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DSPTRD(1)		 LAPACK routine (version 3.2)		     DSPTRD(1)

NAME
       DSPTRD  -  reduces  a  real symmetric matrix A stored in packed form to
       symmetric tridiagonal form T by an orthogonal similarity transformation

SYNOPSIS
       SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )

	   CHARACTER	  UPLO

	   INTEGER	  INFO, N

	   DOUBLE	  PRECISION AP( * ), D( * ), E( * ), TAU( * )

PURPOSE
       DSPTRD reduces a real symmetric matrix A stored in packed form to  sym‐
       metric  tridiagonal  form T by an orthogonal similarity transformation:
       Q**T * A * Q = T.

ARGUMENTS
       UPLO    (input) CHARACTER*1
	       = 'U':  Upper triangle of A is stored;
	       = 'L':  Lower triangle of A is stored.

       N       (input) INTEGER
	       The order of the matrix A.  N >= 0.

       AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
	       On entry, the upper or lower triangle of the  symmetric	matrix
	       A,  packed  columnwise in a linear array.  The j-th column of A
	       is stored in the array AP as follows: if UPLO  =	 'U',  AP(i  +
	       (j-1)*j/2)  =  A(i,j)  for  1<=i<=j;  if	 UPLO  =  'L',	AP(i +
	       (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.  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 subdiag‐
	       onal, with the array TAU, represent the orthogonal matrix Q  as
	       a  product  of  elementary  reflectors. See Further Details.  D
	       (output) DOUBLE PRECISION array,	 dimension  (N)	 The  diagonal
	       elements of the tridiagonal matrix T: D(i) = A(i,i).

       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 AP,  overwrit‐
       ing A(1:i-1,i+1), and tau is stored 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 AP,  overwrit‐
       ing A(i+2:n,i), and tau is stored in TAU(i).

 LAPACK routine (version 3.2)	 November 2008			     DSPTRD(1)
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