NAME

       dgghd3.f -

SYNOPSIS

   Functions/Subroutines
       subroutine dgghd3 (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK, LWORK,
           INFO)
           DGGHD3

Function/Subroutine Documentation

   subroutine dgghd3 (character COMPQ, character COMPZ, integer N, integer ILO, integer IHI,
       double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, *
       ) B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision,
       dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK,
       integer INFO)
       DGGHD3

       Purpose:

            DGGHD3 reduces a pair of real matrices (A,B) to generalized upper
            Hessenberg form using orthogonal 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 orthogonal matrix Q to the left side
            of the equation.

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

            The orthogonal 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**T = (Q1*Q) * H * (Z1*Z)**T

                 Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

            If Q1 is the orthogonal matrix from the QR factorization of B in the
            original equation A*x = lambda*B*x, then DGGHD3 reduces the original
            problem to generalized Hessenberg form.

            This is a blocked variant of DGGHRD, using matrix-matrix
            multiplications for parts of the computation to enhance performance.

       Parameters:
           COMPQ

                     COMPQ is CHARACTER*1
                     = 'N': do not compute Q;
                     = 'I': Q is initialized to the unit matrix, and the
                            orthogonal matrix Q is returned;
                     = 'V': Q must contain an orthogonal matrix Q1 on entry,
                            and the product Q1*Q is returned.

           COMPZ

                     COMPZ is CHARACTER*1
                     = 'N': do not compute Z;
                     = 'I': Z is initialized to the unit matrix, and the
                            orthogonal matrix Z is returned;
                     = 'V': Z must contain an orthogonal matrix Z1 on entry,
                            and the product Z1*Z is returned.

           N

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

           ILO

                     ILO is INTEGER

           IHI

                     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 DGGBAL; 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

                     A is DOUBLE PRECISION 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

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

           B

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

           LDB

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

           Q

                     Q is DOUBLE PRECISION array, dimension (LDQ, N)
                     On entry, if COMPQ = 'V', the orthogonal matrix Q1,
                     typically from the QR factorization of B.
                     On exit, if COMPQ='I', the orthogonal matrix Q, and if
                     COMPQ = 'V', the product Q1*Q.
                     Not referenced if COMPQ='N'.

           LDQ

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

           Z

                     Z is DOUBLE PRECISION array, dimension (LDZ, N)
                     On entry, if COMPZ = 'V', the orthogonal matrix Z1.
                     On exit, if COMPZ='I', the orthogonal matrix Z, and if
                     COMPZ = 'V', the product Z1*Z.
                     Not referenced if COMPZ='N'.

           LDZ

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (LWORK)
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The length of the array WORK.  LWORK >= 1.
                     For optimum performance LWORK >= 6*N*NB, where NB is the
                     optimal blocksize.

                     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.

           INFO

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           January 2015

       Further Details:

             This routine reduces A to Hessenberg form and maintains B in
             using a blocked variant of Moler and Stewart's original algorithm,
             as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
             (BIT 2008).

Author

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