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
Example 1. 2. Consider the convergent iteration p0=0.5 ano Ph+1=e pk for k=0, 1 The first 10 terms are obtained by the calculations P1=e-0.500000.606531 72=e-060=0.545239 e-0.5452390.579703 0.566409 pg =e 0.567560 0.567560 P10 =0.566907
Theorem 1. 2 Assume that g E Cla, b If the range of the mapping y=9(x) satisfies y∈[,列 for all a∈[a,列,then g has a fixed point in[,列 (1.3) Furthermore, suppose that g ()is defined over(a, b)and that a positive constant K< 1 exists with lg (a)< K<1 for all E(a, b), then g has a (1.4) unique fixed point P in a, b
Example 1.3. Apply Theorem 1.2 to rigorously show that g(a)=cos(a )has a unique fixed point in 0, 1]
Theorem 1.3(Fixed-point Theorem). Assume that(i)g,g E ca, b1, (i)KI is a positive constant,i)∈(a,b),and(iv)9(x)∈a,列 for all a∈[a,列 If lg(a)s K1 for all x E [ a, b, then the iteration Pn 9(pn-1) will not con verge to P. In this case, P is said to be a repelling fixed point and the iter-(1.7) ation exhibits local divergence
Corollary 1. 1. Assume that g satisfies the hypothesis given in(1.6) of Theorem 1. 3. Bounds for the error involved when using pn to approximate P are given by P-pn|≤K|P- pol for all n≥1, (113) ane P-p|≤ KmP for all m≥1 (1.14