[hem, vecka 1, vecka 2, vecka 3, vecka 4, vecka 5, vecka 6, vecka 7]

ALA-A, studio 5.1, 2006

The fixed point algorithm, 2.

1. We know that the fixed point iteration does not always converge. Therefore it is important that your program stops when the iteration diverges. Put code in your program fixpoint.m so that the program stops after 1000 iterations and exits with x=[] and displays the message

"The fixed point iteration did not converge after 1000 iterations."

2. Try to solve the equation x²-x-2=0 by rewriting it as a fixed point equation in the following four ways:

(a) x=x²-2

(b) x=(x+2)^1/2

(c) x=2/x+1

(d) x=(x²+2)/(2x-1)

In each case try to estimate the Lipschitz constant as

L_g=max |g'(x)|

where the maximum is taken over a suitable interval. Is L_g<1?

3. Download the interactive program fixpunktsdemo.m and use it to experiment with the fixed point algorithm. Note that you change between the different functions g by uncommenting the appropriate alternative (alt1, alt2, alt3, alt4) in the program code. You must change in two places: in the plotting and in the iteration.

4. (Advanced) Read the program fixpunktsdemo.m and learn how it works.

/stig