associated eigenvectors of $T$. Finally F08GGF (DOPMTR) is called to transform the eigenvectors to those of $A$.

### 10.1  Program Text

Program Text (f08ggfe.f90)

### 10.2  Program Data

Program Data (f08ggfe.d)

### 10.3  Program Results

Program Results (f08ggfe.r)