http://interval.louisiana.edu/courses/556/556spring2001assignments.html
Math. 55601, Spring, 2001 Assignments
/ Assignments /
This list is updated as the assignments are made and exam dates are
set.
Home
page for the course;
R. Baker Kearfott

Due Friday, January 26:

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.)

Due Monday, February 19:

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 GaussSeidel 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.

Due Friday, March 2:

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

Due Monday, March 5:

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.

Due Monday, March 26: All of these problems are from the online
course.

Exercise
set 1 from the ODE book, 1.1a, 1.1h, 1.2 a.

Exercise
set 2 from the ODE book, 2.1f

Exercise
set 3 from the ODE book, 3.4c, 3.4g, and 3.7.

Due Monday, April 9:

Consider the matrix A formed by central difference discretization
(with the fivepoint 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.)
Page
one of suggested style for answers
Page
two of suggested style for answers
Page
three of suggested style for answers
Mathematica
notebook with computations for answer

Due Monday, April 23:

Program the method of characteristics for the model problem u_{xx}u_{tt}
= 0; u(x,0) = 1  x^{2} 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 threedimensional 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.

Due Wednesday, May 2:

Perform four predictor / corrector steps of arclength continuation,
with a Newton corrector step perpendicular to the predictor direction,
for the problem H(x,l) = x^{2}
l, starting with the point (1,1), and proceeding,
with predictor step size of 0.1, in the direction of increasing l.