slahrd(3F)

NAME
     SLAHRD - reduce the first NB columns of a real general n-by-(n-k+1)
     matrix A so that elements below the k-th subdiagonal are zero

SYNOPSIS
     SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

         INTEGER        K, LDA, LDT, LDY, N, NB

         REAL           A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

PURPOSE
     SLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix
     A so that elements below the k-th subdiagonal are zero. The reduction is
     performed by an orthogonal similarity transformation Q' * A * Q. The
     routine returns the matrices V and T which determine Q as a block
     reflector I - V*T*V', and also the matrix Y = A * V * T.

     This is an auxiliary routine called by SGEHRD.


ARGUMENTS
     N       (input) INTEGER
             The order of the matrix A.

     K       (input) INTEGER
             The offset for the reduction. Elements below the k-th subdiagonal
             in the first NB columns are reduced to zero.

     NB      (input) INTEGER
             The number of columns to be reduced.

     A       (input/output) REAL array, dimension (LDA,N-K+1)
             On entry, the n-by-(n-k+1) general matrix A.  On exit, the
             elements on and above the k-th subdiagonal in the first NB
             columns are overwritten with the corresponding elements of the
             reduced matrix; the elements below the k-th subdiagonal, with the
             array TAU, represent the matrix Q as a product of elementary
             reflectors. The other columns of A are unchanged. See Further
             Details.  LDA     (input) INTEGER The leading dimension of the
             array A.  LDA >= max(1,N).

     TAU     (output) REAL array, dimension (NB)
             The scalar factors of the elementary reflectors. See Further
             Details.

     T       (output) REAL array, dimension (NB,NB)
             The upper triangular matrix T.

     LDT     (input) INTEGER
             The leading dimension of the array T.  LDT >= NB.
     Y       (output) REAL array, dimension (LDY,NB)
             The n-by-nb matrix Y.

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

FURTHER DETAILS
     The matrix Q is represented as a product of nb elementary reflectors

        Q = H(1) H(2) . . . H(nb).

     Each H(i) has the form

        H(i) = I - tau * v * v'

     where tau is a real scalar, and v is a real vector with
     v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i),
     and tau in TAU(i).

     The elements of the vectors v together form the (n-k+1)-by-nb matrix V
     which is needed, with T and Y, to apply the transformation to the
     unreduced part of the matrix, using an update of the form:  A := (I -
     V*T*V') * (A - Y*V').

     The contents of A on exit are illustrated by the following example with n
     = 7, k = 3 and nb = 2:

        ( a   h   a   a   a )
        ( a   h   a   a   a )
        ( a   h   a   a   a )
        ( h   h   a   a   a )
        ( v1  h   a   a   a )
        ( v1  v2  a   a   a )
        ( v1  v2  a   a   a )

     where a denotes an element of the original matrix A, h denotes a modified
     element of the upper Hessenberg matrix H, and vi denotes an element of
     the vector defining H(i).