# f = x^2 - 16
f = x^2 - x -1
# exact = 4
exact = (1+sqrt(1+4))/2
# f = tan(x/4)-1
# exact = pi
(plot(f,(x,0,5))).show()
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# x0 = 1
# x1 = 5
x0 = 1
x1 = 3
points = plot(f,(x,0,x0))
points += point((x0,0),color='red',size=40)+point((x1,0),color='red',size=40)
show(points)
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p0 = x0
p1 = x1
errors = []
Lines = Graphics()
for k in range(6):
fx = f(x=p1)
m = n((fx-f(x=p0))/(p1-p0))
p0 = p1
p1 = p1 - fx/m
pretty_print(p1)
points += point((p1,0),color='red',size=40)
if p1<p0:
Lines += plot(fx+m*(x-p0),(x,p1,p0),color='green')
else:
Lines += plot(fx+m*(x-p0),(x,p0,p1),color='green')
errors.append(n(abs(p1-exact)/exact))
end
show(points+Lines)
pretty_print((p1,f(x=p1)))
pretty_print(' or ',(n(p1),n(f(x=p1))))
pretty_print(errors)
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pretty_print(' or ',(n(p1),n(f(x=p1))))
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Experimentally determine the rate of convergence
# Experiment with r to see if there is a (relatively) constant value for the following ratio of successive error terms
r = 1.62
for k in range(len(errors)-1):
errors[k+1]/errors[k]^r
0.427067837796310 0.919356447350661 0.540333683081218 0.754621891141515 0.632238125507041 0.427067837796310 0.919356447350661 0.540333683081218 0.754621891141515 0.632238125507041 |
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