MTH510-W2011Final

MTH510-W2011Final - RYERSON UNIVERSITY DEPARTMENT or...

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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 closed-book 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) Non-programmable scientific calculators. MTH 510 - FINAL - Page 2 - WINTER 2011 1. (3 marks) Consider the system 4x1+z2+2m3 = I $1+5$2 == 2 21+ 22:2 - 102:3 == —1. Do two iterations of Ga Seidel iteration starting with an initial guess of [1 0.5 CIT. Compute the approximatgélative error at each iteration. 1“ L L 2. Kiwi: 4,U~)<A:—RX3) x: -- + < am . UL LH ‘ LH (3— +Q‘b(+«1+>(,+£2x&\ . g, 1, L L Xl >< 74$ _ ‘04 O l O. 5” O __ .L ~— 1 #9 (x 0) £(Q'fi) t(l+§+wl ' iv”! fab " g : 12; = i1 ) ya I Va: :7, if 3 g — r ‘0 {(00% 6( (2 7L __ ’ W571 ~wi?oo%, 33.3%,looz .L 3 gang .. . _L ._ .L Q «ft/6 aft) 3}; t) IOC‘WWWStoozl 3. m, 2.0%} ' «o ' ‘82) t 660 - . v 007 :l :q38?r I o a: aw {1,00 \WHS)W EMU)“ ’1 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 0Qm%%o ;( 39,0 ME? 3 wk ’& 396%“ g Wax V4 :Vak fiagacéwjuiq O O Q/w O MERGE} 00 X3: my; «w 2 (C: ‘ré59" AXR+X3) V 74L, “Q’Q+O)ev/ 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 ,3}... %, :7 EL wax ax'é 95% % X .a-Eaxe‘fi-f‘ 6’9“ X6? 3: mo % r 2‘33 L I; \: 3 E M 6+ 0:9 fig 08 O §;€’l+lv\ cry; "3/ 1/5 , o . :ew: ° W /“O/)<:(.%8 (ot o =>< - M4" 7 oLQ+ 31 V ":33 6346+; 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 \m Mksrkq \M O Ck : 3C¥U3.><a3~m“8‘t3) O M mam) k3 Cs BQCMXJ‘ (33393] 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 . : Qfiafl-éngacg-a 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 =~% 0(3sz -4_ ~ \ 3"“3‘ 13%) V (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)$(2! 2)) (3v “2), (4, 1)} (a) Using a Table for Divided Differences, find the Newton interpolating polynomial for the data. Do not simplify the expression, and leave it in factored form. (b) Write down, but do not simplify, the term of the Lagrange Interpolating polynomial that has f(4) as a. factor. «A \s‘r ad 3 % 1": ’ifié—r—Vg/A §*% 3'0 4,,0 ‘* 3:3 :4- Erica 55 Ava 9x A“ MTH 510 - FINAL - Page 7 - WINTER 2011 6. (8 marks) Consider the integral 3 [[23 +4COS$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””? ‘ y 96c ] CL) jimmy bag-Mm 4mm QHxaH-‘V’COQX . +,\C(x4} :tUL\\+4~CO/fl+ QM?) ‘ t +4 H370.) \— M3 : t Eamon» + 71° (SWIWFI- 4;, 5708 &5307~ 3.8m W64”? '3‘? m Ham] 5 M‘HSMOS‘Q G én> 3‘18; @ nfll‘i @ 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 fourth-order RungeKutta method (RK4) with step size h = 0.1, to solve Estimateyatt=0.h to:o‘(ao:;\)htoq} G) igl‘déolgo}; 4Com: oa+a~+z~a@ 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’\) :— ?(u )l.7r8‘1%a§) ‘~= *Awoaafl' (D ‘ a: cam :2, (61:30 +%(L(+Ag+acg+ H) 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 :(o4‘4‘ 6&‘t \6.+‘/\81£I‘xu(d:) : 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)1-F6 == 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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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.

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MTH510-W2011Final - RYERSON UNIVERSITY DEPARTMENT or...

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