Lecture20101014_2

Lecture20101014_2 - Numerical Analysis II Dr Abigail Wacher...

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Numerical Analysis II Dr Abigail Wacher 1 (1) O O O x n+1 = ax n +b f:= x->x; g:= x->2*x-2; plot([f,g],0. .5,colour = [red, blue]); f := x / g := / 2 K 2 1 2 3 4 5 K 2 0 2 4 6 8 If the sequence converges it's limit is the fixed point of the function g(x)=ax+b. This is the point of intersection of the line y=ax+b and y=x. Note : for a positive interger slope 0<a<1 the convergence will be monatonic. For negative then the convergence oscillates from both sides. g:= x-> cos(x) + 0.3*sin(2*x) + 0.2*sin(6*x); f:= x-> x; := / cos C 0.3 sin 2 C 0.2 sin 6 := / plot([f,g], 0. .1.5, colour=[red,blue]);
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Numerical Analysis II Dr Abigail Wacher 2 O 0 0.5 1 1.5 0 0.5 1 1.5 For this we get a spiral rather than a staircase (due to the negative slope) Suppose we want to find the zeros of f(x)=x 2 -2x-3 by fixed point iteration. (note zeros are x=-1,3) We want to solve x^2-2x-3=0 f:= x->x^2-2*x-3; plot(f,-2. .4); f := x / 2 K 2 K 3
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Numerical Analysis II Dr Abigail Wacher 3 K 2 K 1 0 1 2 3 4 K 4 K 3 K 2 K 1 1 2 3 4
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Lecture20101014_2 - Numerical Analysis II Dr Abigail Wacher...

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