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 decomposi-tion 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 com-puted 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.
1
