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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 subdiag-
     onal 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 )



     #include <sunperf.h>

     void slahrd(int n, int k, int nb, float *sa, int lda,  float
               *tau, float *t, int ldt, float *sy, int ldy) ;

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  simi-
     larity  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 subdiago-
               nal 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).

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