MCS 471 Lecture Thirty two

Review for Exam II

The questions below are meant to stimulate the preparation of the exam. 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.

Exam 2 in Spring 2005:

Exam 2 in Fall 2003:

Exam 2 in Spring 2002:

You may use a formula sheet: postscript version , and version in pdf format .

Show that

               n
              ---
      p(x) =  >     l (x) f
              ---    i     i
              i=0

with

               n    x - x
              ---        j
      l (x) = | |   -----
       i      j=0   x - x
              j/=i   i   j

satisfies


      p(x ) = f    for i = 0,1,..,n.
         i     i

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

The Maclaurin expansion of tan(x) is

                       1   3    2   5    17   7     62   9      10
         tan(x) = x + --- x  + --- x  + ---- x  + ----- x  + O(x  )
                       3        15       315       2835

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

Consider
```
            2
           x  + 3 x - 1
   r(x) = --------------
              2
             x   -  1
```
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).
1. Explain why (f(x+h) - f(x))/h is a first-order approximation for f'(x).
2. Explain why (f(x+h) - f(x-h))/(2h) is a second-order approximation for f'(x).
1. Use forward differences to compute a first-order approximation for the third derivative of f(x).
2. Use central differences to compute a second-order approximation for the third derivative of f(x).
1. Derive the formulas for Richardson extrapolation to compute the first derivative of f(x) with forward differences.
2. Derive the formulas for Richardson extrapolation to compute the first derivative of f(x) with central differences.
Write pseudo-code for Richardson extrapolation to compute a table with n rows using one of the formulas you derived above.
Consider
```
                1
                /  x
           I = |  e  dx
               /
                0
```
Write your answers to the questions below with six decimal places.
1. Apply the composite trapezoidal rule with 4 subintervals to approximate I.
2. Apply Romberg integration to obtain a sixth-order approximation of I.
3. How many decimal places are correct in the final answer from Romberg integration? Explain.

The midpoint rule for

             b
             /                        ( a+b )
            |  f(x) dx    is  (b-a) f (-----)
            /                         (  2  )
             a

Give the formula for the composite midpoint rule that applies the midpoint rule to n equally sized subintervals of [a,b].

Simpson's rule on an interval [a,b] approximates
```
       b
       /           b-a 
       | f(x) dx = --- ( f(a) + 4*f((a+b)/2) + f(b) )
       /            6 
       a
```
1. Write a composite formula to integrate f(x) over [a,b], with Simpson'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 (b-a)/n.
Consider the approximation of
```
             3a
             /   
            |   f(x) dx
            /
             a 
```
by the rule w_1 f(a) + w_2 f(2a) + w_3 f(3a).
1. Determine the weights w_1, w_2, and w_3 so that the rule has the highest possible algebraic degree of precision.
2. Can we reach a higher algebraic degree of precision to approximate the integral? Explain how. What is the highest possible algebraic degree of precision we can reach with three function evaluations?

Please note the policy on skipping exams:

If an exam is missed, then greater weight will be placed on the final exam, especially on the material covered on the missing exam.

What this means is that if you decide not to take one midterm exam, your final exam will be weighted for one hundred points more.

What it does NOT mean is that you can drop the score of a midterm exam. If you take the midterm, then your score counts. So, please be prepared when you show up for the exam.