Chapter 1 The Solution of Nonlinear Equations f(=0 1.1 Iteration for Solving x=g(x)
Chapter 1 The Solution of Nonlinear Equations f(x)=0 1.1 Iteration for Solving x=g(x)
1=9(0) P2=9(1) Pk=9(k-1 Pk+1=9(Pk)
Example 1.1. The iterative rule po l and pk+1= 1.001pk for k=0, 1,..pro- duces a divergent sequence. The first 100 terms look as follows 1.00120=(1.001)(1.000001.00100 P2=1.0011=(1.0101001000=1.00201 P3=1.001p2=(1.001)(1.002001)=1.003003 p100=1.00109=(1.001)(1.104012)=1.105116
1.1.1 Finding Fixed Points Definition 1.1 (Fixed Point). A ficed point of a function g(a)is a real number P such that P=9(P) Geometrically, the fixed points of a function y=g(r) are the points of intersection of y=g( and Definition 1.2 (Fixed-point Iteration). The iteration Pn+1=g(pn)forn=0,1 is called ficed-point iteration
1.1.1 Finding Fixed Points
Theorem 1. 1. Assume that g is a continuous function and that ipn ln_o is a se quence generated by fixed-point iteration. If limn-ooPn=P, then P is a fixed point