zhgeqz(3)

NAME

ZHGEQZ - implement a single-shift version of the QZ method

for finding the generalized eigenvalues 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 triangular) 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 single-shift version of the QZ method

for finding the generalized eigenvalues 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 triangular) 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),...,AL

PHA(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 General

ized Matrix

Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.

ARGUMENTS

JOB (input) CHARACTER*1

= 'E': compute only ALPHA and BETA. A and B will

not necessarily be put into generalized 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:ILO-1 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 N-by-N 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 tri

angular form. If JOB='E', then on exit A will have been de

stroyed.

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 N-by-N 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 tri

angular form. If JOB='E', then on exit B will have been de

stroyed.

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 normalized so that BE

TA(1),...,BETA(N) are non-negative 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 conjugate 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 unitary transformations which are ap

plied 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 i-th argument had an ille

gal value
= 1,...,N: the QZ iteration did not converge.

(A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=IN

FO+1,...,N should be correct. = N+1,...,2*N: the shift calcula

tion failed. (A,B) is not in Schur form, but ALPHA(i) and BE

TA(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(3)