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zhptrd

 NAME
      ZHPTRD - reduce a complex Hermitian matrix A stored in
      packed form to real symmetric tridiagonal form T by a uni-
      tary similarity transformation

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

          CHARACTER      UPLO

          INTEGER        INFO, N

          DOUBLE         PRECISION D( * ), E( * )

          COMPLEX*16     AP( * ), TAU( * )

 PURPOSE
      ZHPTRD reduces a complex Hermitian matrix A stored in packed
      form to real symmetric tridiagonal form T by a unitary simi-
      larity transformation: Q**H * 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) COMPLEX*16 array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermi-
              tian 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 overwrit-
              ten by the corresponding elements of the tridiagonal
              matrix T, and the elements above the first superdi-
              agonal, with the array TAU, represent the unitary
              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
              unitary matrix Q as a product of elementary reflec-
              tors. 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) COMPLEX*16 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 complex scalar, and v is a complex vector
      with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit
      in AP, overwriting 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 complex scalar, and v is a complex vector
      with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit
      in AP, overwriting A(i+2:n,i), and tau is stored in TAU(i).