Extra Credit – Math 104B

Due on Friday, June 12th, 2020

Instructor: Carlos J. García Cervera

1. (40 points) When discretizing the differential equation

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(e). (5 points) Use formula (9) and the spectral radius obtained earlier to show that

the method converges, and to estimate the number of iterations necessary for

the error to be less than a given  as a function of the number of grid points

used, n. You should end up with a formula of the form Iter(n) = O(n

α) for

some α. Hint: At the end you might need to use Taylor to estimate α.

(f). (5 points) To verify the value of α numerically, fix  = 10䛈6

. Consider an

initial random vector x

(0), and solve the system of equations

using Jacobi’s method. Use the values n = 10, 20, 40, 80, 160, 320. Do a

log-log plot of the number of Jacobi iterations necessary for the error to satisfy

How does the computed number of iterations compare with the theoretical one

obtained earlier?

(g). (5 points) Repeat the previous part with Gauss-Seidel’s method. How much

faster is it?

2. (20 points) In this problem, we analyze the Gauss-Seidel method in the same way

we studied Jacobi’s method above:

(a). (10 points) For each k = 1, 2, . . . , n, show that the vector u