[PDF] Clicker Question Bank for Numerical Analysis (Version 1.0 – May 14

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SIMON FRASER UNIVERSITY M. Alamgir Hossain

FACULTY OF SCIENCEJohn M. Stockie

DEPARTMENT OF MATHEMATICSClicker Question Bank for Numerical Analysis (Version 1.0 { May 14, 2020)This teaching resource (including L ATEX source, graphical images and Matlab code) is made available under the

Creative Commons \CC BY-NC-SA" license. This license allows anyone to reuse, revise, remix and redistribute the

databank of clicker questions provided that it is not for commercial purposes and that appropriate credit is given

to the original authors. For more information, visithttp://creativecommons.org/licenses/by-nc-sa/4.0.1. Introduction

Q1{1

1.Select the best denition for \numerical analysis":

(A) the study of round-o errors (B) the study of algorithms for computing appro ximatesolutions to problems from con tinuousmathematics (C)

the study of quan titativeappro ximationsto the solutions of mathematical pr oblemsincluding consider-

ation of and bounds for the errors involved (D) the branc hof m athematicsthat d ealswith the dev elopmentand use of n umericalmetho dsfor solving problems (E)

the branc hof mathematics dealing with metho dsfor obtaining appro ximaten umericalsolutions of math-

ematical problems Answer: (B). All 5 denitions are valid in some sense since they re ect some aspect of the eld (most are

pulled o the internet). But my favourite denition is (B) because it contains three very important keywords

underlined below: the study of algorithmsfor computing approximatesolutions to problems from continuousmathematics [ algorithms()computing, approximate() oating point arithmetic, continuous()solutions are smooth f'ns ] fSource: JMSg

1a. Floating Point Arithmetic and Error

Q1a{1

2.How many signicant digits does the

oating point number 0:03140103have? (A) 6 (B) 5 (C) 4 (D) 3

Answer: (C).

Q1a{2

3.Suppose that a hypothetical binary computer stores

oating point numbers in 16-bit words as shown:1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 s exp mantissaClicker Question Bank for Numerical Analysis (Version 1.0) 1/91

Bit 1 is used for the sign of the number, bit 2 for the sign of the exponent, bits 3-4 for the magnitude of the

exponent, and the remaining twelve bits for the magnitude of the mantissa. What is machine epsilon for this

computer? (A) 2 16 (B) 2 12 (C) 2 8 (D) 2 4 Answer: (B). Assume that rounding is used and recall that"Mis essentially the same as unit round-o erroru=12 B1t, whereB= 2is the base andtis the number of signicant digits. The number of digits stored in the mantissa ist= 12and so"M12

2112= 212.

Q1a{3

4.You are working with a hypothetical binary computer that stores integers as unsigned 4-bit words. What

is the largest non-negative integer that can be represented on this computer? (A) 64
(B) 63
(C) 31
(D) 15 (E) 7

Answer: (D).(1111)2= 120+ 121+ 122+ 123= 15.

Q1a{4

5.In 1958 the Russians developed a ternary (base-3) computer calledSetun, after the Setun River that

ows

near Moscow State University where it was built. In contrast with today's binary computers, this machine

used \trits" (ternary bits) whose three possible states can be represented asf0;1;2g. Its oating-point number

system was based on 27-trit numbers, with 9 trits reserved for the exponent and 18 for the mantissa. What

was the value of machine epsilonMfor theSetun? (A) 3 19 (B) 3 18 (C) 3 9 (D) 13 218

Answer: (B).

Apply the formulaM=Btfrom the notes, where

B= 3is the base andt= 18is the number dig-

its in the mantissa. You may have noticed that I didn't mention a \sign trit" for the mantissa. In actual fact, the oating-point representation onSe- tunwas more complicated than this and the sign of a number came from interpreting one specic trit as f1;0;+1ginstead.Setun { Moscow State University fSource: JMS, plus info fromhttp://homepage.divms.uiowa.edu/~jones/ternary/numbers.shtmlg Q1a{5

6.In Canada, the total for any store purchase paid in cash is rounded to the nearest 5 cents, whereas no

rounding is done if the payment is by credit/debit card. Suppose that when you return home after purchasing

your groceries with cash, you notice that your bill was $10:07. What is the absolute error in your actual cash

payment?Clicker Question Bank for Numerical Analysis (Version 1.0) 2/91 (A)2 cen ts (B)

3 cen ts

(C)

4 cen ts

(D)

5 cen ts

Answer: (A).

