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NAME
     clatrd - reduce NB rows and columns of a  complex  Hermitian
     matrix A to Hermitian tridiagonal form by a unitary similar-
     ity 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, * )



     #include <sunperf.h>

     void clatrd(char uplo, int n, int nb, complex *ca, int  lda,
               float *e, complex *tau, complex *w, int ldw) ;

PURPOSE
     CLATRD reduces NB rows and columns of  a  complex  Hermitian
     matrix A to Hermitian tridiagonal form by a unitary similar-
     ity 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 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
               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 diago-
               nal 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
               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 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

        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 transforma-
     tion 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).

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