High Order Integration Schemes

Integration schemes based on interpolation with polynomials of degree
greater than 2 are seldom used in practice, but one area of application
is in the fields of dynamic simulation, controls, and digital filtering.
For example, if the variables x(t) and y(t) are related according to 
the Nth order differential equation

      N       d^k x         N       d^k y
    SUM   a_k -----   =   SUM   b_k -----       a0, b0, aN <> 0    (1)
     k=0      d t^k        k=0      d t^k

then the exact root-matched recurrence relation for y(t) at discrete
time intervals T based on interpolating x(t) with a nth degree 
polynomial is

            S_A  Y_n = S_A  M  A^(-1)  B  M^(-1)  X_n

where X_n and Y_n are the column vectors containing the values of
x(kT) and y(kT) for k = n-N,..., n-1, n.  Knowing all the values of the
input function x(kT) and all the PAST values of y(kT), this recurrence
formula allows us to compute the current output value y(nT).

S_A is the row vector whose components are the elementary 
symmetric functions of the exponentials of the roots of the 
characteristic polynomial of the left side of (1).  A, B, and M
are square matrices such that the elements in the jth columns of
the kth rows (where j,k range from 0 to N) are given by

                      /   (j!/k!) a_(j-k)   if j >= k 
           A_{k,j} = |
                      \     0              if j < k


                      /   (j!/k!) b_(j-k)   if j >= k 
           B_{k,j} = |
                      \     0              if j < k


                      /   (kT)^j        if (k+j) > 0
           M_{k,j} = |
                      \     1           if (k+j) = 0


It's interesting that this formulation results from treating x(t) as
the independent variable and y(t) as the dependent variable, whereas 
equation (1) is actually symmetrical.  If the two variables are treated
symmetrically the resulting discrete-ized recurrence yields a (slightly)
different response.  The difference goes to zero as the discrete time
increment T goes to zero.  This convergence at T->0 leads to some
very interesting relations involving the Bernoulli numbers, and is 
suggestive of the observed/unobserved dichotomy in quantum mechanics.
Specifically, it suggests that the "collapse of the wave function" when 
a measurement is taken is a result of a discretization of time, and if 
time were truly continuous there would be no distinction between 
"observed" and "unobserved" (or "isolated" and "not-isolated") physics.