MATH 640 – Numerical Analysis
Homework # 2: Iterative Techniques
DUE: Friday, March 5th, 2004
1. For each of the iterative methods we’ve discussed:
(a) Jacobi,
(b) Gauss-Seidel,
(c) SOR,
(d) Conjugate Gradient Method
create an m-file (function) that, given the input of
(a) a matrix, A,
(b) a vector, b,
(c) an initial guess vector, x(0)
(d) an ωvalue (for SOR only)
(e) a tolerance, tol, and
(f) a maximum number of iterations,
will output two things if sucessful:
(a) the approximate solution, ˆ
x, to the problem Ax=b, and
(b) the number of iterations used to find ˆ
x
else the appropriate error statement if it is not successful. Turn in a copy of these documented
m-files.
2. Use the first three (not including Conjugate Gradient) MATLAB functions to approximate the
solution to our room temperature problem from Homework #1
Amat ∗u=bvec
using the upwinding scheme for m=n= 4,8,16,32, a tolerance of 10−4and a maximum of 15
iterations. For the SOR method, use at least two different values of ω. Create a table comparing
the methods used (specify ωvalues with the SOR), and the number of iterations required to solve
each size problem.
3. Solve
Amat ∗u=bvec
from Homework #1 again, using the SOR method, but with each of the following sets of velocity
values:
(a) Case 1:vx = 3, vy = 3 (see above)
(b) Case 2:vx =25
ln((x−0)2+ (y−7.5)2+e)−22
ln((x−25)2+e), vy =25
ln((y−12)2+e)
(c) Case 3:vx =y−12.5, vy =−(x−12)
(d) Case 4:vx = 3 cos(xy), vy = 3 sin(xy)
Give the graphs for m=n= 8,16 for each case.
