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dlahrd examples
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Man Page Details
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NAME
       DLAHRD  -  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 DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

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

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

PURPOSE
       DLAHRD  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 OBSOLETE auxiliary routine.
       This routine will be 'deprecated' in a  future release.
       Please use the new routine DLAHR2 instead.


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

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

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

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
               On entry, the n-by-(n-k+1) general matrix A.  On exit, the ele-
               ments 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  reflec-
               tors.  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) DOUBLE PRECISION array, dimension (NB)
               The  scalar  factors  of the elementary reflectors. See Further
               Details.

       T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
               The upper triangular matrix T.

       LDT     (input) INTEGER
               The leading dimension of the array T.  LDT >= NB.

       Y       (output) DOUBLE PRECISION 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 unre-
       duced  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 modi-
       fied element of the upper Hessenberg matrix H, and vi denotes  an  ele-
       ment of the vector defining H(i).

