AMATH242/CS371 Spring 2021 Assignment 1
Instructor: Maryam Ghasemi
Due date: May 25, 11:59 pm via Crowdmark
1. [5 marks] Consider a fictitious floating point number system F [3, 3, e] corresponding to base 3,
three digits for the normalized mantissa, and −1 ≤ e ≤ 1, that uses chopping when storing real
numbers. Nonzero numbers have the form:±0.d1d2d3 × 3e. For each of the following questions,
provide a brief explanation.
(a) What is the largest positive normalized value that can be stored in F?
(b) What is the smallest positive normalized value that can be stored in F?
(c) What is the value of machine, i.e., the smallest value x such that fl(1 + x) > fl(1) in F?
(d) How many different nonzero numbers (positive and negative) can be stored in F?
(e) Give concrete examples of real values (using base 10) that will cause overflow (over the
maximum value) and underflow (below the minimum value) when converted to the floating
point number system F .
2. [3 marks] Let x = 2. Consider evaluating
z =
√
x2 + 1
using the floating point number system F as defined in Question 1. (Note that x is a floating
point number in F .)
(a) Use a calculator to compute z and then show that the real number z is smaller than the
largest floating point number in F . Thus, both x and z are within range of representable
floating point numbers.
(b) If one uses the floating point number system F to compute z using the formula above,
what would happen to the floating point representation for x2? Derive another formula for
z such that such an error would not occur. Briefly explain how your formula works.
(c) Use the floating point number system F to compute z using the formula derived in (b).
Show all your calculations.
3. [3 marks] Suppose two points (x0, y0) and (x1, y1) are on a straight line, where y0 and y1 are
different. Two formulas are available to find the x-intercept of the line:
x =
x0y1 − x1y0
y1 − y0 and x = x0 −
(x1 − x0)y0
y1 − y0
(a) Show that both formulas are algebraically equivalent.
(b) Using the points (x0, y0) = (1.31, 3.24) and (x1, y1) = (1.93, 4.76) and three-digit rounding
arithmetic, compute the x-intercept both ways. Show your workings, and show where there
is a loss of information due to rounding in your calculations.
(c) Which method is better and why? Will it better for all values of (x0, y0) and (x1, y1)?
Explain your answer.
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4. [11 marks] Consider the following problem:
The sum of two numbers is 20. Add each number to its square root, and multiply together. The
product of the two sums is approximately 155.55.
(a) Define a function f(x) such that finding a root of f(x) solves the above problem. You may
assume x is a number between 1 and 9, inclusive.
(b) Write Matlab code to find a root of your function f(x)=0 using the Newton’s method.
Continue the iterations until |f(x)| ≤ 10−6. Include a table of the iterate values, when
starting from x0 = 1 and x0 = 5.
(c) Without implementing the bisection algorithm:
(i) Prove that [1,9] is an appropriate initial bracket for a root of f.
(ii) Determine the approximate number of iterations required by the bisection algorithm
to find the interval [ak, bk] of length≤ 10−6.
5. [8 marks] Moler’s Numerical Computing with Matlab website
(http://www.mathworks.com/moler/ncmfilelist.html) includes many useful Matlab functions,
including powersin.m (which can be downloaded):
1 f unc t i on s = powers in ( x )
2 %POWERSIN Power s e r i e s f o r s i n ( x ) .
3 % y = POWERSIN( x ) t r i e s to compute s i n ( x ) from i t s power s e r i e s .
4 %
5 % Copyright 2014 Cleve Moler
6 % Copyright 2014 The MathWorks , Inc .
7
8 s = 0 ;
9 t = x ;
10 n = 1 ;
11 whi le s+t ˜= s ;
12 s = s + t ;
13 t = =x . ˆ 2 / ( ( n+1)*(n+2) ) .* t ;
14 n = n + 2 ;
15 end
This function computes an approximation to using its power series expansion (which is infinite):
sin(x) =
∑∞
n=1(−1)n+1 x
2n−1
(2n−1)!
(a) The while loop terminates when s+t does not differ from s. Is this the same as continuing
while t is not 0?
(b) Modify powersin so that it returns three values corresponding to the calculation of the
sum s; function [s, maxterm, numterms] = powersin(x) where maxterm is the largest
term (value of t) calculated, and numterms is the number of terms of the power series used
to calculate sum s (note that the first term, x, is calculated before the loop begins). Include
a listing of the modified powersin function with your submitted solution.
(c) Call the new version of powersin for x = pi/2, 11pi/2, 21pi/2, and 31pi/2. For each value of
x, show the final values of s, maxterm, and numterms, and calculate the absolute value of
the relative error in the computed sum for each x.
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(d) Based on your results, what can you conclude about the use of the powersin algorithm for
various values of x? Does the distance of x from 0 affect the results?
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