SGGHRD (3lapack)

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, * )

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 subdiagonal of A are overwritten with
          the upper Hessenberg 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 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.)