- Binary digits: Determine the binary representation of the decimal number 9.4.
Compute the first 8 binary digits of the famous number sqrt(2). Please show your
- Can you beat Newton?
Enter the difference equation x1 -> x1 - 0.21*(x1*x1 - 2)
into PHASER. Iterate the initial condition x0 =2.24 and observe that
it approaches to sqrt(2).
Compare the rate of convergence of this method of
computing sqrt(2) with that of Newton's method x1-> 0.5*(x1 + 2/x1).
To make a fair comparison, use the same initial condition
in both iterations. How many iterations are required to get 13 correct digits
of sqrt(2) in each iteration?
- Small angle approximation:
In physics, it is often convenient to use the 'small angle approximation' by
replacing sin(x) with x when x is small. Use Newton-Raphson method
to find a value of x which satisfies the equation sin(x) - x = 0.016.
How many such x can you find? Draw the stair-step diagram of the iteration
process. How many fixed points do you see?
- Trouble with Newton:
Using Newton's method, with starting value x = 1, try to find
a root of the function f(x) = -x^4 + 3x^2 + 2.
Does the iteration converge? Can you explain what is going wrong?
Can you find other initial conditions that converge to a root?
How many (real) roots can you find?
- Your favorite computer trouble:
Write a short report (half a page?) about your favorite, or scariest,
hardware, software, security, etc. problem. Please indicate your sources.