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Unformatted text preview: RYERSON UNIVERSITY
DEPARTMENT or MATHEMATICS
MTH 510  Numerical Analysis FINAL EXAMINATION  WINTER 201 1
Last Name (print): 9 E )MS First Name (print):
Student ID Number:
Signature: Date: April 21, 2011, 11:30 am Time Allowed: 3 hours INSTRUCTIONS: For instructor’s use only. Verify that the test contains all 13 pages, including this cover page. Use a pen or pencil and write leginy 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 closedbook test. When not speciﬁed, six signiﬁcant digits of accuracy is
sufficient. The last page is for rough work, and also contains
a Table of Values for Quadrature Rules. Permitted Aids: i) Two handwritten 8.5 x 11 inch Formula sheets (both
sides), ii) Nonprogrammable scientiﬁc calculators. MTH 510  FINAL  Page 2  WINTER 2011
1. (3 marks) Consider the system
4x1+z2+2m3 = I
$1+5$2 == 2
21+ 22:2  102:3 == —1.
Do two iterations of Ga Seidel iteration starting with an initial guess of [1 0.5 CIT.
Compute the approximatgélative error at each iteration.
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aﬂme HE; 3315; Email wC‘Hﬂ awn“, MTH 510  FINAL '2. (8 marks) Use Gauﬂan elimination with partial pivoting and backward substitution Page3 to solve the system of equations 3 A” a «l /&
l ~—> o t «A ¢
1 RS: ( 40 3
4 1 wt 4 {\4— a 4 «V
o/{ a (90% mil/a
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000 $0M 4:9 He &%M .3 $1“! / 0] WINTER 2011 MTH 510  FINAL — Page 4  WINTER 2011 3. (.0 marks) Determine the solution of the simuitaneous nonlinear equations me”—x+y—1 = 0,
e”2:c=0. Do one iteration of the Newtoanaphson method using the initial guess 2 = 1 and y = 1. “‘6
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1" I “ =0 '3‘ e343; /\/‘\)
‘ o %°a£° 2%? , h (633%? {WWW MTH 510  FINAL  Page 5  WINTER 2011 4. (12 marks) Find the natural cubic spline to ﬁt the data {(—2, —— 1), (0, 1), (2, 3), (3,0 l8». XI:_3‘ FQ:I‘ ‘hg: X3VX(:3\ XA;O ‘Cg; 143‘: X3'X_\= A >953" {3 53 kgsayxsr; £453 £43313 (EMA x : (3,9. t 111. Cw \ ‘3 XA,)(‘ 0+3 )( 2; L "g 3“ C a.
3 a] ﬁg; 3 7,0 l ¥\:)<4'\5(31: (Ci“Cg :"(8‘:__3 : ~Q‘ V X’r">‘3 ' ~—~——————~ \ O O O ‘ O
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O O O 1 Cr 0 \ O O 0 Ct O ._ O m a 8 a O R , (Z i 0 a (o \ Cg "09 O o O l Cw O C :0 :0 I . :
Qﬁaﬂéngacga O 0 (9% C§°"4C°~ ';@(0:
QCA+QDC3+ﬂ2 élCal (SIKA ~. @ hex ‘Qﬁxmxal— bfmcﬁcs);(~33(;1(3)la):5‘
0 1% :mwgw bg(acwﬂ) = Wm ~‘;(awa))= ~Is mu: ' O
, v w?) \ .3. ~.. J—
C d" gig? " 3(a\ “ A
a: 1 = T‘R‘i =~%
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(arm/trap 619d 5fUAJL v2 —(2<+M + ‘51 (XHP‘ mum @m: wa) mum} rr/Qéwﬂ xeCOM 3 '(3 (X'&) ﬂak (chnl +lﬂ )(v3\3 ‘ XECQ‘SB lad C3wa MTH 510  FINAL  Page 6 — WINTER 2011 5. marks) Consider the following data: {(07 1)$(2! 2)) (3v “2), (4, 1)} (a) Using a Table for Divided Differences, ﬁnd the Newton interpolating polynomial for
the data. Do not simplify the expression, and leave it in factored form. (b) Write down, but do not simplify, the term of the Lagrange Interpolating polynomial
that has f(4) as a. factor. «A
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Ava 9x A“ MTH 510  FINAL  Page 7  WINTER 2011
6. (8 marks) Consider the integral 3
[[23 +4COS$I dam; A” (8) Use Simpmn’s 1/3 rule with n = 4 to estimate the integral.
