dlabrd(3S)

NAME
     DLABRD - reduce 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' * A * P, and returns the matrices X and Y which are
     needed to apply the transformation to the unreduced part of A

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

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

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

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DLABRD 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' * 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 DGEBRD


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

     TAUQ    (output) DOUBLE PRECISION array dimension (NB)
             The scalar factors of the elementary reflectors which represent
             the orthogonal matrix Q. See Further Details.  TAUP    (output)
             DOUBLE PRECISION array, dimension (NB) The scalar factors of the
             elementary reflectors which represent the orthogonal matrix P.
             See Further Details.  X       (output) DOUBLE PRECISION 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 >= M.

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

     LDY     (output) INTEGER
             The leading dimension of the array Y. LDY >= 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'  and G(i) = I - taup * u * u'

     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' 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' - X*U'.

     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).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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