Math 801 Problems

• HWK 1 Due Sep 26: Problem 2 and 5 in the list below

• HWK 2 Due Oct 31: Write your own simple versions of GMRES and QMR as discussed in class using matlab's built-in least squares solver (\). Your codes should be defined as matlab functions. Test the algorithm on some random matrices. Compare the convergence of these 2 algorithms as well as CG applied to the normal equations A^H A x = A^H b for reasonably big matrices. ( see the demos ).

• HWK 3 Due Nov 14: Modify the CG function given in the demos to work with and without preconditioning (using "nargin"). The preconditioner must be of the form LL^H and your code should call two successive triangular solves to solve Mz=r: (1) Ly=r, (2) L^H z=y. Invent one or more test problem(s) consisting of Poisson's equation on a square discretized using centered finite differences (matlab can generate the grid and the matrix A, see "numgrid", "delsq" or "del2"). Determine how many iterations are necessary for convergence, as a function of the linear dimension. Try preconditioning using incomplete Cholesky (see "cholinc").

• HWK 4: Due R Nov 16: Problem 11 below.

1. Plot the curve 5 y⁵ -y⁶ -x ⁴=0 . Explain your solution strategy (don't simply plug in a smart math package).
2. Resonant traces
3. Derive the equation governing the motion of a bead on a circular hoop rotating about a vertical axis through the center of the loop. Include gravity and damping proportional to the velocity. Nondimensionalize your equation and rewrite as a first order system. Find all the equilibria of the system.
4. Determine the rate of convergence of the secant method for a nonlinear equation with one variable i.e. x_k+1 = x_k -f_k/d_k where d_k =(f_k-f_k-1)/(x_k -x_k-1) .
5. Implement Newton's method and the secant method (i.e. Broyden's update) for the 2 by 2 problem: x² + y²= 2, e^(x-1)+y³=2. Compare the convergence of both methods starting from a few (e.g. 3) initial guesses.
6. Show that the constrained minimization problem used to define the updated Jacobian in the secant method for more than one variable may have multiple solutions if the 2-norm is used instead of the Frobenius norm (construct a 2 by 2 example).
7. Prove that the Jacobi iteration to solve Ax=b converges if A is diagonally dominant.
8. Modify the steepest descent and conjugate gradients algorithms given in the demos page so they can be used for any nonsingular matrix A by applying them to the system A^H A x = A^H b . Construct a 2 by 2 example as in the demos (but now non-symmetric) and compare the 2 algorithms.
9. Show that the biorthogonal Lanczos process requires only two coupled 3-term recurrences.
10. Find the eigenvalues and eigenvector of the (smart) centered difference discretization of -u''=f with u'=0 at x=0 and u=0 at x=L.
11. Find the eigenvalues and eigenvectors of the Gauss-Seidel iteration matrix for the centered finite difference discretization of -u''=f with u=0 at x=0,L. (Hint: the discrete PDE has constant coefficients in both x and t so Fourier analysis can be used, i.e. look for solution of the form exp(-i omega t) exp(ikx) or directly lambdaⁿ mu^j. Find lambda and mu to satisfy the discrete equations and the boundary conditions). Compare to the eigenvalues of the Jacobi iteration matrix.