A— ‘ (b) How many terms are required to estimate the integral using Simpson’s 1/3 rule with
an accurac of 10””? ‘
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MM V\ QUED" Y5 SO 5W“:‘EE>J MTH 510  FINAL  Page 8 — WINTER 2011 [ég’gi’bd Do two iterations of Romberg integration. After each iteration, calculate the approximate
percent relative error. 7. (IO marks) Consider the integral m \: Xi mm mm» 1quu¢b<g + #093]
z 4— 633416“ + «16 + Reiij
>° ﬁflaAHBI’rS $ MTH 510  FINAL  Page 9 — WINTER 2011 8. (6 marks) Use the fourthorder RungeKutta method (RK4) with step size h = 0.1,
to solve Estimateyatt=0.h
to:o‘(ao:;\)htoq} G) igl‘déolgo}; 4Com: [email protected]
KA: H6014: \ go wtgtﬂzz (2(a0§’9+,_$_ (.013) = H.053 M )
: 6605—) A '4’ {oﬁ'q'
= v 9.0%? Q k3 : Ha +m ) L50 +§g \ : may, a+.0§(a.0‘1?5/)
=¢(.os, LMmﬂ
1" £40013” @ (+3 «C(hﬁr’lﬂ )%0+le’\)
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3 1077 00003108 \ \ MTH 510  FINAL  Page 10  WINTER 2011 9. (6‘ marks) Use the Euler method with h = 0.2 to solve im = x+&+3ct 1; 'pUcpékké‘ dt
dz; 3 = 3+ w 5%(‘5Muﬂ z(1)=0, y(1) = 1. What are the approximate values for J:(1.4) and y(1.4)?
XL“: ){L'HA QCJLLQY‘M‘N)
3m C ‘3;+ \A‘gQnglgg) X&:>(‘+)A~CC£HXH%,)
== 0J0 +Qa) Hm, 0.61 {KM
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: z.a+g9&)ob(l.sé.a,¢o&)
= \.5b “MW Ask” Mr, 130.4) A aka/Xe MTH 51o — FINAL — Page 11  WINTER 2011 Part B  Multiple Choice Questions 10. (2 marks) Let 6 denote the machine epsilon in 3—digit rounding. Which of the following
statements is true: (B)(§+§)+§=e
((3)1F6 == l'6 (D) None of the above statements are true.
ANSWER: 10. 2 2 11. (2 marks) The following commands have been entered in Matlab:
clear;
A [2 3; 4 1] ;
3 [3:5] 3
The result of the command xahtb
leads to (A) An error message
(B) A value for x equivalent to x=[21; 7] in Matlab
(C) A value for x equivalent to x==[26 4] in Matlab (D) None of the above.
ANSWER: 11. 12. (2 marks) The following commands have been entered in Matlab: x=[1 2 3 7] 3
y=[ 2 7 1 7];
zzspline(x,y,3); Which of the following statements is true: (A) z is the natural cubic spline for the data stored in x and y, (B) 2 is a clamped cubic spline for the data stored in x and y with zero derivatives at
the end nodes, (C) Zzl,
(D) None of the above statements are true. C
ANSWER: 12. 13. (2 marks) Consider the following matrix A and its inverse A“: 124 ~11 4 [41:330. A”: $553.45
2 427 3—3115 Assuming that the matrix A has been properly entered in Matlab, which of the following
statements is true: (A) Typing cond(A,1) in Matlab gives approximately 11.7333 (B) Typing cond(A,1) in Matlab gives approximame 2.2 (C) Typing condnum(A,1) in Matlab gives approximately 2.2 (D) None of the above statements are true. A ANSWER: 13. MTH 510  FINAL — Page 12  WINTER 2011 14. (2 marks) Suppose that the centered ﬁnitedifference approodmation for f’ at a: = 0.1
with h = 0.2 gives f’ (0.1) m 2.034. Suppose the true error made in this approximation is
approximately 0.238. Based on the given information, which of the following statements is
true: (A) Dividing h by three leads to a true error of approximately (1238/2. (B) Dividing h by three leads to a true error of approximately 0.238/4. (C) Nothing can be said about the true error when dividing h by 3. (D) None of the above statements are true. 3> ANSWER: 14. 15. (2 marks) Suppose you wish to use the Simpson’s 3/8 rule to estimate the integral
2
/ [I + 9:3] d1.
0 Which of the following statements is true: (A) The maximum of the absolute value of the true error is 73%. (B) The integration rule is exact (eg. the maximum of the absolute value of the true
error is zero). (C) The maximum of the absolute value of the true error is g‘g—g.
(D) Nothing can be said about the true error using Simpson’s 3/8 rule. 9)
ANSWER: 15 16. (2 marks) The following commands have been entered in Matlab: xwfl 3 4];
y=[3 —1 1]; Prpolyfit(x,y,2);
xxalinspaceﬁ ,4) ; Which of the following statements is true: (A) P(1)+P(2) *xx+P(2)*xx.‘2 is the best quadratic polynomial to fit the
data in x and y (B) P(3)+P(2)*xx+P(1)*xx.‘2 is the best quadratic polynomial to fit the
data in x and y (C) P is the interpolating polynomial evaluated at 1:82 (D) None of the above statements are true. E ANSWER: 16. ...
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 Winter '12
 Dr.SilvanaIlie
 Numerical Analysis

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