2.2 Fixed-Point Iteration
2.2 Fixed-Point Iteration
1
Basic Definitions
? A number is a fixed point for a given function if =
? Root finding = 0 is related to fixed-point iteration =
? Given a root-finding problem = 0, there are many with fixed points at :
Example: - + 3 ...
If has fixed point at , then = - () has a zero at
2
Why study fixed-point iteration?
1. Sometimes easier to analyze 2. Analyzing fixed-point problem can help us find
good root-finding methods
A Fixed-Point Problem Determine the fixed points of the function = 2 - 2.
3
When Does Fixed-Point Iteration Converge?
Existence and Uniqueness Theorem a. If [, ] and [, ] for all
[, ], then has a fixed-point in [, ] b. If, in addition, () exists on (, ) and a
positive constant < 1 exists with , for all , ,
then there is exactly one fixed point in , . Note:
1. , - is continuous in , 2. , - takes values in ,
4
Proof
? If = , or = , then has a fixed point at the endpoint.
? Otherwise, > and < . ? Define a new function = -
? = - > 0 and = - < 0 ? is continuous
? By intermediate value theorem, there exists (, ) for which = .
? Thus, = .
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