ZLAHRD  - reduce the first NB columns of a complex general
       n-by-(n-k+1) matrix A so that elements below the k-th sub­
       diagonal are zero


SYNOPSIS

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

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

           COMPLEX*16     A( LDA, * ), T( LDT, NB ), TAU(  NB  ),
                          Y( LDY, NB )


PURPOSE

       ZLAHRD  reduces  the first NB columns of a complex general
       n-by-(n-k+1) matrix A so that elements below the k-th sub­
       diagonal are zero. The reduction is performed by a unitary
       similarity transformation Q' * A * Q. The routine  returns
       the  matrices V and T which determine Q as a block reflec­
       tor I - V*T*V', and also the matrix Y = A * V * T.

       This is an auxiliary routine called by ZGEHRD.


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) COMPLEX*16 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) COMPLEX*16 array, dimension (NB)
               The scalar factors of the  elementary  reflectors.
               See Further Details.

               The upper triangular matrix T.

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

       Y       (output) COMPLEX*16 array, dimension (LDY,NB)
               The n-by-nb matrix Y.

       LDY     (input) INTEGER
               The leading dimension  of  the  array  Y.  LDY  >=
               max(1,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 complex scalar, and v is a  complex  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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