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DSYTD2(3F)							    DSYTD2(3F)

NAME
     DSYTD2 - reduce a real symmetric matrix A to symmetric tridiagonal form T
     by an orthogonal similarity transformation

SYNOPSIS
     SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )

	 CHARACTER	UPLO

	 INTEGER	INFO, LDA, N

	 DOUBLE		PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )

PURPOSE
     DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T
     by an orthogonal similarity transformation: Q' * A * Q = T.

ARGUMENTS
     UPLO    (input) CHARACTER*1
	     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.	 N >= 0.

     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 diagonal and first
	     superdiagonal of A are overwritten by the corresponding elements
	     of the tridiagonal matrix T, and the elements above the first
	     superdiagonal, with the array TAU, represent the orthogonal
	     matrix Q as a product of elementary reflectors; if UPLO = 'L',
	     the diagonal and first subdiagonal of A are over- written by the
	     corresponding elements of the tridiagonal matrix T, and the
	     elements below the first subdiagonal, 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 >= max(1,N).

     D	     (output) DOUBLE PRECISION array, dimension (N)
	     The diagonal elements of the tridiagonal matrix T:	 D(i) =
	     A(i,i).

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DSYTD2(3F)							    DSYTD2(3F)

     E	     (output) DOUBLE PRECISION array, dimension (N-1)
	     The off-diagonal elements of the tridiagonal matrix T:  E(i) =
	     A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

     TAU     (output) DOUBLE PRECISION array, dimension (N-1)
	     The scalar factors of the elementary reflectors (see Further
	     Details).

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value.

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

	Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
     A(1:i-1,i+1), and tau in TAU(i).

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

	Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and
     tau in TAU(i).

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

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

       (  d   e	  v2  v3  v4 )		    (  d		  )
       (      d	  e   v3  v4 )		    (  e   d		  )
       (	  d   e	  v4 )		    (  v1  e   d	  )
       (	      d	  e  )		    (  v1  v2  e   d	  )
       (		  d  )		    (  v1  v2  v3  e   d  )

     where d and e denote diagonal and off-diagonal elements of T, and vi
     denotes an element of the vector defining H(i).

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DSYTD2(3F)							    DSYTD2(3F)

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