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Root Finding Methods
Adrian Down
February 02, 2006
1 Review: Second-order convergence
1.1 Review
We found that we could expect better than first order convergence if g′(α) =
Assume g′(α) = 0, and g′′(α) ∈ (a, b) has
|g′′(x)| ≤ 2 M ∀x ∈ (a, b)
Define the iterative series xν+1 = g(xν ). We showed that the error is bounded,
|xν − α| ≤ (M |x 0 − α|)^2
ν (^) − 1 |x 0 − α|
The series converges if M |x 0 − α| < 1, in particular if M |b − a| < 1.
1.2 Significant figures
We would like to know the number of iterations required for ≥ m significant figures. i.e.,
|xν − α| ≤ 10 −m
We have a limit on the error, |xν − α|,
|xν − α| ≤ (M |b − a|)^2
ν (^) − 1 |b − a|
To examine powers of 10, take the logarithm, expressing the negative explic- itly.
(M |b − a|)^2
ν (^) − 1 |b − a| = 10−(
ν (^) −1) log( (^) M |b^1 −a| )+log |b−a| ≤ 10 −m
(2ν^ − 1) log
M |b − a|
≥ m + log |b − a|
Take this as a rough equality and solve for m,
m = 2ν^ log
M |b − a|
Hence the number of significant figures doubles with each iteration. This is called second order convergence.
2 Root finding iterations
2.1 General method
Recall that x in some interval I, is a root of f (x) if and only if x is a fixed point of g(x) = x − φ(x)f (x), where φ(x) 6 = 0 ∀x ∈ I. We examine several choices of m which yield some standard methods.
2.2 Chord method
2.2.1 Definition
The chord method takes
φ(x) =
m
= constant
⇒ g(x) = x −
m
f (x)
The iteration is convergent for x 0 in some interval about a fixed point α if
|g′(α)| =
∣^1 −^
f ′(α) m
∣ <^1
This method yields second order convergence. To check this, we examine the derivative of the equivalent g,
g(x) = x −
f (x) f ′(x)
g′(x) = 1 − 1 +
f (x)f ′′(x) (f ′(x))^2
f (x)f ′′(x) (f ′(x))^2
f (α) = 0 by the definition of the fixed point as a 0 of f (x). Hence, assuming f ′(α) 6 = 0, g′(α) = 0 and the method provides second order convergence.
Note. To check second-order convergence of a root-finding method, it is suf- ficient to check that g′(α) = 0 for the equivalent g of the method.
Consider the special case of f ′(α) = 0. Then f (x) ∼ a(x − a)p^ as x → α, where p is a positive integer > 1. We must evaluate the derivative by using an expansion, equivalent to using L’Hopital’s rule.
g′(x) =
f (x)f ′′(x) (f ′(x))^2
∼
a(x − α)pap(p − 1)(x − a)p−^2 a^2 p^2 (x − α)^2 p−^2
= 1 −
p
as x → a. Hence 0 < g′(α) = 1 − (^1) p < 1 implies first order convergence.
2.3.2 Second-order iterator
Suppose we have an iteration defined by xν+1 = g(xν ), 0 < |g′(α)| < 1, and the series converges to α in the first order. We would like to construct a series that converges more quickly. Write the fixed point as a sum of the intervals that are successively shorter.
α = xν + (xν+1 − xν ) + (xν+2 − xν+1) +...
Each of these terms is an error term, which are governed by the factor ˆλ = g′(α),
α ≈ xν + (ν+1 − ν )
1 + ˆλ + λˆ^2 +...
≈ xν +
xν+1 − xν 1 − ˆλ
We take as a proposal,
xν+1 = xν +
g(xν ) − xν 1 − g′(xν )
= γ(xν )
with the definition,
γ(x) = x +
g(x) − x 1 − g′(x)
Taking the derivative,
γ′(x) = 1 − 1 + (g(x) − x)
g′′(x) (1 − g′(x))^2
At α,
γ′(α) = (g(α) − α) ︸ ︷︷ ︸ 0
g′′(α) (1 − g′(α))
Hence the iterator is second order.
