Review of Chapters 3, 4, and 5

The questions below are meant to stimulate the preparation of the final exam, concerning chapters 3, 4, and 5. They cover some of the most important but not all topics. You must review all course materials: textbook, homework exercises, lecture notes (also consult the course web site), and quizzes. Also please review the computer projects.

1. Consider the polynomial p(x) = x^2 + x - 5.

   1. Construct the Newton form of p(x) by divided differences, using the points (x_i,p(x_i)), with x_i = i, for i=0,1,2,3.

   2. Explain why the last element f_0123 in the table of divided differences you constructed above is (or should have been) zero.

   3. Apply Neville's algorithm to evaluate the interpolating polynomial at 0.5.

   4. Approximate p(x) with the linear function that minimizes the squares of the errors, using the points (x_i,p(x_i)), with x_i = i, for i=0,1,2,3.

2. The Maclaurin expansion of e^(x^2) is

             2
            x          2        4        6         8      10
           e   =  1 + x  + 1/2 x  + 1/6 x  + 1/24 x  + O(x  )
   
   

   Use this Maclaurin expansion to construct a Padé approximation for e^(x^2) as a quotient of two quadrics.

3. The answer to the previous question is

                                                 2
                                        1 + 1/2 x
                                   r =  ----------
                                                 2
                                        1 - 1/2 x
   

   1. Compute a continued-fraction representation of r(x).

   2. Count the number of arithmetical operations you need to evaluate the continued-fraction representation of r(x). Compare this number with the number of arithmetical operations needed to evaluate r(x) if you would use the Horner form of the numerator and denominator of r(x).

4. Explain why the formulas for Richardson extrapolation to approximate the derivative with forward differences are different from those that use central differences. In particular, why do we see only even powers of the ratio r in the extrapolation formulas that use central differences?

5. Consider a function whose values are tabulated below:

        +---------+----------------+
        |    x    |      f(x)      |
        ============================
        |  0.000  |  1.0000000000  |
        |  0.125  |  0.9921976672  |
        |  0.250  |  0.9689124217  |
        |  0.375  |  0.9305076219  |
        |  0.500  |  0.8775825619  |
        |  0.625  |  0.8109631195  |
        |  0.750  |  0.7316888689  |
        |  0.875  |  0.6409968582  |
        |  1.000  |  0.5403023059  |
        +---------+----------------+
   

   Suppose we are interested in the derivative of f(x) at x = 0.5.

   1. One way to compute the derivative would be to compute the interpolating polynomial through all these points and then to take its derivative for x = 0.5. Explain why numerically this is not such a good idea.

   2. Compute the most accurate approximation for f'(0.5). Estimate the accuracy in your answer.

6. Consider

                1
                /  x      
               |  e  cos(2 Pi x) dx    
               / 
                0
   

   1. Apply the composite trapezoidal rule using four subintervals of equal length to approximate this integral. Write your answer with six decimal places.

   2. Estimate the accuracy of the numerical approximation you just computed. How many decimal places can be correct?

   3. Apply Romberg integration to obtain a sixth-order approximation of the integral. Use six decimal places to denote the answers. What is the accuracy of your final answer?

7. Simpson's rule on an interval [a,b] approximates

                b
                / 
               |  f(x) dx 
               / 
                a
   
   

   by

             b - a  (           ( a+b )        )
            ------- ( f(a) + 4 f(-----) + f(b) )
               3    (           (  2  )        )
   

   1. Write a composite formula to integrate

                   b
                   /        
                  |  f(x) dx    
                  / 
                   a
      

      with Simpon's rule, using seven function evaluations.

   2. Give the formula for the general composite Simpson's rule, over n subintervals of [a,b], of length h = (b-a)/n.

8. Consider the approximation of

                2a
                /        
               |  f(x) dx    
               / 
                0
   

   by the rule w_1 f(0) + w_2 f(a/2) + w_3 f(a).

   1. Determine the weights w_1, w_2, and w_3 so that the rule has the highest possible algebraic degree of precision.

   2. What is the highest possible algebraic degree of precision we can reach with three function evaluations? Give the nonlinear system in the weights w_1, w_2, w_3 and abscisses x_1, x_2, x_3 to determine the quadrature rule w_1 f(x_1) + w_2 f(x_2) + w_3 f(x_3).

9. The Fourier series of a function f(t) appear in two different forms: either as

                         infinity
                         --------
                 1       \
        f(t) =  --- a   + >      a cos(2 Pi k t) + b sin(2 Pi k t)
                 2   0   /        k                 k
                         --------
                          k = 1
   

   or as

                infinity
                --------
                \           i 2 Pi k t
        f(t) =   >      c e
                /        k     
                --------
              k = -infinity
   

   Derive the relationship between the coeffients c_k and (a_k, b_k).

FINAL EXAM is in BH 0309 on Monday 29 April 2002, from 1 till 3PM.