NAME

       sgghrd.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
           SGGHRD

Function/Subroutine Documentation

   subroutine sgghrd (characterCOMPQ, characterCOMPZ, integerN, integerILO, integerIHI, real, dimension( lda, *
       )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( ldq, * )Q, integerLDQ, real,
       dimension( ldz, * )Z, integerLDZ, integerINFO)
       SGGHRD

       Purpose:

            SGGHRD 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 SGGHRD reduces the original
            problem to generalized Hessenberg form.

       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 SGGBAL; 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 REAL 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 REAL 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 REAL 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 REAL 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.

           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:
           November 2011

       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.)

       Definition at line 207 of file sgghrd.f.

Author

       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2                                    Wed Feb 26 2014                                     sgghrd.f(3)