ZHETD2  - reduce a complex Hermitian matrix A to real sym­
       metric tridiagonal form T by a unitary  similarity  trans­
       formation


SYNOPSIS

       SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, LDA, N

           DOUBLE         PRECISION D( * ), E( * )

           COMPLEX*16     A( LDA, * ), TAU( * )


PURPOSE

       ZHETD2  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*16 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) 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 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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