NAG Library Routine Document

F08FGF (DORMTR)

+− 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

F08FGF (DORMTR) multiplies an arbitrary real matrix

C

by the real orthogonal matrix

Q

which was determined by F08FEF (DSYTRD) when reducing a real symmetric matrix to tridiagonal form.

2 Specification

SUBROUTINE F08FGF (

SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)

INTEGER	M, N, LDA, LDC, LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), TAU(), C(LDC,*), WORK(max(1,LWORK))
CHARACTER(1)	SIDE, UPLO, TRANS

The routine may be called by its LAPACK name dormtr.

3 Description

F08FGF (DORMTR) is intended to be used after a call to F08FEF (DSYTRD), which reduces a real symmetric matrix

A

to symmetric tridiagonal form

T

by an orthogonal similarity transformation:

A = Q T Q^{T}

. F08FEF (DSYTRD) 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

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

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 F08FEF (DSYTRD).

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: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array A must be at least

\max (1, M)

SIDE ='L'

and at least

\max (1, N)

SIDE ='R'

On entry: details of the vectors which define the elementary reflectors, as returned by F08FEF (DSYTRD).

7: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08FGF (DORMTR) is called.

Constraints:

if $SIDE ='L'$ , $LDA \geq \max (1, M)$ ;
if $SIDE ='R'$ , $LDA \geq \max (1, N)$ .

8: 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 F08FEF (DSYTRD).

9: 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.

10: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which F08FGF (DORMTR) is called.

Constraint:

LDC \geq \max (1, M)

11: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

WORK (1)

contains the minimum value of LWORK required for optimal performance.

12: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08FGF (DORMTR) is called.

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,

LWORK \geq N \times nb

SIDE ='L'

and at least

M \times nb

SIDE ='R'

, where

nb

is the optimal block size.

Constraints:

if $SIDE ='L'$ , $LWORK \geq \max (1, N)$ or $LWORK = - 1$ ;
if $SIDE ='R'$ , $LWORK \geq \max (1, M)$ or $LWORK = - 1$ .

13: 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 F08FUF (ZUNMTR).

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}) .

Here

A

is symmetric and must first be reduced to tridiagonal form

T

by F08FEF (DSYTRD). The program then calls F08JJF (DSTEBZ) to compute the requested eigenvalues and F08JKF (DSTEIN) to compute the associated eigenvectors of

T

. Finally F08FGF (DORMTR) is called to transform the eigenvectors to those of

A

NAG Library Routine DocumentF08FGF (DORMTR)

+− 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

F08FGF (DORMTR)