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NAME
     sgghrd - reduce 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

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, * )



     #include <sunperf.h>

     void sgghrd(char compq, char compz, int n, int ilo, int ihi,
               float  *sa, int lda, float *sb, int ldb, float *q,
               int ldq, float *sz, int ldz, int *info) ;

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:  Q' *
     A * Z = H and Q' * B * Z = T, where H is upper Hessenberg, T
     is upper triangular, and Q and Z are orthogonal, and ' means
     transpose.

     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' = (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
               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         (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.
               < 0:  if INFO = -i, the i-th argument had an ille-
               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  Hopkins
     Press.)

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