University of Waterloo
Faculty of Engineering
Department of Electrical and Computer Engineering
ECE 204A
Pre-Laboratory 5
Prepared by
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2A Electrical/Computer Engineering
8 February 2016
5.1.1a For each row in Table 1, convert the given numbers to the other two formats.
Table 1. Binary, decimal, and hexadecimal numbers.
Binary
10001
Decimal
Hexadecimal
?
?
?
23
?
?
101010
?
?
?
?
2B
65
?
?
?
AA
5.1.1b Add the following hexadecimal numbers without converting them to decimal first.
A3825
+ 1F539
?
38925
+ 93295
?
ACFEB
+ CDAEF
?
5.1.2a If the first hexadecimal digit of a double-precision floating-point number is A, is
that number positive or negative?
5.1.2.1a The sign bit indicating a positive number is
0 or 1.
5.1.2.2a If the first three hexadecimal digits of a double are 3E7, what is the
corresponding power of 2?
Your answer.
5.1.2.2a If the first three hexadecimal digits of a double are C03, what is the
corresponding power of 2?
Your answer.
5.1.2.3a Why is it unnecessary to store the leading 1?
Your explanation here.
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5.1.2.4a One justification for denormalized numbers is that for two doubles, x and y, the
difference x – y is never zero unless x = y. Confirm this by seeing the output of the
following code:
>> format hex
>> 5e-300
>> 5e-300 - 5.00000000001e-300
Copy and paste your Matlab commands and the output here.
5.1.2.4b Another justification for denormalized numbers is that it would be absurd to
have a double x on the order of 2-1023 to be stored with 53 bits of precision, but to then
have x/2 equal zero. The cost, however, is that real numbers less than 2-1022 are no
longer stored with 53 bits of precision. For example, how many bits of precision are used
to store 10-320 (be sure to use format hex)?
Your answer here.
5.1.2.5a Fill in the second row of Table 2 with NaN or Inf.
Caclulation
Result
Table 2. Various Matalb calculations with infinity.
Inf - Inf Inf + Inf Inf/0
Inf/Inf
?
?
?
?
Inf^2
?
5.1.2.5b Is it ever possible to perform a calculation with Inf and get a value that is
neither infinity nor not-a-number? If yes, give an example.
Your answer here.
5.1.2.6a Which of the following doubles (represented using hexadecimal digits) is larger?
52832e8d36bf5eed or 52832e8d86bf5eed
Your answer here.
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5.1.2.7a Consider the following:
>> M = randn( 3 )
M =
2.7694
0.7254
-1.3499
-0.0631
3.0349
0.7147
>> M >= 0
ans =
1
1
0
0
0
0
1
1
1
-0.2050
-0.1241
1.4897
Use this to devise a test on a matrix M to determine
1. whether or not all entries in the matrix are positive, and
2. whether or not there is at least one negative entry (< 0) in the matrix.
Copy and paste your Matlab commands here.
5.1.2.8a Create a while loop that finds the first integer k such that 1/k < /1000.
Copy and paste your Matlab commands and the output here.
5.2.1a The exact value of x2 + 1574359x – 1.912846 when x = 0.000001215 is
0.000000185001476225. What is the relative error of Matlab’s approximation when you
calculate
>> x = 0.000001215;
>> x^2 + x*1574359 - 1.912846
Copy and paste your Matlab commands and the output here.
5.2.2a What is the Newton polynomial that passes through the three points (3.2, 0.5),
(5.4, 2.7) and (8.3, 3.6)?
? + ?(x – 3.2) + ?(x – 3.2)(x – 5.4)
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5.2.2b Why does the Newton polynomial in Question 5.2.2a not have a term of the form
(x – 3.2)(x – 5.4)(x – 8.3)?
Your answer here.
5.2.3a Save the program newtonpoly and use it to find the Newton polynomial passing
through:
1.
2.
3.
4.
The point (3.2, 4.9);
The points (3.2, 4.9) and (5.3, 8.1);
The points (3.2, 4.9), (5.3, 8.1) and (6.4, 7.0); and
The points (3.2, 4.9), (5.3, 8.1), (6.4, 7.0) and (10.5, 2.8).
Copy and paste your Matlab commands and the output here.
5.2.4a Write a Matlab function that implements Horner’s rule as described in Section
5.2.4.
function [y] = horners( a, x )
% enter your implementation here
end
Your implementation must, in addition to evaluating the polynomial defined by a at the
point x, allow the user to evaluate the polynomial at each point of a vector or matrix.
Your function must have the following output:
>> format long
>> horners( [1 2 3], 3.2 )
ans =
19.640000000000001
>> horners( [1 2 3], 3.5 )
ans =
22.250000000000000
Your function should also be able to perform the following:
>> format
>> horners( [1 2 3], 0:0.1:1 )
ans =
3.00 3.21 3.44 3.69 3.96 4.25 4.56 4.89 5.24 5.61 6.00
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If we define xs to be 100 equally spaced points on the interval [ –3, 3], we can also plot
the polynomial on that interval:
xs = linspace( -3, 3, 100 );
ys = horners( [1 2 3], xs );
plot( xs, ys );
ylim( [min( ys ) - 1, max( ys ) + 1] );
title('The polynomial x^2 + 2x + 3 plotted on the interval [-3, 3]');
The output of this sequence of commands should be a plot similar to Figure 1.
Figure 1. The output of Matlab when plotting a polynomial.
You will modify this function in the laboratory.
5.2.4b What is the purpose of the statement
ylim( [min( ys ) - 1, max( ys ) + 1] );
?
Your answer here.
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