NAG Library Routine Document

F08WEF (DGGHRD)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

1 Purpose

F08WEF (DGGHRD) reduces a pair of real matrices

(A, B)

, where

B

is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations.

2 Specification

SUBROUTINE F08WEF (

COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)

INTEGER	N, ILO, IHI, LDA, LDB, LDQ, LDZ, INFO
REAL (KIND=nag_wp)	A(LDA,), B(LDB,), Q(LDQ,), Z(LDZ,)
CHARACTER(1)	COMPQ, COMPZ

The routine may be called by its LAPACK name dgghrd.

3 Description

F08WEF (DGGHRD) is the third step in the solution of the real generalized eigenvalue problem

A x = λ B x .

The (optional) first step balances the two matrices using F08WHF (DGGBAL). In the second step, matrix

B

is reduced to upper triangular form using the

Q R

factorization routine F08AEF (DGEQRF) and this orthogonal transformation

Q

is applied to matrix

A

by calling F08AGF (DORMQR).

F08WEF (DGGHRD) reduces a pair of real matrices

(A, B)

, where

B

is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations. This two-sided transformation is of the form

\begin{matrix} Q^{T} A Z = H \\ Q^{T} B Z = T \end{matrix}

where

H

is an upper Hessenberg matrix,

T

is an upper triangular matrix and

Q

and

Z

are orthogonal matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices

Q_{1}

and

Z_{1}

, so that

\begin{matrix} Q_{1} A Z_{1}^{T} = (Q_{1} Q) H {(Z_{1} Z)}^{T}, \\ Q_{1} B Z_{1}^{T} = (Q_{1} Q) T {(Z_{1} Z)}^{T} . \end{matrix}

4 References

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

Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

5 Parameters

1: COMPQ – CHARACTER(1)Input

On entry: specifies the form of the computed orthogonal matrix

Q

$COMPQ ='N'$: Do not compute $Q$ .
$COMPQ ='I'$: The orthogonal matrix $Q$ is returned.
$COMPQ ='V'$: Q must contain an orthogonal matrix $Q_{1}$ , and the product $Q_{1} Q$ is returned.

Constraint:

COMPQ ='N'

'I'

'V'

2: COMPZ – CHARACTER(1)Input

On entry: specifies the form of the computed orthogonal matrix

Z

$COMPZ ='N'$: Do not compute $Z$ .
$COMPZ ='I'$: The orthogonal matrix $Z$ is returned.
$COMPZ ='V'$: Z must contain an orthogonal matrix $Z_{1}$ , and the product $Z_{1} Z$ is returned.

Constraint:

COMPZ ='N'

'I'

'V'

3: N – INTEGERInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

N \geq 0

4: ILO – INTEGERInput

5: IHI – INTEGERInput

On entry:

i_{lo}

and

i_{hi}

as determined by a previous call to F08WHF (DGGBAL). Otherwise, they should be set to

1

and

n

, respectively.

Constraints:

if $N > 0$ , $1 \leq ILO \leq IHI \leq N$ ;
if $N = 0$ , $ILO = 1$ and $IHI = 0$ .

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

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

\max (1, N)

On entry: the matrix

A

of the matrix pair

(A, B)

. Usually, this is the matrix

A

returned by F08AGF (DORMQR).

On exit: A is overwritten by the upper Hessenberg matrix

H

7: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08WEF (DGGHRD) is called.

Constraint:

LDA \geq \max (1, N)

8: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the upper triangular matrix

B

of the matrix pair

(A, B)

. Usually, this is the matrix

B

returned by the

Q R

factorization routine F08AEF (DGEQRF).

On exit: B is overwritten by the upper triangular matrix

T

9: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08WEF (DGGHRD) is called.

Constraint:

LDB \geq \max (1, N)

10: Q(LDQ, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

COMPQ ='I'

'V'

and at least

1

COMPQ ='N'

On entry: if

COMPQ ='V'

, Q must contain an orthogonal matrix

Q_{1}

COMPQ ='N'

, Q is not referenced.

On exit: if

COMPQ ='I'

, Q contains the orthogonal matrix

Q

COMPQ ='V'

, Q is overwritten by

Q_{1} Q

11: LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08WEF (DGGHRD) is called.

Constraints:

if $COMPQ ='I'$ or $'V'$ , $LDQ \geq \max (1, N)$ ;
if $COMPQ ='N'$ , $LDQ \geq 1$ .

12: Z(LDZ, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

COMPZ ='I'

'V'

and at least

1

COMPZ ='N'

On entry: if

COMPZ ='V'

, Z must contain an orthogonal matrix

Z_{1}

COMPZ ='N'

, Z is not referenced.

On exit: if

COMPZ ='I'

, Z contains the orthogonal matrix

Z

COMPZ ='V'

, Z is overwritten by

Z_{1} Z

13: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08WEF (DGGHRD) is called.

Constraints:

if $COMPZ ='I'$ or $'V'$ , $LDZ \geq \max (1, N)$ ;
if $COMPZ ='N'$ , $LDZ \geq 1$ .

14: 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 reduction to the generalized Hessenberg form is implemented using orthogonal transformations which are backward stable.

8 Further Comments

This routine is usually followed by F08XEF (DHGEQZ) which implements the

Q Z

algorithm for computing generalized eigenvalues of a reduced pair of matrices.

The complex analogue of this routine is F08WSF (ZGGHRD).

9 Example

See Section 9 in F08XEF (DHGEQZ) and F08YKF (DTGEVC).

F08WEF (DGGHRD) (PDF version)

F08 Chapter Contents

F08 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentF08WEF (DGGHRD)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine Document

F08WEF (DGGHRD)