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dsytd2

 NAME
      DSYTD2 - reduce a real symmetric matrix A to symmetric tri-
      diagonal 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 tridi-
      agonal 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 con-
              tains 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 tri-
              angular 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 dimen-
              sion 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).

      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).