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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 sym‐
       metric 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 elementary 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 col‐
	       umns 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 Further 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)
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