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CHETD2(1)		 LAPACK routine (version 3.2)		     CHETD2(1)

NAME
       CHETD2  - reduces a complex Hermitian matrix A to real symmetric tridi‐
       agonal form T by a unitary similarity transformation

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 symmetric tridiago‐
       nal form T by a unitary similarity transformation: 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 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 ele‐
	       ments 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, rep‐
	       resent the unitary matrix Q as a product of elementary  reflec‐
	       tors. See Further Details.  LDA	   (input) INTEGER The leading
	       dimension of the array A.  LDA >= max(1,N).

       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
       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).

 LAPACK routine (version 3.2)	 November 2008			     CHETD2(1)
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