Jeffrey Hellrung
Friday, November 23, 2005
Math 269A, Assignment 08
1. Consider the task of solving the fixed point problem x=G(x) using fixed point iteration, e.g., xn+1 =
G(xn) for n= 0,1,2,....
(a) Give conditions on Gand the initial iterate x0that will insure that the fixed point iteration
converges.
(b) Assume one is using Backward Euler to solve the differential equation dy/dt =f(y). If the implicit
equation that arises is solved using fixed point iteration, what is the restriction on the timestep
that will ensure that the iteration converges?
(c) Assume one is using the trapezoidal method to solve the differential equation dy/dt =f(y). If
the implicit equation that arises is solved using fixed point iteration, what is the restriction on
the timestep that will ensure that the iteration converges?
(d) What conclusion can you draw about the utility of using implicit ODE methods for stiff systems
when the implicit equations that arise are solved using fixed point iteration?
Solution
(a) Let x∗be a fixed point of G(supposing one exists), i.e., G(x∗) = x∗. Then fixed point iteration
converges if Gis a contraction mapping within a neighborhood of x∗and the initial iterate x0lies
in this neighborhood. Explicitly, we want
|xn+1 −x∗| ≤ α|xn−x∗|
for some 0 < α < 1. In this case, we have
¯
¯
¯
¯
G(xn)−G(x∗)
xn−x∗¯
¯
¯
¯
≤α < 1,
and by the Mean Value Theorem, the difference quotient is the value of |G′|somewhere between
xnand x∗. It follows that if G′is bounded in absolute value by some α < 1 in some neighborhood
of x∗, then if the initial iterate x0lies in this neighborhood, fixed point iteration will converge,
i.e., xn→x∗.
Indeed, if |G′|< α uniformly, then there exists a unique fixed point, and fixed point iteration will
converge for any initial iterate.
(b) The Backward Euler method takes the form
yk+1 =yk+hf(yk+1),
hence we wish to solve
x=G(x) = yk+hf(x).
By the comments above, this will have a unique fixed point, and fixed point iteration will converge
for any initial iterate, if
|G′(x)|=h¯
¯
¯
¯
∂f
∂y (x)¯
¯
¯
¯
≤α < 1
for all x∈R.
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