NAG Library Routine Document

F08GGF (DOPMTR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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 = Q T Q^{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

Q C, Q^{T} C, C Q ​ or ​ C Q^{T},

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

Q Z

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

Q^{T}

is to be applied to

C

$SIDE ='L'$: $Q$ or $Q^{T}$ is applied to $C$ from the left.
$SIDE ='R'$: $Q$ or $Q^{T}$ is applied to $C$ from the right.

Constraint:

SIDE ='L'

'R'

2: UPLO – CHARACTER(1)Input

On entry: this must be the same parameter UPLO as supplied to F08GEF (DSPTRD).

Constraint:

UPLO ='U'

'L'

3: TRANS – CHARACTER(1)Input

On entry: indicates whether

Q

Q^{T}

is to be applied to

C

$TRANS ='N'$: $Q$ is applied to $C$ .
$TRANS ='T'$: $Q^{T}$ is applied to $C$ .

Constraint:

TRANS ='N'

'T'

4: M – INTEGERInput

On entry:

m

, the number of rows of the matrix

C

;

m

is also the order of

Q

SIDE ='L'

Constraint:

M \geq 0

5: N – INTEGERInput

On entry:

n

, the number of columns of the matrix

C

;

n

is also the order of

Q

SIDE ='R'

Constraint:

N \geq 0

6: AP( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array AP must be at least

\max (1, M \times (M + 1) / 2)

SIDE ='L'

and at least

\max (1, N \times (N + 1) / 2)

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

\max (1, M - 1)

SIDE ='L'

and at least

\max (1, N - 1)

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

\max (1, N)

On entry: the

m

n

matrix

C

On exit: C is overwritten by

Q C

Q^{T} C

C Q

C Q^{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:

LDC \geq \max (1, M)

10: WORK( $*$ ) – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array WORK must be at least

\max (1, N)

SIDE ='L'

and at least

\max (1, M)

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:

$INFO < 0$: 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

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

8 Further Comments

The total number of floating point operations is approximately

2 m^{2} n

SIDE ='L'

and

2 m n^{2}

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 = (\begin{array}{r} 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 \end{array}),

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

NAG Library Routine DocumentF08GGF (DOPMTR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08GGF (DOPMTR)