MME 523

October 14, 2003

Numerical Explorations

 

 

Now that you have some experience with numerical methods, we can try out some different problems, more “on your own” than in the past.

 

Problem One            Quadratic Equations

 

Everyone knows the quadratic equation:  the solutions to   ax2 + bc + c = 0   are given by

 

                                   

(exercise: can you derive this???)

 

so if we have any quadratic, unless there are imaginary or complex roots, this should be pretty easy.

 

Let’s limit things a little for sake of discussion.  Suppose  b> 0.  Also that  b2  is a lot greater than | 4ac|  (which is usually written  b2 >> |4ac|   )

 

As an example,  form the quadratic  which has one root as 100,000  and the other as  .002.  Use Maple to generate a quadratic with these as roots. Then

·        use the quadratic equation above to determine the positive root

·        note the error

 

Now lets use a little algebra to improve things.  Take the solution for the positive root,

 

                                   

and multiply top and bottom by the conjugate of the numerator,  ,  and simplify.  Use this equivalent equation to generate a new solution to the same problem and compare results.

 

Now…why was the second result more accurate???

 

Generate 3 other examples illustrating the comparison.

 

Problem 2:   Integrals

 

We know from earlier experience that the Trapezoidal Rule provides pretty good results.  We also know that Taylor polynomials can do a good job approximating functions.  So the natural question for inquiring minds is:

 

            how good of an approximation is the integral of a Taylor polynomial?

 

In other words if  P(x)  is the Taylor polynomial for f(x)  then how close of an approximation is

 

                           to             ?  

 

To explore this we need specific problems.

 

1)  Consider  

                                   

How many terms in the Taylor polynomial about a=0  do we need to get 2 decimal place accuracy in the approximation obtained by integrating  P(x) ?  How about 3??

 

2) compared to what you got in problem 1, how do things change if we change the limits to be

 

                                             ?

  More specifically, how many terms do you now need for 2 and 3 decimal places??

 

Problem 3:  the hunt for p.  Part One – off the porch.

 

Have Maple and Taylor generate polynomials for  Arctan(x) about a=0.  Note the pattern of terms.

 

Now let’s do it by hand:

 

            a)  write the basic geometric series formula down. Use x as your variable (instead of the usual r)

 

            b)  replace x with –x

 

            c) now replace x with x2

 

            d) now antidifferentiate both sides

 

Setting x=1  you then have a method of approximating  p. Take 3 different orders. How good is each??

(Use Maple for this work…)