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NAME
     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, * )



     #include <sunperf.h>

     void slatrd(char uplo, int n, int nb, float  *sa,  int  lda,
               float *e, float *tau, float *w, int 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 supplied.

     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
               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 orthogonal matrix Q as  a  pro-
               duct  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

        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  symmetric
     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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