22S:166
Lecture 10
Sept. 22, 2006
Root-finding
1
Root finding algorithms
•problem: to find values of variable xthat satisfy
f(x) = 0 for given function f
•solution is called “zero of f” or “root of f”
•when is this an important problem in statistics?
2
The bisection method
•also called “binary-search method”
•conditions for use
–fcontinuous, defined on interval [a, b]
–f(a) and f(b) of opposite sign
•by Intermediate Value Theorem, there exists a
p, a < p < b, such that f(p) = 0
•procedure works when f(a) and f(b) of opposite sign
and more than one root in [a, b]
•for simplicity, we’ll assume unique root in interval
•method consists of
–repeated halving of subintervals of [a, b]
–at each step, locating half containing p
•requires following inputs
–endpoints a,b
–tolerance T OL
–maximum number of iterations N0
3
Fixed-point iteration
•solution to
g(x) = x
is called fixed point of function g
•Theorem
–conditions
∗gcontinuous on [a, b]
∗g(x)∈[a, b]∀x∈[a, b]
–conclusions
∗ghas a fixed point in [a, b]
–if further
∗g0(x) exists on (a, b) and a positive constant
k < 1 exists such that
|g0(x)| ≤ k < 1,∀x∈(a, b)
–then
∗ghas a unique fixed point pin [a, b]
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