196 Ordinary Differential Equations
Solving the set of equations (1) for the a's, we getan= bn-l
an-l = bn-2 - bn-lt
a2 = b1 - b2t
a1 = bo - bitao = R-bot
Substituting in p(x) and rearranging algebraically, we find for any t
(R - bot)Defining q(x) = bn-1Xn-l + bn-2Xn-^2 + · · · + bix + bo, we see that p(x) =
(x -t)q(x) + R. That is, when p(x) is divided by x-t, the quotient is q(x) and
the remainder is R. If tis a root of p(x), then R = 0 and p(x) = (x - t)q(x).
Additional roots of p( x) can then be found by finding roots of q( x). The
process of computing the coefficients bn-l, bn-2, ... , bo of the polynomial
q(x)- which is of degree n - 1- from the coefficients an, an-I, ... , ao when
t is a root of p( x) is called deflation.
Homer's method- which is actually the much more ancient Chinese method,
fan fa- for finding the roots of a polynomial consists of applying the Newton-
Raphson method to polynomials. To apply this method we must be able to
evaluate both p(x) and p'(x) at any point t. Since p(x) = (x - t)q(x) + R ,
p'(x) = (x - t)q'(x) + q(x) and therefore p'(t) = q(t). Now p(t) = R can
be evaluated using equation (1) and p'(t) = q(t) can be calculated from the
analogous set of equations for q, n amely
(2) Cn-2 = bn-l
Cn-3 = Cn-2t + bn-2
Co = C1t + b1
S =cot+ bo = q(t) = p'(t)
So Homer's method for computing a root of p(x) is as follows: