cs249 Homework 1 Solutions

1.1
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1)

>> absoluteError = 22/7 - pi

absoluteError =    0.0013

>> relativeError = absoluteError / pi

relativeError =   4.0250e-04  =  .0402 %


2)

a) .800 + .333 = 1.33

absoluteError =   0.00333333333333

relativeError =   0.00294117647059


b) 0.8 * 0.333 = 0.266

absoluteError =   6.666666666666488e-04

relativeError =   0.00250000000000


c) .333 - .273 = .060

   .060 + .150 = .210

absoluteError =   6.060606060606100e-04

relativeError =   0.00287769784173


d) .333 - .273 = .060

   .060 - .150 = -.090

absoluteError =   6.060606060606100e-04

relativeError =  -0.00677966101695


The difference between (c) and (d) is the difference between addition
and subtraction.  Subtracting two similar values tends to increase
relative errors by "losing" significant digits.


1.2
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1)  1 1 2 3 5 8 13 21 34 55 89

    F(10) = 89

2 and 3) Here is the content of fibonacci1.m

t1 = 1 / sqrt (5);
t2 = (1 + sqrt (5)) / 2;
t3 = (1 - sqrt (5)) / 2;
f = t1 * (t2^(n+1) - t3^(n+1))

>> n = 10;
>> fibonacci1

f =  89.00000000000003

>> n=100;
>> fibonacci1

f =  5.731478440138190e+20



1.3
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1. I couldn't find a value of x that yielded different results.

2. This one is easy to break to varying degrees:

>> x = 1e-6; 1 - sin(x)^2 - cos(x)^2

ans =    1.110223024625157e-16

>> x = pi/2; 1 - sin(x)^2 - cos(x)^2

ans =    -3.749399456654644e-33


3. This one is harder to break.

> x = 1e-6; sinh(x/2) - sqrt (1/2 * (cosh(x) - 1))

ans =    -2.222465162926600e-11


4. I couldn't break this one.