CLAQZ0 computes the eigenvalues of a matrix pair (H,T),


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


 using the double-shift QZ method.


 Matrix pairs of this type are produced by the reduction to


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


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


 as computed by CGGHRD.


 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, P and S are an upper triangular


 matrices.


 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 CGGHRD that reduced


 the matrix pair (A,B) to generalized upper 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 values (alpha,beta), where alpha is


 complex and beta real.


 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.


 Eigenvalues 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.


 Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ


      Algorithm with Aggressive Early Deflation', SIAM J. Numer.


      Anal., 29(2006), pp. 199--227.


 Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift,


      multipole rational QZ method with aggressive early deflation'

WANTS

          WANTS is CHARACTER*1


          = 'E': Compute eigenvalues only;


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

WANTQ

          WANTQ 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 (A,B) is returned;


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


                 the product Q1*Q is returned.

WANTZ

          WANTZ is CHARACTER*1


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


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


                 of right Schur vectors of (A,B) is returned;


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


                 the product Z1*Z is returned.

N

          N is INTEGER


          The order of the matrices A, B, Q, and Z.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI mark the rows and columns of A 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.

A

          A is COMPLEX array, dimension (LDA, N)


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


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


          matrix S from the generalized Schur factorization.


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


          the rest of A is unspecified.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max( 1, N ).

B

          B is COMPLEX array, dimension (LDB, N)


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


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


          matrix P from the generalized Schur factorization.


          If JOB = 'E', the diagonal of B matches that of P, but


          the rest of B is unspecified.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max( 1, N ).

ALPHA

          ALPHA is COMPLEX array, dimension (N)


          Each scalar alpha defining an eigenvalue


          of GNEP.

BETA

          BETA is COMPLEX array, dimension (N)


          The scalars beta that define the eigenvalues of GNEP.


          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 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 (A,B), 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 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 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 REAL array, dimension (N)

REC

          REC is INTEGER


             REC indicates the current recursion level. Should be set


             to 0 on first call.

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.  (A,B) is not


                     in Schur form, but ALPHA(i) and


                     BETA(i), i=INFO+1,...,N should be correct.

Thijs Steel, KU Leuven

May 2020