SLABRD(3)      LAPACK routine of NEC Numeric Library Collection      SLABRD(3)



NAME
       SLABRD

SYNOPSIS
       SUBROUTINE SLABRD (M, N, NB, A, LDA, D, E, TAUQ, TAUP, 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)
                     M is INTEGER
                     The number of rows in the matrix A.

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

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

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

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

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

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

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

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

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

           LDY       (input)
                     LDY is 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 routine                  31 October 2017                      SLABRD(3)