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

NAG Library Routine Document


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.

+ Contents

    1  Purpose
    7  Accuracy

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

REAL (KIND=nag_wp)  AP(*), TAU(*), C(LDC,*), WORK(*)
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=QTQT. 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 QT is to be applied to C.
Q or QT is applied to C from the left.
Q or QT is applied to C from the right.
Constraint: SIDE='L' or 'R'.
2:     UPLO – CHARACTER(1)Input
On entry: this must be the same parameter UPLO as supplied to F08GEF (DSPTRD).
Constraint: UPLO='U' or 'L'.
3:     TRANS – CHARACTER(1)Input
On entry: indicates whether Q or QT is to be applied to C.
Q is applied to C.
QT is applied to C.
Constraint: TRANS='N' or 'T'.
4:     M – INTEGERInput
On entry: m, the number of rows of the matrix C; m is also the order of Q if SIDE='L'.
Constraint: M0.
5:     N – INTEGERInput
On entry: n, the number of columns of the matrix C; n is also the order of Q if SIDE='R'.
Constraint: N0.
6:     AP(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1, M × M+1 / 2  if SIDE='L' and at least max1, N × N+1 / 2  if SIDE='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 max1,M-1 if SIDE='L' and at least max1,N-1 if SIDE='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 max1,N.
On entry: the m by n matrix C.
On exit: C is overwritten by QC or QTC or CQ or CQT 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: LDCmax1,M.
10:   WORK(*) – REAL (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least max1,N if SIDE='L' and at least max1,M if SIDE='R'.
11:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If 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 ε is the machine precision.

8  Further Comments

The total number of floating point operations is approximately 2m2n if SIDE='L' and 2mn2 if SIDE='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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012