Q1a{6

7.Let ^xbe some approximation ofx. Which of the following error denitions is correct?

(A) absolute error = jx^xj, relative error =jx^xjjxj (B) absolute error = jx^xjjxj, relative error =jx^xj (C) absolute error = jx^xjjxj;x6= 0, relative error =jx^xj (D) absolute error = jx^xj, relative error =jx^xjjxj;x6= 0

Answer: (D).

Q1a{7

8.For a base-10 (decimal)

oating point numberxhavingtsignicant digits, the relative error satises R x=jxf`(x)jjxj6u whereudenotes unit round-o error. Which of the following is true aboutu? (A)u=(

101t;chopping

12

101t;rounding

(B)u=( 12

101t;chopping

10

1t;rounding

(C)u=( 12

101t;rounding

10

1t;chopping

(D)u=(

101t;rounding

12

101t;chopping

Answer: (A).

Q1a{8

9.Fill in the blank:Iff(x) is a real-valued function of a real variable, then theerror in the

dierence approximation for the derivativef0(x)f(x+h)f(x)h goes to zero ash!0. (A) absolute (B) relativ e (C) cancellation (D) truncation

Answer: (D). Strictly, response (A) is also correct since truncation error is an (absolute) dierence from

the exact derivative. Q1a{9

10.The two solutions of the quadratic equationax2+bx+c= 0 given by

x=bpb

24ac2a

are computed using

oating point arithmetic. Which of the statements below is TRUE?Clicker Question Bank for Numerical Analysis (Version 1.0) 3/91

(A)F orsome v aluesof th eco ecients,this form ulacan ge neratecancellation errors. (B) If the co ecientsa,bandcare very small or very large, thenb2or 4acmay over ow or under ow. (C)

The expression x=2cbpb

24acis an alternative formula forxthat avoids truncation error.

(D)

All of the ab ove.

Answer: (D).

Q1a{10

11.In oating-point arithmetic, which of the following operations on two positive oating-point numbers can produce an over ow? (A) addition (B) subtraction (C) m ultiplication (D) division

Answer: (A). But (C) and (D) are also valid responses. Letxbe the largest number that can be represented.

Then the operationsx+ 1:0,x2:0andx0:3all generate an over ow. fSource: Heath [4], Review Question 1.29, p. 40g

Q1a{11

12.In oating-point arithmetic, which of the following operations on two positive oating-point numbers can produce an under ow? (A) addition (B) subtraction (C) m ultiplication (D) division

Answer: (C). But (D) is also a valid response. Letxbe the smallest positive number that can be represented.

Then the operationsx0:5andx2:3both generate an under ow. fSource: Heath [4], Review Question 1.30, p. 40g

Q1a{12

13.Letfxkgbe a decreasing sequence of positive numbers withxk+1< xkfork= 1;2;:::. In what order

should the sum NX k=1x kbe computed so as to minimize round-o error? (A)

Order the xkfrom largest to smallest (1;2;:::;N).

(B)

Order the xkfrom smallest to largest (N;:::;2;1).

(C)

Sum the terms in random order.

(D)

It do esn'tmatter.

Answer: (B).

fSource: Heath [4], adapted from Review Question 1.45, p. 41g

Q1a{13

14.True or False:If two real numbers can be represented exactly as

oating-point numbers, then the result of a real arithmetic operation on them can also be represented exactly as a oating-point number. Answer: FALSE. As a counterexample, letx1="M(machine epsilon) andx2= 2, which are exact in any other binary oating point system (like the IEEE standard). Thenx1=x2has no oating point representation. fSource: Heath [4], Review Question 1.7, p. 39g

Q1a{14

15.Below are four

oating point approximations, each accompanied by its corresponding exact value. Which approximation is the most accurate? (A)

315700, exact v alue315690 Clicker Question Bank for Numerical Analysis (Version 1.0) 4/91

(B)0 :0005500, exact value 0:0005510 (C)

8 :7362105, exact value 8:7743105

(D)"M(machine epsilon), exact value 0

Answer: (A). Accuracy is measured either by counting signicant digits or computing relative error. Answer

(A) has the most signicant digits of accuracy (4 after rounding), whereas choices (B) and (C) have 2 and

3 signicant digits. The accuracy of the answer from (D) can't be compared because relative error formula is

undened when the exact answer is zero.quotesdbs_dbs9.pdfusesText_15

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