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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "MCS 471 Project One: Numer
ical Evaluation of Rational Expressions" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 309 "The goal of this project is to investiga
te the numerical evaluation of rational expressions.  For this project
 we will use Maple 7 (although older versions of Maple will probably w
ork as well) because one interesting feature of Maple is that we can s
et the working precision to any arbitrary number of digits." }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 30 "0. Random Rational Expressions" }}
{PARA 0 "" 0 "" {TEXT -1 170 "We will generate a quotient of two rando
m polynomials.  The command randpoly in Maple generates a polynomial o
f degree five with integer coefficients between -99 and +99." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "randomize():        # random
ize seed" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p := randpoly(x
);   # numerator" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,.*$)%\"xG\"
\"&\"\"\"!#S*&\"\"$F*)F(\"\"%F*!\"\"*&\"\"'F*)F(F-F*F0*&\"#;F*)F(\"\"#
F*F**&\"#uF*F(F*F0F7F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "q
 := randpoly(x);   # denominator" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"qG,.*$)%\"xG\"\"&\"\"\"\"#O*&\"\"%F*)F(F-F*F**&\"#'*F*)F(\"\"$F*F**&
\"#GF*)F(\"\"#F*!\"\"*&\"#?F*F(F*F7\"#JF7" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 51 "r := p/q: rf := unapply(r,x);   # rational function
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rfGf*6#%\"xG6\"6$%)operatorG%&a
rrowGF(*&,.*$)9$\"\"&\"\"\"!#S*&\"\"$F2)F0\"\"%F2!\"\"*&\"\"'F2)F0F5F2
F8*&\"#;F2)F0\"\"#F2F2*&\"#uF2F0F2F8F?F8F2,.F.\"#O*&F7F2F6F2F2*&\"#'*F
2F;F2F2*&\"#GF2F>F2F8*&\"#?F2F0F2F8\"#JF8F8F(F(F(" }}}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 30 "1. Evaluation at Random Points" }}{PARA 0 "" 0 
"" {TEXT -1 119 "We evaluate the rational function at random points, u
sing floating-point arithmetic.  We compare with the exact answer." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Digits := 4;     # working p
recision is four digits" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG
\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x0 := stats[random
,normald](1);  # random point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0
G$\"%V5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "rfx0 := rf(x
0);  # approximate function evaluation" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%rfx0G$!%.;!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "
x1 := convert(x0,'rational');  # rational representation of point" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G#\"#C\"#B" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "rfx1 := rf(x1);  # exact function evaluation" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rfx1G#!*5T,$y\"*$=O\"*[" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "err := abs(rfx0-rfx1);  # er
ror" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$errG$\"\"#!\"$" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 15 "Assignment One:" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 345 "Repeat the calculation above for increasing values for
 Digits.  Make a table with the error between the approximate and exac
t function evaluation for these values of Digits.   Take enough values
 (at least four) for Digits until you see the general relation between
 precision (number of digits) and the error for arbitrary large values
 of Digits." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "2. Evaluation a
t a Special Points" }}{PARA 0 "" 0 "" {TEXT -1 100 "Now we generate a \+
special point and compare again the approximate function value with th
e exact one." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Consider the follo
wing experiment:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "Digits := 4;   # working precision is again four" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"\"%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 37 "x0 := fsolve(q,x);  # a special point" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"%8\")!\"%" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 65 "x1 := convert(x0,'rational');  # rational r
epresentation of point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G#\"#I
\"#P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "rf(x1); # exact ari
thmetic" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"+M,h`[\"(no[(" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 15 "Assignment Two:" }}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 297 "Repeat the experiment above for increasing values for
 Digits.  Make a table for values for rf(x1) for all values for Digits
 you choose.  Again, take enough values (at least four) for Digits unt
il you see a general pattern emerging.  What happens to rf(x1) as you \+
let Digits grow larger and larger?" }}}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 23 "3. Sensitivity Analysis" }}{PARA 0 "" 0 "" {TEXT -1 360 "
In the first experiment we took a random point and compared the result
 obtained with floating-point arithmetic with the result obtained with
 exact arithmetic.  In the second experiment we took the rational repr
esentation of a special point and computed its exact function evaluati
on.  We repeated this experiment with increasing precision of the spec
ial point." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Assignment Three:" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 318 "In the experiments above we kno
w both the random and the special point only up to a limited number of
 digits.  So in both cases there is an error of magnitude 10^(-Digits)
 on the points.  What is the influence of such an error on the functio
n evaluation in these points?  In particular, answer the following que
stions:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "1. \+
Given an error of 10^(-8) on a random point, what is the error on the \+
function value?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 66 "    Do an experiment with enough digits to illustrate your answ
er." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "2. Give
n an error of 10^(-8) on a special point, what is the error on the fun
ction value?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
66 "    Do an experiment with enough digits to illustrate your answer.
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "3. Is the
re a difference in your answer between the random and special point?  \+
Explain why there is or why there is no difference." }{MPLTEXT 1 0 0 "
" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "4. Deadline is Friday 25 J
anuary, at 1PM" }}{PARA 0 "" 0 "" {TEXT -1 71 "Bring your project solu
tion to class.  It should contain the following:" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 70 "1. The tables with numerical values for the experime
nts you have done." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 103 "2. Answers to the questions in the assignments.  Please \+
write complete grammatically correct sentences." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 60 "3. A print out of the Maple worksheet with your expe
riments." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "If
 you have questions, comments, or difficulties, feel free to come to m
y office for help." }{MPLTEXT 1 0 0 "" }}}}}{MARK "5 2 0 0" 17 }
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