
ZHGEQZ(l) ) ZHGEQZ(l)
NAME
ZHGEQZ  implement a singleshift version of the QZ method for finding the generalized ei
genvalues w(i)=ALPHA(i)/BETA(i) of the equation det( A  w(i) B ) = 0 If JOB='S', then
the pair (A,B) is simultaneously reduced to Schur form (i.e., A and B are both upper tri
angular) by applying one unitary tranformation (usually called Q) on the left and another
(usually called Z) on the right
SYNOPSIS
SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z,
LDZ, WORK, LWORK, RWORK, INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( *
), Z( LDZ, * )
PURPOSE
ZHGEQZ implements a singleshift version of the QZ method for finding the generalized ei
genvalues w(i)=ALPHA(i)/BETA(i) of the equation det( A  w(i) B ) = 0 If JOB='S', then the
pair (A,B) is simultaneously reduced to Schur form (i.e., A and B are both upper triangu
lar) by applying one unitary tranformation (usually called Q) on the left and another
(usually called Z) on the right. The diagonal elements of A are then
ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary transformations used to
reduce (A,B) are accumulated into the arrays Q and Z s.t.:
Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241256.
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHA and BETA. A and B will not necessarily be put into gen
eralized Schur form. = 'S': put A and B into generalized Schur form, as well as
computing ALPHA and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the conjugate transpose of the unitary
tranformation that is applied to the left side of A and B to reduce them to Schur
form. = 'I': like COMPQ='V', except that Q will be initialized to the identity
first.
COMPZ (input) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the unitary tranformation that is
applied to the right side of A and B to reduce them to Schur form. = 'I': like
COMPZ='V', except that Z will be initialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is already upper triangular in rows
and columns 1:ILO1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0,
if N=0.
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the NbyN upper Hessenberg matrix A. Elements below the subdiagonal
must be zero. If JOB='S', then on exit A and B will have been simultaneously
reduced to upper triangular form. If JOB='E', then on exit A will have been
destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the NbyN upper triangular matrix B. Elements below the diagonal must
be zero. If JOB='S', then on exit A and B will have been simultaneously reduced
to upper triangular form. If JOB='E', then on exit B will have been destroyed.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA (output) COMPLEX*16 array, dimension (N)
The diagonal elements of A when the pair (A,B) has been reduced to Schur form.
ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues.
BETA (output) COMPLEX*16 array, dimension (N)
The diagonal elements of B when the pair (A,B) has been reduced to Schur form.
ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues. A and B are normal
ized so that BETA(1),...,BETA(N) are nonnegative real numbers.
Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or 'I', then the conju
gate transpose of the unitary transformations which are applied to A and B on the
left will be applied to the array Q on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >=
N.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or 'I', then the uni
tary transformations which are applied to A and B on the right will be applied to
the array Z on the right.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >=
N.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = 1, then 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 by XERBLA.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not in Schur form, but
ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct. = N+1,...,2*N: the shift
calculation failed. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO
N+1,...,N should be correct. > 2*N: various "impossible" errors.
FURTHER DETAILS
We assume that complex ABS works as long as its value is less than overflow.
LAPACK version 3.0 15 June 2000 ZHGEQZ(l) 
