sgghrd.f(3)

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

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