F08GGF (DOPMTR) (PDF version)
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:     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:     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:     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:     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:     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:     AP($*$) – 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:     TAU($*$) – 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:     C(LDC,$*$) – 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:     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:   WORK($*$) – 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:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
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  Further Comments

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).

## 9  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 $A$.

### 9.1  Program Text

Program Text (f08ggfe.f90)

### 9.2  Program Data

Program Data (f08ggfe.d)

### 9.3  Program Results

Program Results (f08ggfe.r)

F08GGF (DOPMTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual