DLATRD  -  reduce  NB rows and columns of a real symmetric
       matrix A to symmetric tridiagonal form  by  an  orthogonal
       similarity  transformation  Q'  *  A  * Q, and returns the
       matrices V and W which are needed to apply the transforma­
       tion to the unreduced part of A


SYNOPSIS

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

           CHARACTER      UPLO

           INTEGER        LDA, LDW, N, NB

           DOUBLE         PRECISION  A(  LDA, * ), E( * ), TAU( *
                          ), W( LDW, * )


PURPOSE

       DLATRD reduces NB rows and columns  of  a  real  symmetric
       matrix  A  to  symmetric tridiagonal form by an orthogonal
       similarity transformation Q' * A  *  Q,  and  returns  the
       matrices V and W which are needed to apply the transforma­
       tion to the unreduced part of A.  If UPLO  =  'U',  DLATRD
       reduces the last NB rows and columns of a matrix, of which
       the upper triangle is supplied;
       if UPLO = 'L',  DLATRD  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 DSYTRD.


ARGUMENTS

       UPLO    (input) CHARACTER
               Specifies whether the upper  or  lower  triangular
               part of the symmetric 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) DOUBLE PRECISION array, dimension
               (LDA,N)
               On entry, the symmetric 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 orthogonal matrix Q as a  prod­
               uct  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
               orthogonal matrix Q as  a  product  of  elementary
               reflectors.  See Further Details.  LDA     (input)
               INTEGER The leading dimension of the array A.  LDA
               >= (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) DOUBLE PRECISION 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)   DOUBLE   PRECISION   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 real scalar, and v is a real 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).


          H(i) = I - tau * v * v'

       where tau is a real scalar, and v is a real 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 symmet­
       ric 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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