PDLAHRD(l)					      LAPACK auxiliary routine (version 1.5)						PDLAHRD(l)

NAME

       PDLAHRD	-  reduce the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the
       k-th subdiagonal are zero

SYNOPSIS

       SUBROUTINE PDLAHRD( N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK )

	   INTEGER	   IA, IY, JA, JY, K, N, NB

	   INTEGER	   DESCA( * ), DESCY( * )

	   DOUBLE	   PRECISION A( * ), T( * ), TAU( * ), WORK( * ), Y( * )

PURPOSE

       PDLAHRD reduces the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-
       th subdiagonal are zero. The reduction is performed by an orthogo- nal similarity transformation Q' * A * Q. The routine returns the matri-
       ces 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 PDGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1).

ARGUMENTS

       N       (global input) INTEGER
	       The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ).	N >= 0.

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

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

       A       (local input/local output) DOUBLE PRECISION pointer into
	       the local memory to an array of dimension (LLD_A, LOCc(JA+N-K)). On entry, this array contains the the local pieces of the N-by-(N-
	       K+1)  general  distributed  matrix  A(IA:IA+N-1,JA:JA+N-K). 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 distributed 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(IA:IA+N-1,JA:JA+N-K) are
	       unchanged. See Further Details.	IA	(global input) INTEGER The row index in the global array A indicating  the  first  row	of
	       sub( A ).

       JA      (global input) INTEGER
	       The column index in the global array A indicating the first column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
	       The array descriptor for the distributed matrix A.

       TAU     (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-2)
	       The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A.

       T       (local output) DOUBLE PRECISION array, dimension (NB_A,NB_A)
	       The upper triangular matrix T.

       Y       (local output) DOUBLE PRECISION pointer into the local memory
	       to  an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD_Y >=
	       LOCr(IA+N-1).

       IY      (global input) INTEGER
	       The row index in the global array Y indicating the first row of sub( Y ).

       JY      (global input) INTEGER
	       The column index in the global array Y indicating the first column of sub( Y ).

       DESCY   (global and local input) INTEGER array of dimension DLEN_.
	       The array descriptor for the distributed matrix Y.

       WORK    (local workspace) DOUBLE PRECISION array, dimension (NB)

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(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).

       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(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V').

       The contents of A(ia:ia+n-1,ja:ja+n-k) 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(ia:ia+n-1,ja:ja+n-k),	h  denotes  a  modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining
       H(i).

LAPACK version 1.5						    12 May 1997 							PDLAHRD(l)