WPI - Worcester Polytechnic Institute

 

MME 523

Analysis For Applications

Fall 2003

instructor:  John Goulet         (goulet@wpi.edu)

Kaven Hall, room 202

 

Purposes of course:

 

  1. study various numerical methods
  2. use Maple and Excel as numerical tools
  3. review analysis
  4. study π

 

Each class will:

    1. review mathematical basics as needed
    2. develop a numerical algorithm(s)
    3. implement them in Maple and Excel

Misc Files:

                        roundoff error   (9/9)

                       

 

 

Outline of Topics

 

     1. Root Finding

bisection, Newton’s Method

differentials

            here is the demo from class on Sept 2:  (excel file)

            Newton's Method in Maple  (.mws Maple file)

 

2. Integration                     

 Riemann Sums

                         Trapezoidal             follow this link for the first assignment  (9/23)

 Simpson’s Rule

                         Bell Curves; Improper Integrals  follow this link for second assignment (9/30)

 

3. Series and Sequences

            including power series

            follow this link for the assignment for October 7, 2003

            follow this link for the assignment for October 14, 2003

            great animation showing Taylor Series convergence!

 

4. Regression Analysis

          October 21 assignment and notes

            October 28 material

            November 4 assignment   and the products of that assignment

 

5.  The Hunt for Pi  Nov 11

          more stuff about Pi than you ever wanted to know!

 

6. Differential Equations (1st order)   Dec 2, 2003

 Euler’s Method

 

7.  Putting things together

                        Dec 9

 

Assignments:

 

            #1 for Sept 9, please

·        get an Academic computer account

·        bring a floppy disk

·        install Maple at home

·        read through Maple website looking for essential appropriate to your needs

        see especially  this link for both help and downloads

·        work on  Bisection and Newton’s  in both Excel and Maple

·        get a calculus book to serve as a reference (any one will do)

·        Maple capabilities needed

o       set up a function       f:=x ->  x^3 + 2;

o       assignment                x:=3;

o       loops                          while (b-a > e) do   stuff   od;

o       plotting                      plot( f(x), x = 2..5);

o       convert to decimal    evalf(%);

            #2  (Sept 16, for Sept 23)

                        follow this link.

 

                        A good reference on Newton’s Method is at this link.  

 

Now let’s try solving some problems!!  In each case, we would like to get the answer to 4 decimal places with  Bisection .  Please make a note of how many iterations in each it takes to achieve this. Let’s use Excel for Bisection and Maple for Newton’s.  We will work on Newton’s Method as well as Maple on 9/9 .

 

1)    x3 + 2x – 6 = 0 ; the root in   [1,2]

 

2)   x4 + 5x3 + 1 = 0  ; the root in [-1, 0 ]

 

3)    7x3 + x – 5 = 0   (the only root, please)

 

4)    the root of  cos(x) = 2x

 

5)   x – 2 + 2 ln(x) = 0

 

A question lurking in the background is this:  if you want 4 decimal place accuracy, how small does the

eventual range for the root have to be???  In other words, how do you translate “decimal places” into

“tolerance” ???