ZLATRD - 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 ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )

           CHARACTER      UPLO

           INTEGER        LDA, LDW, N, NB

           DOUBLE         PRECISION E( * )

           COMPLEX*16     A( LDA, * ), TAU( * ), W( LDW, * )


PURPOSE

       ZLATRD  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', ZLATRD reduces
       the last NB rows and columns of a  matrix,  of  which  the
       upper triangle is supplied;
       if  UPLO  =  'L',  ZLATRD  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 ZHETRD.


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*16 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) DOUBLE PRECISION 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*16 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*16 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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