SSPTRD - reduce a real symmetric matrix A stored in packed
       form to symmetric tridiagonal form T by an orthogonal sim­
       ilarity transformation


SYNOPSIS

       SUBROUTINE SSPTRD( UPLO, N, AP, D, E, TAU, INFO )

           CHARACTER      UPLO

           INTEGER        INFO, N

           REAL           AP( * ), D( * ), E( * ), TAU( * )


PURPOSE

       SSPTRD  reduces a real symmetric matrix A stored in packed
       form to symmetric tridiagonal form T by an orthogonal sim­
       ilarity transformation: Q**T * A * Q = T.


ARGUMENTS

       UPLO    (input) CHARACTER*1
               = 'U':  Upper triangle of A is stored;
               = 'L':  Lower triangle of A is stored.

       N       (input) INTEGER
               The order of the matrix A.  N >= 0.

       AP      (input/output) REAL array, dimension (N*(N+1)/2)
               On  entry, the upper or lower triangle of the sym­
               metric matrix A, packed  columnwise  in  a  linear
               array.   The  j-th  column  of  A is stored in the
               array AP  as  follows:  if  UPLO  =  'U',  AP(i  +
               (j-1)*j/2)  =  A(i,j)  for 1<=i<=j; if UPLO = 'L',
               AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.   On
               exit, if UPLO = 'U', the diagonal and first super­
               diagonal of A are overwritten by the corresponding
               elements of the tridiagonal matrix T, and the ele­
               ments above  the  first  superdiagonal,  with  the
               array  TAU, represent the orthogonal matrix Q as a
               product of elementary reflectors; if UPLO  =  'L',
               the  diagonal and first subdiagonal of A are over-
               written  by  the  corresponding  elements  of  the
               tridiagonal  matrix  T, and the elements below the
               first subdiagonal, with the array  TAU,  represent
               the orthogonal matrix Q as a product of elementary
               reflectors. See Further Details.  D       (output)
               REAL array, dimension (N) The diagonal elements of
               the tridiagonal matrix T: D(i) = A(i,i).

       E       (output) REAL array, dimension (N-1)
               The  off-diagonal  elements  of  the   tridiagonal
               matrix  T:  E(i)  = A(i,i+1) if UPLO = 'U', E(i) =
               A(i+1,i) if UPLO = 'L'.

               The scalar factors of  the  elementary  reflectors
               (see Further Details).

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value


FURTHER DETAILS

       If UPLO = 'U', the matrix Q is represented as a product of
       elementary reflectors

          Q = H(n-1) . . . H(2) H(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+1:n)  =  0 and v(i) = 1; v(1:i-1) is stored on exit in
       AP, overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

       If UPLO = 'L', the matrix Q is represented as a product of
       elementary reflectors

          Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on  exit  in
       AP, overwriting A(i+2:n,i), and tau is stored in TAU(i).


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