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Unformatted text preview: O1 RYERSON UNIVERSITY DEPARTMENT OF MATHEMATICS
MTH 510 — Numerical Analysis
MIDTERM  WINTER 2011 WQIKQE )3 Last Name (print):
First Name (print):
Student ID Number:
Signature: Date: March 7, 2011, 1:15 pm INSTRUCTIONS: Time Allowed: 90 minutes Verify that the test contains all 9 pages, including this cover page. Use a. pen or pencil and write legibly for full marks. The examination has two parts: Part A consists of full
solution questions with the mark for each full solution
as indicated. Answer all questions in the space provided.
Clearly explain your methods, and show all relevant steps
in the solution. An answer consisting only of the ﬁnal result
will be given little or no credit. Part B contains Multiple Choice questions. Clearly
write your answer in the space provided. No part marks
will be given and no marks will be deducted for incorrect
answers. If more than one answer is given, a mark of zero
will be assigned to that question. This is a closed—book test. When not speciﬁed, six signiﬁcant digits of accuracy is
sufﬁcient. The last page is for rough work.
DO NOT SEPARATE THE PAGES Permitted Aids:
1) One handwritten 8.5 x 11 inch Formula sheet (both sides),
ii) Nonprogrammable scientiﬁc calculators. For instructor’s use only. Question(s)  Value Mark] MTH 510  MIDTERM  Page 2  WINTER 2011 1. (a) (4 marks) Express the decimal number 13.8125 as a binary fraction. (b) (kmaﬂcs) What is the largest positive number in a computer that uses a total of 16
bits to store numbers? Assume that the length of the mantissa is 7 bits. (1&0 eég‘SMKkt/aay‘5(
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5303(893‘388‘1 K O MTH 510 ‘ MIDTERM — Page 3  WINTER 2011 2. (6 marks) Do two iterations of the NewtonRaphson method with 9:0 2})?0 approxi—
mate one of the roots of f(2:) = 43inzr+ Inx. Calculate the value of tea! at each iteration. QKV" ﬁrsinx {ILAX ‘ '
PDQ; AFLOSX + J): “T 1, XL mg) . gm) \eoJ 0 m mme __ b‘i __ _ m
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( .E?‘+>=55WT */ “00% ‘1’ 91.3%
3 104% I a 0&4” alWioHS, ><,. c: X&$,&54’336?’+R MTH 510 — MIDTERM .  Page 4  WINTER 2011 3. (8 marks) (a) Do three iterations of the Bisection method to estimate the root of
ﬁre) = x3 — er
on the interval [1.5, 2}. What is the maximum true error in your approximation of the root? (b) How many iterations would be required for the true error to be less than 1045? .. , A4 E W’(
(M A x;‘ X“ c; m pm PM V1 [SLY .10668‘10? 339$); .6lo‘14zclo;
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h> m(;~(,Y\—'U¥\UO )é48ﬂ MTH 510  MIDTERM 4. Consider the matrix A given by (a) (6 marks) Find the LU decomposition of A = LU the space below 100
«AtO
5'3‘3/Irl L:  Page 5  andU: mm“ 7 (“39“ng ﬂaw {ls/0+“
/ WINTER 2011 Uin RLO oax
00/‘2r (b) (4 marks) Use the LU decomposition of A in (a), to solve the system Ax = b where b=[237}T. MTH 510  MIDTERM  Page 6 — WINTER 2011 5. Use naive Gaussian elimination with backward substitution to solve the system 11:1 + 2:132 — $3 =
2171+£E2 = 3, I2+£II3 I 51 ~! g 1 O 3 ‘9 O *3 3x
O 1 I ’01 O l {I
1 <1 4 l 0 W3 A l o 0 % 473 “gig x! :2 [w £1>g+><3
[+3v"! v
V {a Is 30 m SM‘A MTH 510  MIDTERM  Page 7  WINTER 2011 6. (5 marks) Use Jacobi iteration to solve 2,
1,
4. $1 + 25172 + 2:53
2.731 + 41132 + 173 H [I ll 3:1 *1? “333 D0 ONE iteration with initial guess [1 2 3?, and give the absolute value of the approximate
relative error. ..
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>43: L U" “I X3) I A~axyk~ﬁ‘_%3) VléﬂrHV'M WE Elk/33%))
:1‘4“6 :,. z«SZB ; :J “/ ~ 8 L513 j’m MTH 510 — MIDTERM  Page 8  WINTER 2011 Part B  Multiple Choice Questions 7. (2 marks) Consider the matrix 2 4 —1
A z —1 1 0
4 0 4
Which of the following statements is true:
(A) llAlll = 7,
(B) HAHl = 5,
(C) llAlh : 8, (D) None of the above. ANSWER: 6. A 8. (2 marks) Let r = linspace(0, 1). To plot y : :rsina: over this range in 2:, which of the
following commands will work: (A) ploth,x*sin(x));
(B) plot(x,@(x) x.*sin(x));
(C) plot(x,@(x) x*sin(x));
(D) None of the above ANSWER: 7. 1) 9. (2 marks) The following commands have been entered in MATLAB: k=2;
a=[1 2 3]; The result of the command
a.‘2+k
is (A) [3 4 9]
(B) [3 6 11]
(C) ??? Error using ==> mpower
Inputs must be a scalar and a square matrix.
(D) None of the above ANSWER: s. L 10. (2 marks) Typing the commands x=linspace(1 ,3 ,4);
y=3*x gives the following result for y
(A) 1/ = l3 6 9}
(B) y z [3 4‘5 6 7.5 9}
(C) y : [3 5 7 9}
(D) None of the above C
ANSWER: 9. ' ...
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This note was uploaded on 03/07/2012 for the course MTH MTH510 taught by Professor Dr.silvanailie during the Winter '12 term at Ryerson.
 Winter '12
 Dr.SilvanaIlie
 Numerical Analysis

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