zlabrd.f(3)

NAME
zlabrd.f

SYNOPSIS

  Functions/Subroutines
subroutine zlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Function/Subroutine Documentation

  subroutine zlabrd (integer M, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16, dimension( * ) TAUQ, complex*16, dimension( * ) TAUP, complex*16, dimension( ldx, * ) X, integer LDX, complex*16, dimension( ldy, * ) Y, integer LDY)

Purpose:

     ZLABRD reduces the first NB rows and columns of a complex general
     m by n matrix A to upper or lower real bidiagonal form by a unitary
     transformation Q**H * A * P, and returns the matrices X and Y which
     are needed to apply the transformation to the unreduced part of A.
     If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
     bidiagonal form.
     This is an auxiliary routine called by ZGEBRD

Parameters
    M

              M is INTEGER
              The number of rows in the matrix A.

    N

              N is INTEGER
              The number of columns in the matrix A.

    NB

              NB is INTEGER
              The number of leading rows and columns of A to be reduced.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the m by n general matrix to be reduced.
              On exit, the first NB rows and columns of the matrix are
              overwritten; the rest of the array is unchanged.
              If m >= n, elements on and below the diagonal in the first NB
                columns, with the array TAUQ, represent the unitary
                matrix Q as a product of elementary reflectors; and
                elements above the diagonal in the first NB rows, with the
                array TAUP, represent the unitary matrix P as a product
                of elementary reflectors.
              If m < n, elements below the diagonal in the first NB
                columns, with the array TAUQ, represent the unitary
                matrix Q as a product of elementary reflectors, and
                elements on and above the diagonal in the first NB rows,
                with the array TAUP, represent the unitary matrix P as
                a product of elementary reflectors.
              See Further Details.

    LDA

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

    D

              D is DOUBLE PRECISION array, dimension (NB)
              The diagonal elements of the first NB rows and columns of
              the reduced matrix.  D(i) = A(i,i).

    E

              E is DOUBLE PRECISION array, dimension (NB)
              The off-diagonal elements of the first NB rows and columns of
              the reduced matrix.

    TAUQ

              TAUQ is COMPLEX*16 array, dimension (NB)
              The scalar factors of the elementary reflectors which
              represent the unitary matrix Q. See Further Details.

    TAUP

              TAUP is COMPLEX*16 array, dimension (NB)
              The scalar factors of the elementary reflectors which
              represent the unitary matrix P. See Further Details.

    X

              X is COMPLEX*16 array, dimension (LDX,NB)
              The m-by-nb matrix X required to update the unreduced part
              of A.

    LDX

              LDX is INTEGER
              The leading dimension of the array X. LDX >= max(1,M).

    Y

              Y is COMPLEX*16 array, dimension (LDY,NB)
              The n-by-nb matrix Y required to update the unreduced part
              of A.

    LDY

              LDY is INTEGER
              The leading dimension of the array Y. LDY >= max(1,N).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The matrices Q and P are represented as products of elementary
      reflectors:
         Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
      Each H(i) and G(i) has the form:
         H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
      where tauq and taup are complex scalars, and v and u are complex
      vectors.
      If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
      A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
      A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
      A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
      A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
      The elements of the vectors v and u together form the m-by-nb matrix
      V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
      the transformation to the unreduced part of the matrix, using a block
      update of the form:  A := A - V*Y**H - X*U**H.
      The contents of A on exit are illustrated by the following examples
      with nb = 2:
      m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
        (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
        (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
        (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
        (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
        (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
        (  v1  v2  a   a   a  )
      where a denotes an element of the original matrix which is unchanged,
      vi denotes an element of the vector defining H(i), and ui an element
      of the vector defining G(i).

Definition at line 210 of file zlabrd.f.

Author

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