chptrd(3F)

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

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

         CHARACTER      UPLO

         INTEGER        INFO, N

         REAL           D( * ), E( * )

         COMPLEX        AP( * ), TAU( * )

PURPOSE
     CHPTRD reduces a complex Hermitian matrix A stored in packed form to real
     symmetric tridiagonal form T by a unitary similarity 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 array, dimension (N*(N+1)/2)
             On entry, the upper or lower triangle of the Hermitian 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 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
             reflectors. See Further Details.  D       (output) REAL array,
             dimension (N) The diagonal elements of the tridiagonal matrix T:
             D(i) = A(i,i).

     E       (output) REAL 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 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).