CSC 210

Week 1



  1. Binary digits: Determine the binary representation of the decimal number 9.6. Compute the first 8 binary digits of the famous number sqrt(3). Please show your computational steps.

  2. Is faster better?: Consider the map x1 -> 0.375*x1 + 1.5/x1 - 0.5/(x1*x1*x1). Iterate this map (derived by Alkalsadi around 1450AD) with the starting value 2.0 until 15 digits settle down. How many iterations are required? Now, count the number of all the additions, subtractions, multiplications and divisions you have made to obtain your approximation to sqrt(2).
    Next do the same with Newton's method using x1 ->0.5*(x1 + 2.0/x1).
    Which method requires fewer number of iterations?
    Which method requires fewer number of arithmetical operations to compute sqrt(2) to the same precision?

  3. Newton's cubic: Using the modern version of Newton-Raphson method described in the lectures, find an approximation to a root of Newton's cubic polynomial y^3 - 2y -5 = 0 starting with the initial guesses y_0 = 2.0, and also y_0 = 1.4. How many iterations do you have to make to get Newton's finding 2.09455148? (See the Newton's original paper above.) How good is this answer as an approximate root? To to this problem, you first need to determine which function to iterate; then enter this function into Phaser as a Custom Equation (MAP). Does Newton's polynomial have any other real root? (Hint: Using a function plotter, plot the graph of the cubic polynomial.)

  4. 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?