MTH510-Midterm-W2011-Solns

MTH510-Midterm-W2011-Solns - O1 RYERSON UNIVERSITY...

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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) Non-programmable 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( 3L3, cam; :or -—>o Rfl l .S’”»< a =( ~a / L/ O c( (3 Blame, : H 0t. HO! Wirzm'. 7~§§i§ 51 g l 1 931:2) : 6 bl‘k} TOW? Ha b7“ ‘ z gm- 9 cm. \wj W q $0M”) AZ: ’> 0 ¢ Ham“ 9 “(37(15’PC W8 ‘ w) wmbegelsfl can W&~ Qflva Mr {5 L x3”; (., x, 2113} =[H’LMW‘L335K W 3-. mmmr X33“ 1 3'3 5303(893‘388‘1 K O MTH 510 ‘ MIDTERM — Page 3 - WINTER 2011 2. (6 marks) Do two iterations of the Newton-Raphson method with 9:0 2})?0 approxi— mate one of the roots of f(2:) = 43inzr+ Inx. Calculate the value of tea! at each iteration. QKV" firsinx {ILAX ‘ ' PDQ; AFLOSX + J): “T 1, XL mg) . gm) \eoJ 0 m mme __ b‘i __ _ m I ' ' ‘ flSJrOiLQ'ZMY‘ . (WyowM—Jr ( .E?‘+>=55WT */ “00% ‘1’ 91.3% 3 104% I a 0&4” alWioHS, ><,. c: X&$,&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; g ms— <0 :oroggfi >0 3 1er 1.876“ {.6m’ 00’ My" 3 LWAS, Kr%[.8(&r [DEM W Mam/or fi(l.8'+f'lo%'j=-oogay For Eald< 104; W I VIE. 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. .. .. Hi L, C K| ‘ a vaxa“ax> QM ~ Cw " >43: L U" “I X3) I A~axyk~fi‘_%3) Vléflr-HV'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.

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MTH510-Midterm-W2011-Solns - O1 RYERSON UNIVERSITY...

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