COALITION MATH 151, FALL 1997

        Group II (S. A. Fulling, assisted by Cary Lasher)

                              Day 9.1


STARTED BY FINISHING THE RELATED RATES PROBLEM THAT HAD BEEN LEFT 
HANGING ON DAY 8.1.  OTHER COMPLICATIONS WITH TEST AND CAPA.

Numerical integration:

    1.  One almost never evaluates integrals by directly taking 
    the limit (strip width -> 0) in the definition.  Instead, 
    either 

        A.  Find antiderivative and use fundamental theorem.

        B.  Approximate by a particular Riemann sum with small 
        strip width, or some better approximation to the strip 
        areas.  This is today's topic.

    2.  Sometimes one needs to run the fundamental theorem 
    backwards, to find antiderivatives numerically!  Necessary if

        A.  Don't know formula for antiderivative.

        B.  Know f itself only at isolated points (empirical data 
        or another computer calculation).

    3.  Midpoint and trapezoidal rules.  [Finish Maple demo.]
NEITHER CARY NOR I HAD TIME FOR THE MAPLE DEMO!
    Compare for convenience (trap) and accuracy (midpoint -- 
    state without details for now).
    
    4.  Geometrical interpretations of trapezoidal rule.
    
        A.  Area of trapezoids.  (Integrates linear functions 
        exactly.)  You should be able to remember or reconstruct 
        the trapezoidal rule on a test.

        B.  Midpoint-type Riemann sum with end strips of half 
        width.

        C.  Average of left and right sums (presumably better 
        than either).
GOT THIS FAR.

    5.  Simpson's rule.

        A.  The formula, and remark on efficient evaluation 
        (factor the constants out!).   [Do we need to memorize 
        this?]

        B.  Parabolic interpretation (without algebra details).
I GUIDED CLASS THROUGH THE CONSTRUCTION AND SOLUTION OF THE 3 
EQUATIONS IN 3 UNKNOWNS.

        C.  Exactly integrates quadratic -- and, by accident,
        cubic -- functions.
GOT THIS FAR ON DAY 9.2.

    6.  Error bounds

        A.  State the error bound formulas for midpoint, trap, 
        and Simpson rules.  (overwhelming)

        B.  How much of this junk is important?  What does it 
        mean?  In DECREASING order of importance:

            i.  h^2 vs. h^4:  For a better method, error goes 
            away faster as you decrease strip width.  [Do an 
            example where exact answer is known.]

            ii.  f'' vs. f'''':  Possible trouble if f is not 
            smooth.

            iii.  Numerical coefficients: Let you choose h to 
            guarantee error less than desired epsilon.
GOT THIS FAR ON DAY 9.3; REST OMITTED.

        C.  If function behaves differently in different regions,
        different h's may be needed to get both efficiency and 
        accuracy.

        D.  If time permits, point out Romberg's formula relating 
        Simpson to trap with two values of h (and similar formula 
        relating Simpson to trap and mid).  [Demonstrate with the 
        example from (i) above.]