DLATRD(l)								 )								 DLATRD(l)

NAME

       DLATRD  -  reduce NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation
       Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A

SYNOPSIS

       SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

	   CHARACTER	  UPLO

	   INTEGER	  LDA, LDW, N, NB

	   DOUBLE	  PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )

PURPOSE

       DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q'
       *  A  * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.  If UPLO = 'U', DLATRD
       reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied;
       if UPLO = 'L', DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

       This is an auxiliary routine called by DSYTRD.

ARGUMENTS

       UPLO    (input) CHARACTER
	       Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
	       = 'U': Upper triangular
	       = 'L': Lower triangular

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

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

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
	       On entry, the symmetric matrix A.  If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular  part
	       of the matrix A, and the strictly lower triangular part of A is not referenced.	If UPLO = 'L', the leading n-by-n lower triangular
	       part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is  not  referenced.	On
	       exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal
	       elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as	a  product  of	elementary
	       reflectors;  if	UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the
	       diagonal elements of A; the elements below the diagonal with the array TAU, represent the  orthogonal matrix Q as a product of ele-
	       mentary reflectors.  See Further Details.  LDA	  (input) INTEGER The leading dimension of the array A.  LDA >= (1,N).

       E       (output) DOUBLE PRECISION array, dimension (N-1)
	       If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb)
	       contains the subdiagonal elements of the first NB columns of the reduced matrix.

       TAU     (output) DOUBLE PRECISION array, dimension (N-1)
	       The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.	 See  Fur-
	       ther  Details.	W	 (output) DOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced
	       part of A.

       LDW     (input) INTEGER
	       The leading dimension of the array W. LDW >= max(1,N).

FURTHER DETAILS

       If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

	  Q = H(n) H(n-1) . . . H(n-nb+1).

       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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

       If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

       The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part
       of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

       The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

       if UPLO = 'U':			    if UPLO = 'L':

	 (  a	a   a	v4  v5 )	      (  d		    )
	 (	a   a	v4  v5 )	      (  1   d		    )
	 (	    a	1   v5 )	      (  v1  1	 a	    )
	 (		d   1  )	      (  v1  v2  a   a	    )
	 (		    d  )	      (  v1  v2  a   a	 a  )

       where  d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an
       element of the vector defining H(i).

LAPACK version 3.0						   15 June 2000 							 DLATRD(l)