Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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

SYNOPSIS

       SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

           INTEGER        LDA, LDX, LDY, M, N, NB

           REAL           A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y( LDY,
                          * )

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

ARGUMENTS

        M       (input) INTEGER
                The number of rows in the matrix A.

        N       (input) INTEGER
                The number of columns in the matrix A.

        NB      (input) INTEGER
                The number of leading rows and columns of A to be reduced.

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

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

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

        TAUQ    (output) REAL array dimension (NB)
                The scalar factors of the elementary reflectors which
                represent the orthogonal matrix Q. See Further Details.
                TAUP    (output) REAL array, dimension (NB)
                The scalar factors of the elementary reflectors which
                represent the orthogonal matrix P. See Further Details.
                X       (output) REAL array, dimension (LDX,NB)
                The m-by-nb matrix X required to update the unreduced part
                of A.

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

        Y       (output) REAL array, dimension (LDY,NB)
                The n-by-nb matrix Y required to update the unreduced part
                of A.

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

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).

 LAPACK auxiliary routine (version 3.3.1)   April 2011                            SLABRD(3lapack)