Homework due Wednesday 2/3
Please come see me or send me e-mail if you're
having any problems mathematically, computationally, or
otherwise.
Chapter 5
Problems 23, 26, 27, 32, 42, 43
For problem #42, you don't have to do exactly their example. But
choose a function f and a point x and make the analogue of Table
5.25.
Problem 1:
Find a quadrature rule for a triangle where the triangle
has vertices (-1,0), (1,0), and (0,1). The rule will use the function
values at the vertices and at the midpoints of the edges of the
triangle. Demand that your quadrature rule gets the integral exactly
for as high order a polynomial in x and y as you can. Now consider an
arbitrary triangle, with vertices (a_1,b_1), (a_2,b_2), and (a_3,b_3).
Figure out how to transform the quadrature rule you found to one that
applies for this triangle. Check that you did it right by integrating
f(x,y) = xy on your triangle and comparing your answer to what you get
from the transformed quadrature rule.
Problem 2:
On the integration web-page, there are two adaptive
integration programs, adaptive_trap.m and adaptive_simpsons.m Download
these and try them on the function f_eps(x) = 1/(x^2+eps), integrated
from -10 to 10.
First test the routines on a function whose integral you know
exactly. Present a table showing tol and error for various values of
tol.
Fix tol and eps and compare the trapezoidal
method to the simpsons method --- why does one need so many more
intervals than the other.
Study the effect of changing eps on n_adaptive (the
number of intervals used) and the ratio of n_uniform/n_adaptive (where
n_uniform is the number of intervals you'd have to use if they were
all the same length).
Study the effect of changing tol on the
number of intervals used. Plot the interval lengths (plot(h)) and
verify that the itnervals are small where they should be small.