SRC/dgghd3.f(3)

NAME
SRC/dgghd3.f

SYNOPSIS

  Functions/Subroutines
subroutine dgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
DGGHD3

Function/Subroutine Documentation

  subroutine dgghd3 (character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)

DGGHD3

Purpose:

    !>
    !> DGGHD3 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 DGGHD3 reduces the original
    !> problem to generalized Hessenberg form.
    !>
    !> This is a blocked variant of DGGHRD, using matrix-matrix
    !> multiplications for parts of the computation to enhance performance.
    !>

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 DGGBAL; 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK. LWORK >= 1.
    !>          For optimum performance LWORK >= 6*N*NB, where NB is the
    !>          optimal blocksize.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    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.

Further Details:

    !>
    !>  This routine reduces A to Hessenberg form and maintains B in triangular form
    !>  using a blocked variant of Moler and Stewart's original algorithm,
    !>  as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
    !>  (BIT 2008).
    !>

Definition at line 228 of file dgghd3.f.

Author

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