SGGHRD  - reduce a pair of real matrices (A,B) to general­
       ized upper Hessenberg form  using  orthogonal  transforma­
       tions, where A is a general matrix and B is upper triangu­
       lar


SYNOPSIS

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

           CHARACTER      COMPQ, COMPZ

           INTEGER        IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N

           REAL           A(  LDA, * ), B( LDB, * ), Q( LDQ, * ),
                          Z( LDZ, * )


PURPOSE

       SGGHRD reduces a pair of real matrices (A,B)  to  general­
       ized  upper  Hessenberg  form using orthogonal transforma­
       tions, where A is a general matrix and B is upper triangu­
       lar:  Q'  * A * Z = H and Q' * B * Z = T, where H is upper
       Hessenberg, T is upper triangular, and Q and Z are orthog­
       onal, and ' means transpose.

       The orthogonal matrices Q and Z are determined as products
       of Givens rotations.  They may either  be  formed  explic­
       itly, or they may be postmultiplied into input matrices Q1
       and Z1, so that

            Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
            Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'


ARGUMENTS

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

       ILO     (input) INTEGER
               IHI     (input) INTEGER It is assumed  that  A  is

               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       (input/output) REAL array, dimension (LDA, N)
               On entry, the N-by-N general matrix to be reduced.
               On exit, the upper triangle and the first subdiag­
               onal  of  A are overwritten with the upper Hessen­
               berg matrix H, and the rest is set to zero.

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

       B       (input/output) REAL array, dimension (LDB, N)
               On  entry,  the  N-by-N upper triangular matrix B.
               On exit, the upper triangular matrix T = Q'  B  Z.
               The elements below the diagonal are set to zero.

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

       Q       (input/output) REAL array, dimension (LDQ, N)
               If COMPQ='N':  Q is not referenced.
               If COMPQ='I':  on entry, Q need not be set, and on
               exit it contains the orthogonal matrix Q, where Q'
               is the product of the Givens transformations which
               are applied to A and B on the left.  If COMPQ='V':
               on entry, Q must contain an orthogonal matrix  Q1,
               and on exit this is overwritten by Q1*Q.

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

       Z       (input/output) REAL array, dimension (LDZ, N)
               If COMPZ='N':  Z is not referenced.
               If COMPZ='I':  on entry, Z need not be set, and on
               exit it contains the orthogonal matrix Z, which is
               the product of the  Givens  transformations  which
               are   applied  to  A  and  B  on  the  right.   If
               COMPZ='V':  on entry, Z must contain an orthogonal
               matrix  Z1,  and  on  exit  this is overwritten by
               Z1*Z.

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

       INFO    (output) INTEGER
               = 0:  successful exit.

               gal 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  Hop­
       kins Press.)


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