Krylov and Direct Methods

Part A

Use the program
that I have provided (or build it from the source
code) to examine the convergence rates of several Krylov methods.
Select a 129x129 array of volumes for the conduction problem, and plot
the
logarithm of the maximum residual vs.iteration number for the 3
Krylov methods, and the SOR. You should tell the
program not to generate a direct solution. Accept the default
response to questions on the level of incomplete factorization and the
number of iterations before restart of GMRES. Compare run times
with
those you got for a similar conduction problem with the best
combination of options in the MultiGrid program mgsor. Note that
comparison of residuals from the MultiGrid and Krylov solvers is a bit
like comparing apples and oranges, because the matrices are different
as are the forms of the residual used in convergence tests. I
tryed to account for these differences when setting the convergence
criteria in the two programs.

Run all
methods 5 times to get a feeling for the range of run times.
Construct a table listing the method name, the minimum execution time
in your 5 runs, and the difference between maximum and minimum
execution times. Order your table from low to high execution
time. Try to make your computer runs with as few other
applications active as possible.

Part B

Use the program that I have provided (or build it from the source code) to look at behavior of direct solution for non-diagonally dominant matrices. Plot the log of maximum absolute error vs. coefficient below the diagonal for 100 equations and coefficient values from 0.0 to 3.0 in increments of 0.2. Plot the maximum residual for the same set of coefficients. Aside from incorrect answers, what do you see in the results that should disturb you.