Description
Using the matrices from last assignment (or any others you find interesting),
perform the following study:
1. Generate a right-hand-side b of all ones of appropriate size.
2. Solve Ax = b with a generic linear solver (eg, numpy.linalg.solve
or Matlab’s backslash). Call the resulting vector truth. This is the
vector against which you will compute the error. Run a timing study
with the generic linear solver.
3. Write a function that solves Ax = b using either the LU decomposition or the Cholesky factorization, depending on whether the matrix is
symmetric or not.
4. Write a function that solves Ax = b using the Jacobi method. Run a
timing study with your function.
5. Write a function that solves Ax = b using the Gauss-Seidel method.
Run a timing study with your function.
For parts 3-5, report the relative error compared to the truth that you computed in part 2.
Here are some questions that might help you think about this problem.
• Look up “diagonally dominant.” It relates to the convergence of the
iterative methods.
• Remember your paramedic training. Which methods work better for
which matrices? Why do you think that is? And what do you mean
by “work better”?
• Try interpreting the results from the iterative methods in light of the
theory we know about fixed point methods.
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