dgghrd (l)  Linux Manuals
dgghrd: reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular
NAME
DGGHRD  reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangularSYNOPSIS
 SUBROUTINE DGGHRD(
 COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO )
 CHARACTER COMPQ, COMPZ
 INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )
PURPOSE
DGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem isA*x
and B is typically made upper triangular by computing its QR factorization and moving the orthogonal matrix Q to the left side of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z
and transforms B to another upper triangular matrix T:
Q**T*B*Z
in order to reduce the problem to its standard form
H*y
where y = Z**T*x.
The orthogonal matrices Q and Z are 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
If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then DGGHRD reduces the original problem to generalized Hessenberg form.
ARGUMENTS
 COMPQ (input) CHARACTER*1

= aqNaq: do not compute Q;
= aqIaq: Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = aqVaq: Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned.  COMPZ (input) CHARACTER*1

= aqNaq: do not compute Z;
= aqIaq: Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = aqVaq: Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.  N (input) INTEGER
 The order of the matrices A and B. N >= 0.
 ILO (input) INTEGER
 IHI (input) INTEGER ILO and IHI mark the rows and columns of A which are to be reduced. It is assumed that A is already upper triangular in rows and columns 1:ILO1 and IHI+1:N. ILO and IHI are normally set by a previous call to SGGBAL; otherwise they should be set to 1 and N respectively. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
 On entry, the NbyN general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
 On entry, the NbyN upper triangular matrix B. On exit, the upper triangular matrix T = Q**T B Z. The elements below the diagonal are set to zero.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,N).
 Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
 On entry, if COMPQ = aqVaq, the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ=aqIaq, the orthogonal matrix Q, and if COMPQ = aqVaq, the product Q1*Q. Not referenced if COMPQ=aqNaq.
 LDQ (input) INTEGER
 The leading dimension of the array Q. LDQ >= N if COMPQ=aqVaq or aqIaq; LDQ >= 1 otherwise.
 Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
 On entry, if COMPZ = aqVaq, the orthogonal matrix Z1. On exit, if COMPZ=aqIaq, the orthogonal matrix Z, and if COMPZ = aqVaq, the product Z1*Z. Not referenced if COMPZ=aqNaq.
 LDZ (input) INTEGER
 The leading dimension of the array Z. LDZ >= N if COMPZ=aqVaq or aqIaq; LDZ >= 1 otherwise.
 INFO (output) INTEGER

= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
FURTHER DETAILS
This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and Van Loan (Johns Hopkins Press.)