CXML
        	    
LAPACK version 3.0
clatrd(3)

PURPOSE
  CLATRD - reduce NB rows and columns of a complex Hermitian matrix A to
  Hermitian tridiagonal form by a unitary similarity transformation Q' * A *
  Q, and returns the matrices V and W which are needed to apply the
  transformation to the unreduced part of A

SYNTAX
  SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

      CHARACTER	     UPLO

      INTEGER	     LDA, LDW, N, NB

      REAL	     E( * )

      COMPLEX	     A( LDA, * ), TAU( * ), W( LDW, * )

DESCRIPTION
  CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
  Hermitian tridiagonal form by a unitary similarity transformation Q' * A *
  Q, and returns the matrices V and W which are needed to apply the
  transformation to the unreduced part of A.  If UPLO = 'U', CLATRD reduces
  the last NB rows and columns of a matrix, of which the upper triangle is
  supplied;
  if UPLO = 'L', CLATRD reduces the first NB rows and columns of a matrix, of
  which the lower triangle is supplied.

  This is an auxiliary routine called by CHETRD.

ARGUMENTS
  UPLO	  (input) CHARACTER
	  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.

  NB	  (input) INTEGER
	  The number of rows and columns to be reduced.

  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 last NB columns have been reduced to
	  tridiagonal form, with the diagonal elements overwriting the
	  diagonal elements of A; the elements above the diagonal with the
	  array TAU, represent the unitary matrix Q as a product of
	  elementary reflectors; if UPLO = 'L', the first NB columns have
	  been reduced to tridiagonal form, with the diagonal elements
	  overwriting the diagonal elements of A; the elements below the
	  diagonal 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).

  E	  (output) REAL array, dimension (N-1)
	  If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of
	  the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb)
	  contains the subdiagonal elements of the first NB columns of the
	  reduced matrix.

  TAU	  (output) COMPLEX array, dimension (N-1)
	  The scalar factors of the elementary reflectors, stored in TAU(n-
	  nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.  See Further
	  Details.  W	    (output) COMPLEX array, dimension (LDW,NB) The
	  n-by-nb matrix W required to update the unreduced part of A.

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

FURTHER DETAILS
  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0
  and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in
  TAU(i-1).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(nb).

  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+1:n) is stored on exit in A(i+1:n,i), and tau in
  TAU(i).

  The elements of the vectors v together form the n-by-nb matrix V which is
  needed, with W, to apply the transformation to the unreduced part of the
  matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

  The contents of A on exit are illustrated by the following examples with n
  = 5 and nb = 2:

  if UPLO = 'U':		       if UPLO = 'L':

    (  a   a   a   v4  v5 )		 (  d		       )
    (	   a   a   v4  v5 )		 (  1	d	       )
    (	       a   1   v5 )		 (  v1	1   a	       )
    (		   d   1  )		 (  v1	v2  a	a      )
    (		       d  )		 (  v1	v2  a	a   a  )

  where d denotes a diagonal element of the reduced matrix, a denotes an
  element of the original matrix that is unchanged, and vi denotes an element
  of the vector defining H(i).