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


SYNOPSIS

       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, * )


PURPOSE

       CLATRD  reduces NB rows and columns of a complex Hermitian
       matrix A to Hermitian tridiagonal form by a unitary  simi­
       larity 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  sup­
       plied.

       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 con­
               tains 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 tri­
               angular part of the matrix  A,  and  the  strictly

               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 diago­
               nal elements of A; the elements below the diagonal
               with  the array TAU, represent the  unitary matrix
               Q as a product of elementary reflectors.  See Fur­
               ther 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 superdiag­
               onal  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


       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 transforma­
       tion to the unreduced part of the matrix, using  a  Hermi­
       tian 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 defin­
       ing H(i).


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