CSC 210

Week 1



  1. 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 computational steps.

  2. 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?

  3. 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?

  4. 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?

  5. 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.