18.01 Section, September 16, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Fixed point iteration and Newton’s method
1. Activity about fixed points:
(a) Start with the line y = 12 x + 1. Using high school algebra, find the fixed point.
(b) Now try approximating the fixed point using the fixed point iteration method with an
initial guess of x0 = 1; calculate x1 , x2 , and x3 .
(c) Make a plot of “error vs. time”: i.e. plot points (n, errorn ) where
errorn = (true fixed point found in (a)) − xn .
(d) Do the above three parts again, for the line y = 2x − 1 and initial guess x0 = 0. Does
the initial guess matter for the general shape of the plot?
1
(e) In each case, apply the criterion from class that tells you when a fixed point is attracting
or not.
(f) Start with a linear equation of the form f (x) = α + m(x − α).
(i) What’s the fixed point?
(ii) Write error1 in terms of error0 .
(iii) Write error2 in terms of error1 , and then write error2 in terms of error0 .
(iv) Write error3 in terms of error0 .
(v) Write errorn in terms of error0 .
(vi) How does this explain all the behavior you observed above?
2. f (x) = 2x − 23 has a fixed point at x = −1. Start with an initial guess of x0 = 2. Does the
sequence x0 , x1 , . . . approach x = −1? (You don’t have to calculate exactly; just convince
yourself one way or the other.) It turns out that f 0 (−1) ≈ 0.347. Why is this not a violation
of things said in class about attracting points?
2
3. Use Newton’s method to approximate
√
3. Start with x0 = 1 and calculate x1 , x2 , and x3 .
4. Bonus question: Show that Newton’s method is a special case of fixed point iteration.
Use this to check that Newton’s method converges near the roots of a quadratic equation
(x − a)(x − b).
Review
• Derivative of a polynomial:
d
n
dx ax
= anxn−1
• Linear approximation of f near a:
f (x) ≈ f (a) + f 0 (a)(x − a)
• Newton’s method:
xn+1 =
−f (xn )
+ xn
f 0 (xn )
• Fixed point iteration method: initial guess x0 is given; then x1 = f (x0 ), x2 = f (x1 ), x3 = f (x2 ),
etc.
• Attracting point: fixed point a such that |f 0 (a)| < 1
• Repelling point: fixed point a such that |f 0 (a)| > 1
3