Page 190 - Applied Numerical Methods Using MATLAB

4

NONLINEAR EQUATIONS

4.1 ITERATIVE METHOD TOWARD FIXED POINT

Let’s see the following theorem.

Fixed-Point Theorem: Contraction Theorem [K-2, Section 5.1] . Suppose a function

g(x) is defined and its first derivative g (x) exists continuously on some interval
o
o
o
I = [x − r, x + r] around the fixed point x of g(x) such that
o
g(x ) = x o (4.1.1)

Then, if the absolute value of g (x) is less than or equal to a positive number α
that is strictly less than one, that is,

|g (x)|≤ α< 1 (4.1.2)

the iteration starting from any point x 0 ∈ I

x k+1 = g(x k ) with x 0 ∈ I (4.1.3)

o
converges to the (unique) fixed point x of g(x).


Applied Numerical Methods Using MATLAB , by Yang, Cao, Chung, and Morris
Copyright  2005 John Wiley & Sons, Inc.
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