SRC/slabrd.f(3)            Library Functions Manual            SRC/slabrd.f(3)

NAME
       SRC/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 (integer m, integer n, integer nb, real, dimension( lda,
       * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e,
       real, dimension( * ) tauq, real, dimension( * ) taup, real, dimension(
       ldx, * ) x, integer ldx, real, dimension( ldy, * ) y, integer ldy)
       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.

       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 208 of file slabrd.f.

Author
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LAPACK                          Version 3.12.0                 SRC/slabrd.f(3)