ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),


 where H is an upper Hessenberg matrix and T is upper triangular,


 using the single-shift QZ method.


 Matrix pairs of this type are produced by the reduction to


 generalized upper Hessenberg form of a complex matrix pair (A,B):


    A = Q1*H*Z1**H,  B = Q1*T*Z1**H,


 as computed by ZGGHRD.


 If JOB='S', then the Hessenberg-triangular pair (H,T) is


 also reduced to generalized Schur form,


    H = Q*S*Z**H,  T = Q*P*Z**H,


 where Q and Z are unitary matrices and S and P are upper triangular.


 Optionally, the unitary matrix Q from the generalized Schur


 factorization may be postmultiplied into an input matrix Q1, and the


 unitary matrix Z may be postmultiplied into an input matrix Z1.


 If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced


 the matrix pair (A,B) to generalized Hessenberg form, then the output


 matrices Q1*Q and Z1*Z are the unitary factors from the generalized


 Schur factorization of (A,B):


    A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.


 To avoid overflow, eigenvalues of the matrix pair (H,T)


 (equivalently, of (A,B)) are computed as a pair of complex values


 (alpha,beta).  If beta is nonzero, lambda = alpha / beta is an


 eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)


    A*x = lambda*B*x


 and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the


 alternate form of the GNEP


    mu*A*y = B*y.


 The values of alpha and beta for the i-th eigenvalue can be read


 directly from the generalized Schur form:  alpha = S(i,i),


 beta = P(i,i).


 Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix


      Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),


      pp. 241--256.

JOB

          JOB is CHARACTER*1


          = 'E': Compute eigenvalues only;


          = 'S': Computer eigenvalues and the Schur form.

COMPQ

          COMPQ is CHARACTER*1


          = 'N': Left Schur vectors (Q) are not computed;


          = 'I': Q is initialized to the unit matrix and the matrix Q


                 of left Schur vectors of (H,T) is returned;


          = 'V': Q must contain a unitary matrix Q1 on entry and


                 the product Q1*Q is returned.

COMPZ

          COMPZ is CHARACTER*1


          = 'N': Right Schur vectors (Z) are not computed;


          = 'I': Q is initialized to the unit matrix and the matrix Z


                 of right Schur vectors of (H,T) is returned;


          = 'V': Z must contain a unitary matrix Z1 on entry and


                 the product Z1*Z is returned.

N

          N is INTEGER


          The order of the matrices H, T, Q, and Z.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI mark the rows and columns of H which are in


          Hessenberg form.  It is assumed that A is already upper


          triangular in rows and columns 1:ILO-1 and IHI+1:N.


          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

H

          H is COMPLEX*16 array, dimension (LDH, N)


          On entry, the N-by-N upper Hessenberg matrix H.


          On exit, if JOB = 'S', H contains the upper triangular


          matrix S from the generalized Schur factorization.


          If JOB = 'E', the diagonal of H matches that of S, but


          the rest of H is unspecified.

LDH

          LDH is INTEGER


          The leading dimension of the array H.  LDH >= max( 1, N ).

T

          T is COMPLEX*16 array, dimension (LDT, N)


          On entry, the N-by-N upper triangular matrix T.


          On exit, if JOB = 'S', T contains the upper triangular


          matrix P from the generalized Schur factorization.


          If JOB = 'E', the diagonal of T matches that of P, but


          the rest of T is unspecified.

LDT

          LDT is INTEGER


          The leading dimension of the array T.  LDT >= max( 1, N ).

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)


          The complex scalars alpha that define the eigenvalues of


          GNEP.  ALPHA(i) = S(i,i) in the generalized Schur


          factorization.

BETA

          BETA is COMPLEX*16 array, dimension (N)


          The real non-negative scalars beta that define the


          eigenvalues of GNEP.  BETA(i) = P(i,i) in the generalized


          Schur factorization.


          Together, the quantities alpha = ALPHA(j) and beta = BETA(j)


          represent the j-th eigenvalue of the matrix pair (A,B), in


          one of the forms lambda = alpha/beta or mu = beta/alpha.


          Since either lambda or mu may overflow, they should not,


          in general, be computed.

Q

          Q is COMPLEX*16 array, dimension (LDQ, N)


          On entry, if COMPQ = 'V', the unitary matrix Q1 used in the


          reduction of (A,B) to generalized Hessenberg form.


          On exit, if COMPQ = 'I', the unitary matrix of left Schur


          vectors of (H,T), and if COMPQ = 'V', the unitary matrix of


          left Schur vectors of (A,B).


          Not referenced if COMPQ = 'N'.

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q.  LDQ >= 1.


          If COMPQ='V' or 'I', then LDQ >= N.

Z

          Z is COMPLEX*16 array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the unitary matrix Z1 used in the


          reduction of (A,B) to generalized Hessenberg form.


          On exit, if COMPZ = 'I', the unitary matrix of right Schur


          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of


          right Schur vectors of (A,B).


          Not referenced if COMPZ = 'N'.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1.


          If COMPZ='V' or 'I', then LDZ >= N.

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))


          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is 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

          RWORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value


          = 1,...,N: the QZ iteration did not converge.  (H,T) 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.  (H,T) is not


                     in Schur form, but ALPHA(i) and BETA(i),


                     i=INFO-N+1,...,N should be correct.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

April 2012

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


  overflow.