05 Fixed-Point Iteration

Convert root finding problems into iteration of recurrence relations.Apply theorems about the existence and uniqueness of fixed points and the convergence of recurrence relations.Understand the proofs of the above theorems.

Explore the recurrence relation NiMvJkkieEc2IjYjLCZJIm5HRiYiIiJGKkYqKihJJ2xhbWJkYUdGJkYqJkYlNiNGKUYqLCZGKkYqRi0hIiJGKg== with (1) NiMvSSdsYW1iZGFHNiItSSZGbG9hdEdJKnByb3RlY3RlZEdGKDYkIiM7ISIi and NiMvJkkieEc2IjYjIiIhLUkmRmxvYXRHSSpwcm90ZWN0ZWRHRis2JCIiIiEiIg==, convergence to 0.375(2) NiMvSSdsYW1iZGFHNiItSSZGbG9hdEdJKnByb3RlY3RlZEdGKDYkIiM7ISIi and NiMvJkkieEc2IjYjIiIhLUkmRmxvYXRHSSpwcm90ZWN0ZWRHRis2JCIkdiQhIiQ=, stay at 0.375(3) NiMvSSdsYW1iZGFHNiItSSZGbG9hdEdJKnByb3RlY3RlZEdGKDYkIiNRISIi and NiMvJkkieEc2IjYjIiIhLUkmRmxvYXRHSSpwcm90ZWN0ZWRHRis2JCIiIiEiIg==, chaos (4) NiMvSSdsYW1iZGFHNiItSSZGbG9hdEdJKnByb3RlY3RlZEdGKDYkIiNRISIi and NiMvJkkieEc2IjYjIiIhKiYiIzkiIiIiIz4hIiI=, stay at 14/19 unless there is rounding errorHow is this recurrence relation related to the equation NiMvSSJ4RzYiKihJJ2xhbWJkYUdGJSIiIkYkRigsJkYoRihGJCEiIkYo?lambda:='lambda': solve(x=lambda*x*(1-x),x);g:=x->lambda*x*(1-x); lambda:=3.8: plot({g(x),x},x=0..1);x[0]:=0.1; x[1]:=1.6*x[0]*(1-x[0]); x[2]:=1.6*x[1]*(1-x[1]); g:=x->lambda*x*(1-x); lambda:=38/10: x[0]:=14/19; for n from 0 to 14 do x[n+1]:=g(x[n]); od; points:=[seq([n,x[n]],n=0..15)]: plot(points,style=point);

Definition. NiNJInhHNiI= is a fixed point of NiNJImdHNiI= if NiMvSSJ4RzYiLUkiZ0dGJTYjRiQ= .To find a recurrence relation that will give you the root of the equation NiMvLUkiZkc2IjYjSSJ4R0YmIiIh, write the equation in the form NiMvSSJ4RzYiLUkiZ0dGJTYjRiQ=. Then iterate using the relation NiMvJkkieEc2IjYjLCZJIm5HRiYiIiJGKkYqLUkiZ0dGJjYjJkYlNiNGKQ==.Example 1. NiMvLUkiZkc2IjYjSSJ4R0YmLCYqJiIiIyIiIiokRihGK0YsRixGKCEiIg==Example 2. NiMvLUkiZkc2IjYjSSJ4R0YmLCgqJEYoIiIkIiIiKiYiIiVGLCokRigiIiNGLEYsIiM1ISIig:=x->2*x*(1-x); x[0]:=0.1; for n from 0 to 14 do x[n+1]:=g(x[n]); od;Bisection: need 29 iterations.solve((1/2)^(iterations+1)=1E-9,iterations);

NiMkIismR04oKilHISIp

g:=x->-x^3-4*x^2+x+10: #nonconvergence x[0]:=1.5; for n from 0 to 14 do x[n+1]:=g(x[n]); od;g:=x->sqrt(10/x-4*x): #convergence to a complex root x[0]:=1.5; for n from 0 to 14 do x[n+1]:=g(x[n]); od;g:=x->(1/2)*sqrt(10-x^3): #convergence in 29 iterations x[0]:=1.5; for n from 0 to 40 do x[n+1]:=g(x[n]); od;g:=x->sqrt(10/(x+4)): #convergence in 11 iterations x[0]:=1.5; for n from 0 to 40 do x[n+1]:=g(x[n]); od;g:=x->x-(x^3+4*x^2-10)/(3*x^2+8*x): #convergence in 4 iterations x[0]:=1.5; for n from 0 to 11 do x[n+1]:=g(x[n]); od;

NiM+JkkieEc2IjYjIiIhJCIjOiEiIg==NiM+JkkieEc2IjYjIiIiJCIrTExMdDghIio=NiM+JkkieEc2IjYjIiIjJCIrOj9FbDghIio=NiM+JkkieEc2IjYjIiIkJCIrOStCbDghIio=NiM+JkkieEc2IjYjIiIlJCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiImJCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiInJCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiIoJCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiIpJCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiIqJCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiM1JCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiM2JCIrOCtCbDghIio=NiM+JkkieEc2IjYjIiM3JCIrOCtCbDghIio=

g:=x->sqrt(10/(x+4)): plot(g(x),x=1..2); plot(D(g)(x),x=1..2);

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See pictures of text book pages.g:=x->(1/2)*sqrt(10-x^3): plot(g(x),x=1..1.5); D(g)(x); plot(D(g)(x),x=1..1.5);

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NiMsJComSSJ4RzYiIiIjLCYiIzUiIiIqJEYlIiIkISIiI0YtRicjISIkIiIl

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

Exercise Set 2.2 #4 (10 points), 8 (15 points), 16 (10 points), 10 with no comparison or 14 (5 points), 20a (10 points), 20b (5 points), 24 (10 bonus points).