F08WEF (DGGHRD) (PDF version)
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F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08WEF (DGGHRD)

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
    10  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. F08WEF (DGGHRD) is marked as deprecated by LAPACK; the replacement routine is F08WFF (DGGHD3) which makes better use of level 3 BLAS.

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
Ax=λBx.  
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 QR 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
QTAZ=H QTBZ=T  
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 Q1 and Z1, so that
Q1AZ1T=Q1QHZ1ZT, Q1BZ1T=Q1QTZ1ZT.  

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  Arguments

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 Q1, and the product Q1Q is returned.
Constraint: COMPQ='N', 'I' or '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 Z1, and the product Z1Z is returned.
Constraint: COMPZ='N', 'V' or 'I'.
3:     N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint: N0.
4:     ILO – INTEGERInput
5:     IHI – INTEGERInput
On entry: ilo and ihi as determined by a previous call to F08WHF (DGGBAL). Otherwise, they should be set to 1 and n, respectively.
Constraints:
  • if N>0, 1 ILO IHI N ;
  • if N=0, ILO=1 and IHI=0.
6:     ALDA* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,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: LDAmax1,N.
8:     BLDB* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least max1,N.
On entry: the upper triangular matrix B of the matrix pair A,B. Usually, this is the matrix B returned by the QR 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: LDBmax1,N.
10:   QLDQ* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Q must be at least max1,N if COMPQ='I' or 'V' and at least 1 if COMPQ='N'.
On entry: if COMPQ='V', Q must contain an orthogonal matrix Q1.
If COMPQ='N', Q is not referenced.
On exit: if COMPQ='I', Q contains the orthogonal matrix Q.
If COMPQ='V', Q is overwritten by Q1Q.
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 max1,N ;
  • if COMPQ='N', LDQ1.
12:   ZLDZ* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Z must be at least max1,N if COMPZ='V' or 'I' and at least 1 if COMPZ='N'.
On entry: if COMPZ='V', Z must contain an orthogonal matrix Z1.
If COMPZ='N', Z is not referenced.
On exit: if COMPZ='I', Z contains the orthogonal matrix Z.
If COMPZ='V', Z is overwritten by Z1Z.
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='V' or 'I', LDZ max1,N ;
  • if COMPZ='N', LDZ1.
14:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

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  Parallelism and Performance

F08WEF (DGGHRD) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

This routine is usually followed by F08XEF (DHGEQZ) which implements the QZ algorithm for computing generalized eigenvalues of a reduced pair of matrices.
The complex analogue of this routine is F08WSF (ZGGHRD).

10  Example

See Section 10 in F08XEF (DHGEQZ) and F08YKF (DTGEVC).

F08WEF (DGGHRD) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016