ZGGHRD(3)      LAPACK routine of NEC Numeric Library Collection      ZGGHRD(3)



NAME
       ZGGHRD

SYNOPSIS
       SUBROUTINE ZGGHRD (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ,
           Z, LDZ, INFO)



PURPOSE
            ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper
            Hessenberg form using unitary transformations, where A is a
            general matrix and B is upper triangular.  The form of the
            generalized eigenvalue problem is
               A*x = lambda*B*x,
            and B is typically made upper triangular by computing its QR
            factorization and moving the unitary matrix Q to the left side
            of the equation.

            This subroutine simultaneously reduces A to a Hessenberg matrix H:
               Q**H*A*Z = H
            and transforms B to another upper triangular matrix T:
               Q**H*B*Z = T
            in order to reduce the problem to its standard form
               H*y = lambda*T*y
            where y = Z**H*x.

            The unitary matrices Q and Z are determined as products of Givens
            rotations.  They may either be formed explicitly, or they may be
            postmultiplied into input matrices Q1 and Z1, so that
                 Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
                 Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
            If Q1 is the unitary matrix from the QR factorization of B in the
            original equation A*x = lambda*B*x, then ZGGHRD reduces the original
            problem to generalized Hessenberg form.




ARGUMENTS
           COMPQ     (input)
                     COMPQ is CHARACTER*1
                     = 'N': do not compute Q;
                     = 'I': Q is initialized to the unit matrix, and the
                            unitary matrix Q is returned;
                     = 'V': Q must contain a unitary matrix Q1 on entry,
                            and the product Q1*Q is returned.

           COMPZ     (input)
                     COMPZ is CHARACTER*1
                     = 'N': do not compute Q;
                     = 'I': Q is initialized to the unit matrix, and the
                            unitary matrix Q is returned;
                     = 'V': Q must contain a unitary matrix Q1 on entry,
                            and the product Q1*Q is returned.

           N         (input)
                     N is INTEGER
                     The order of the matrices A and B.  N >= 0.

           ILO       (input)
                     ILO is INTEGER

           IHI       (input)
                     IHI is INTEGER

                     ILO and IHI mark the rows and columns of A which are to be
                     reduced.  It is assumed that A is already upper triangular
                     in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
                     normally set by a previous call to ZGGBAL; otherwise they
                     should be set to 1 and N respectively.
                     1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

           A         (input/output)
                     A is COMPLEX*16 array, dimension (LDA, N)
                     On entry, the N-by-N general matrix to be reduced.
                     On exit, the upper triangle and the first subdiagonal of A
                     are overwritten with the upper Hessenberg matrix H, and the
                     rest is set to zero.

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

           B         (input/output)
                     B is COMPLEX*16 array, dimension (LDB, N)
                     On entry, the N-by-N upper triangular matrix B.
                     On exit, the upper triangular matrix T = Q**H B Z.  The
                     elements below the diagonal are set to zero.

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

           Q         (input/output)
                     Q is COMPLEX*16 array, dimension (LDQ, N)
                     On entry, if COMPQ = 'V', the unitary matrix Q1, typically
                     from the QR factorization of B.
                     On exit, if COMPQ='I', the unitary matrix Q, and if
                     COMPQ = 'V', the product Q1*Q.
                     Not referenced if COMPQ='N'.

           LDQ       (input)
                     LDQ is INTEGER
                     The leading dimension of the array Q.
                     LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

           Z         (input/output)
                     Z is COMPLEX*16 array, dimension (LDZ, N)
                     On entry, if COMPZ = 'V', the unitary matrix Z1.
                     On exit, if COMPZ='I', the unitary matrix Z, and if
                     COMPZ = 'V', the product Z1*Z.
                     Not referenced if COMPZ='N'.

           LDZ       (input)
                     LDZ is INTEGER
                     The leading dimension of the array Z.
                     LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.






FURTHER DETAILS
             This routine reduces A to Hessenberg and B to triangular form by
             an unblocked reduction, as described in _Matrix_Computations_,
             by Golub and van Loan (Johns Hopkins Press).



LAPACK routine                  31 October 2017                      ZGGHRD(3)