% CLASS4OCT.TXT  Transcript of what was done in class on Monday 4 Oct for
%   Math 310.

% demonstrate Newton's method to solve   x = cos(x),  which means using a
%   fixed point iteration with   g(x) = x - (x-cos(x)) / (1+sin(x))
%   because  f(x) = x - cos(x)
format long
p=0.5
for n = 1:7, p = p - (p-cos(p)) / (1+sin(p)), end

% compare to a "naive" fixed point iteration which also solves  x = cos(x)
p=0.5
for n=1:15, p = cos(p), end

% some equivalent fixed point iterations which apparently *do not* converge
p=0.5
for n=1:5, p = acos(p), end

p=0.739
for n=1:10, p = acos(p), end

p=0.5
for n=1:5, p = (p/cos(p))+p-1, end

% and finally one that converges fast
p=0.5
for n=1:10, p = sqrt(cos(p))-sqrt(p)+p, end

% a picture showing why the last iteration converges; note that the slope at
%   the point  x=p, y=p  is small
figure
x=0.5:.001:1;   plot(x,x,x,sqrt(cos(x))-sqrt(x)+x),  axis([0.5 1 0.7 0.8])

% and a picture showing why Newton's method is such a good fixed point iteration
figure
x=0.5:.001:1;   plot(x,x,x,x-(x-cos(x))./(1+sin(x))), axis([0.5 1 0.73 0.745])