NAME

       zhetrd.f -

SYNOPSIS

   Functions/Subroutines
       subroutine zhetrd (UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
           ZHETRD

Function/Subroutine Documentation

   subroutine zhetrd (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA,
       double precision, dimension( * ) D, double precision, dimension( * ) E, complex*16,
       dimension( * ) TAU, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)
       ZHETRD

       Purpose:

            ZHETRD reduces a complex Hermitian matrix A to real symmetric
            tridiagonal form T by a unitary similarity transformation:
            Q**H * A * Q = T.

       Parameters:
           UPLO

                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           A

                     A is 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 contains 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
                     triangular 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 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

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           D

                     D is DOUBLE PRECISION array, dimension (N)
                     The diagonal elements of the tridiagonal matrix T:
                     D(i) = A(i,i).

           E

                     E is 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

                     TAU is COMPLEX*16 array, dimension (N-1)
                     The scalar factors of the elementary reflectors (see Further
                     Details).

           WORK

                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= 1.
                     For optimum performance LWORK >= N*NB, where NB is the
                     optimal blocksize.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

       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**H

             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**H

             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).

Author

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