1. During the quiz you may use your textbook, my notes, and your own notes. 2. No calculators or other electronic devices are permitted. 3. Please make sure your cell phones are quiet during class and off during quizzes. The quiz will cover iterative methods for solving linear systems, including SIMs, Krylov methods, and multigrid. See Chapter 28 of SCCS (or `Iterative Methods for Linear Systems: Following the Meandering Way,") and the notes on the website. For Quiz 5, be able to: -- Apply J, GS, or SOR to a given system of linear equations (algebraically or geometrically) and determine whether it converges. -- Construct a basis for a Krylov subspace and, in particular, contruct an orthogonal basis. -- Explain why the Krylov methods terminate in at most n iterations with the exact solution. -- Count the work per iteration for the Arnoldi algorithm or the cg algorithm. -- Determine the storage requirements for the Arnoldi algorithm or the cg algorithm. -- Use the convergence results for GMRES. Example: Show that if G-hat has only 5 distinct eigenvalues, then GMRES must terminate in at most 5 iterations with the true solution. Example: Show that if G-hat has 5 small clusters of eigenvalues, then after 5 iterations, GMRES produces a good approximate solution. -- Use the convergence results for CG. -- Implement a preconditioning algorithm such as Gauss-Seidel or SOR. -- Transfer values between grids in multigrid, given a restriction operator and an interpolation operator. -- Form a sequence of nested grids. -- Explain or use the V-cycle and nested grids algorithms.