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 |
|
F08WEF (DGGHRD) is the third step in the solution of the real generalized eigenvalue problem
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
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
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
The reduction to the generalized Hessenberg form is implemented using orthogonal transformations which are backward stable.
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).