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

           CHARACTER      UPLO

           INTEGER        LDA, LDW, N, NB

           REAL           A(  LDA, * ), E( * ), TAU( * ), W( LDW,
                          * )


PURPOSE

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


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) REAL 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
               upper triangular part of A is not referenced.   On

               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) 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) REAL 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)  REAL 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).

       Each H(i) has the form


       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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