CHETD2  - reduce a complex Hermitian matrix A to real sym­
       metric tridiagonal form T by a unitary  similarity  trans­
       formation


SYNOPSIS

       SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDA, N

           REAL           D( * ), E( * )

           COMPLEX        A( LDA, * ), TAU( * )


PURPOSE

       CHETD2  reduces  a complex Hermitian matrix A to real sym­
       metric tridiagonal form T by a unitary  similarity  trans­
       formation: Q' * A * Q = T.


ARGUMENTS

       UPLO    (input) CHARACTER*1
               Specifies  whether  the  upper or lower triangular
               part of the Hermitian matrix A is stored:
               = 'U':  Upper triangular
               = 'L':  Lower triangular

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

       A       (input/output) COMPLEX array, dimension (LDA,N)
               On entry, the Hermitian 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 tri­
               angular 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 super­
               diagonal of A are overwritten by the corresponding
               elements of the tridiagonal matrix T, and the ele­
               ments  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.  LDA     (input)
               INTEGER The leading dimension of the array A.  LDA


       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 ille­
               gal 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
       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 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 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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