CXML
        	    
LAPACK version 3.0
zhetrd(3)

PURPOSE
  ZHETRD - reduce a complex Hermitian matrix A to real symmetric tridiagonal
  form T by a unitary similarity transformation

SYNTAX
  SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )

      CHARACTER	     UPLO

      INTEGER	     INFO, LDA, LWORK, N

      DOUBLE	     PRECISION D( * ), E( * )

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

DESCRIPTION
  ZHETRD reduces a complex Hermitian matrix A 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.

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

  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).

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

  LWORK	  (input) 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	  (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
  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).