5.1. Approximation of Roots 1691
r=a+
1
b+ 1
I
c+
d+-+...where a, b, c, d, e,... are integers. Such an expression is called a continued
fraction. More conveniently, we can write this as a+l/b+l/c+l/d+l/e+.. .,
where each fraction bar has everything that follows in the denominator.
Approximations to r can be obtained by taking the part after any of the
plus signs to be zero. Verify that this yields the rational approximations:a, (ab + 1)/b, (abc + a + c)/(bc + l),...For example, if the equation to be solved is t2 - 2 = 0, we know there is
a root between 1 and 2. Set t = 1 + l/s and get s2 - 2s - 1 = 0, which has
a root between 2 and 3. Set s = 2 + l/u and get u2 - 2u - 1 = 0. Deduce
that
Jz=1+1/2+1/2+1/2+1/2+...
and verify that the successive approximations to fi are 1, 312, 715, 17/12,
41129,.... Work these out numerically to see how they approach fi. Com-
pare this sequence of approximations with those obtained from Newton’s
Method (Exercise 5.1.7) and the starting values 1, 3/2, 7/5, 17/12, respec-
tively. (Compare Exploration E.30. When t = 6, the sequence u,, is 0, 1,
6, 35, 204,.... Factor the terms of this sequence.)
Find the continued fraction representations of the roots of t2 - t - 1 = 0
and give the first five rational approximations to these roots.
Show that the substitution t = 1 + l/s in t4 - t3 - t2 - t - 1 = 0 yields
3s4 + 2s3 - 2s2 - 3s - 1 = 0, which has a root between 1 and 2. Substitute
s = 1+1/u to obtain u4-11u3-22u2-1421-3 = 0. Show that this equation
in u has a solution between 12 and 13, and hence obtain the approximation
25/13 for a root of the original equation.
E.46. Continued Fractions: Another Approach for Quadratics. The
quadratic equation t2 + bt + c = 0 can be rewritten as t = -b - (c/t).
Substitute this expression for t into the right side, and repeat the process
to obtain a continued fraction expansion for t. Try this on the polynomials
t2 - 2t - 1, t2 - 2t + 1, t2 - t - 1, t2 - 2, t2 - 3t + 2, t2 -t + 1, t2 -t - 6,
t2 + t - 6. Does the related sequence of approximations always converge?
If so, to which zero does it converge?