Description
1. [15 pts] Use your Simpson’s 1/3 rule code from PS11 to compute R 1.8
1.0
[e
x + 0.5]dx with
h = 0.1 and h = 0.05. Now use one layer of Romberg integration on those results to
get a better answer. Finally, comment on how close each of the three numerical answers
involved in the calculations above — i.e., with h = 0.05, with h = 0.1, and with Romberg
— is to the correct value that you figured out using calculus in PS11.
2. [10 pts] Now compute R 1.8
1.0
[e
x+0.5]dx using three-term Gaussian quadrature. Comment on
the accuracy of the answer and the computational complexity of this approach, compared
to the methods that you have used to solve this same problem in this problem set and in
the previous one.
3. [6 pts] Consider the following system of ordinary differential equations:
x˙(t) = 16(y − x)
y˙(t) = 45x − y − xz
z˙(t) = xy − 4z
Is this system of equations—a famous example from chaos theory called the Lorenz
equations—linear? Explain your reasoning: why or why not? Of what order is this
ODE system?
4. [24 pts] In your favorite programming language, implement the forward Euler method for
solving that system of differential equations numerically.
Your program should take the following inputs:
• an initial condition [x(t = 0), y(t = 0), z(t = 0)]T
• a time step size h
• the number of time steps N over which to solve the ODE system
Generate two 10,000-point-long trajectories using h = 0.01 (that is, N = 10, 000), one
from the initial condition [x, y, z]
T = [1, 1, 1]T and the other from the initial condition
[x, y, z]
T = [1.01, 1.01, 1.01]T
.
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Make time-domain plots of the x coordinates (i.e., x versus t) for both trajectories—
preferably on the same set of axes so you can compare them easily. The initial conditions
are almost identical, but the time course of the x coordinate of the two trajectories should
diverge fairly quickly. That’s one of the main attributes of chaos.
Make state-space plots of both trajectories—preferably in 3D (x vs. y vs. z)—or just
y vs. x if your environment doesn’t support 3D plotting. These two state-space plots
should look pretty much identical to one another. That’s the other main attribute of
chaos.
Please turn in your plots and a copy of your solver code.
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