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CHETRD(l)			       )			     CHETRD(l)

NAME
       CHETRD - reduce a complex Hermitian matrix A to real symmetric tridiag‐
       onal form T by a unitary similarity transformation

SYNOPSIS
       SUBROUTINE CHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )

	   CHARACTER	  UPLO

	   INTEGER	  INFO, LDA, LWORK, N

	   REAL		  D( * ), E( * )

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

PURPOSE
       CHETRD reduces a complex Hermitian matrix A to real symmetric tridiago‐
       nal 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 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).

       WORK    (workspace/output) COMPLEX 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 per‐
	       formance 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).

LAPACK version 3.0		 15 June 2000			     CHETRD(l)
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