CHGEQZ - 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

SUBROUTINE CHGEQZ(

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

REAL RWORK( * )

COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )

CHGEQZ 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),...,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(1), pp. 241--256.

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 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 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 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 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 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 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 BETA(1),...,BETA(N) are non-negative real numbers.

Q (input/output) COMPLEX 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 array, dimension (LDZ, N)

If COMPZ='N', then Z will not be referenced. If COMPZ='V' or 'I', then the unitary 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 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) REAL array, dimension (N)

INFO (output) INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th 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.

We assume that complex ABS works as long as its value is less than overflow.