slabrd.f(3)

NAME
slabrd.f -

SYNOPSIS

  Functions/Subroutines

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

Function/Subroutine Documentation

  subroutine slabrd (integerM, integerN, integerNB, real, dimension( lda, * )A, integerLDA, real, dimension( * )D, real, dimension( * )E, real, dimension( * )TAUQ, real, dimension( * )TAUP, real, dimension( ldx, * )X, integerLDX, real, dimension( ldy, * )Y, integerLDY)

SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

     SLABRD reduces the first NB rows and columns of a real general
     m by n matrix A to upper or lower bidiagonal form by an orthogonal
     transformation Q**T * 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 SGEBRD

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 REAL 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 orthogonal
                matrix Q as a product of elementary reflectors; and
                elements above the diagonal in the first NB rows, with the
                array TAUP, represent the orthogonal 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 orthogonal
                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 orthogonal 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 REAL 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 REAL array, dimension (NB)
              The off-diagonal elements of the first NB rows and columns of
              the reduced matrix.

    TAUQ

              TAUQ is REAL array dimension (NB)
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix Q. See Further Details.

    TAUP

              TAUP is REAL array, dimension (NB)
              The scalar factors of the elementary reflectors which
              represent the orthogonal matrix P. See Further Details.

    X

              X is REAL 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 REAL 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.

Date:
    September 2012

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**T  and G(i) = I - taup * u * u**T

      where tauq and taup are real scalars, and v and u are real 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**T 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**T - X*U**T.

      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 slabrd.f.

Author

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