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


SYNOPSIS

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

           CHARACTER      UPLO

           INTEGER        INFO, N

           DOUBLE         PRECISION AP( * ), D( * ), E( * ), TAU(
                          * )


PURPOSE

       DSPTRD 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)   DOUBLE   PRECISION   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)
               DOUBLE PRECISION array, dimension (N) The diagonal
               elements  of  the  tridiagonal  matrix  T:  D(i) =
               A(i,i).

       E       (output) DOUBLE PRECISION array, dimension (N-1)
               The  off-diagonal  elements  of  the   tridiagonal

               A(i+1,i) if UPLO = 'L'.

       TAU     (output) DOUBLE PRECISION array, dimension (N-1)
               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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