Math. 556-01, Spring, 2001 Assignments

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This list is updated as the assignments are made and exam dates are set. Home page for the course;
R. Baker Kearfott

Do no.'s 1, 3, and 13, p. 189ff of the Neumaier manuscript.
If N(f,x,x) represents a step of a univariate interval Newton method for f over the interval x and with guess point x, then show that, if N(f,x,x) is contained in x, then there is a solution of f(x) = 0 in x, then show that this solution is unique. (Hint: show that there must be a sign change of f between the end points of x. For uniqueness, use that fact that f' must be nonzero everywhere in x. See me if you have any questions.)

Do no. 1 on page 220 of the Neumaier manuscript. You may program it either in Matlab or Fortran. Take note of all (if any or if more than one) solutions you obtain.

Program the interval Newton method with interval Gauss--Seidel iteration, and try it on this nonlinear problem, with initial boxes of various sizes, some of which contain a unique solution, some of which contain no solutions, and some of which contain more than one solution (if there is more than one solution). Explain your results.

Show that each of the formulas (39), (40) and (41) in §2.4.5 of the on-line text reduces to the same formulas as in Algorithm 2.2 of §2.4.3 when the gradient is g(x) = Ax + b.

This is a team project that all the members of the class should do together. I would like one report handed in, but each class member should sign it. Each class member will receive the same grade for this project, and that grade can be anything from 0 to 100%: In part 1 of the assignment from February 19, people implemented the damped Newton method differently, with some implementing Neumaier's "conservative" scheme and others implementing the less conservative scheme. Some implemented it in single precision and some implemented it in double precision. Both MATLAB and Fortran 95 were used. With the less conservative step control, two people observed successful convergence to a solution from starting points at which the undamped Newton method did not converge. The assignment is to explain the differences. The report should discuss each of the eight programs that were handed in, and specifically explain the results.

Consider the matrix A formed by central difference discretization (with the five-point star) on the domain [0,1] × [0,1] of L(u) = uxx+uyy. Assuming there are n interior nodes in the x direction and n interior nodes in the y direction and n interior nodes in the y direction, express the ratio of the largest eigenvalue to the smallest eigenvalue of A as a function of n. (Hint: Related material can be found in Isaacson and Keller, Analysis of Numerical Methods, Wiley, 1966.)

Program the method of characteristics for the model problem u_xx-u_tt = 0; u(x,0) = 1 - x² for -1 < x < 1 and u(x,0) = 0 for all other values of x; u_t(x,0) = 2x for -1 < x < 1 and u_t(x,0) =0 for all other values of x. Take the initial data over the range -2 <x < 2, and take the initial distance between points to be h = 0.1, that is, take the initial points to be (-2,0), (-1.9,0), ..., (1.9,0), (2,0). Advance the program with the method of characteristics from the initial curve for ten steps. (Note: with the initial mesh, this will correspond to t = 0.5; also note that the bounds on x on the final curve will be different from those on the initial mesh.) Plot the solutions on each curve. The general way of doing the plots is by plotting a three-dimensional surface. However, for this particular model problem, the curves upon which the approximate solution values are obtained happen to be lines of constant t, so the solution can also be easily exhibited as a series of eleven plots of u versus x.

Perform four predictor / corrector steps of arclength continuation, with a Newton corrector step perpendicular to the predictor direction, for the problem H(x,l) = x²- l, starting with the point (1,1), and proceeding, with predictor step size of 0.1, in the direction of increasing l.