M235A6Soln

# M235A6Soln - MATH 235 Assignment 6 Solutions Fall 2006 1....

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Unformatted text preview: MATH 235 Assignment 6 Solutions Fall 2006 1. Find the least—squares approximation to a solution of Ax = b by constructing the normal equations for )2 and then solving for fc for the following cases: 2 1 *5 (a)A—[—2 O] ,b=[ 8] 2 3 1 2 2' 3i (b)B= —2 0 ,b= 2i [2 32‘] [24] Note that the normal equations over C for AX = b are A*Ax = A*b. So 3: (Ox) m nominal Lﬁuceiicrm a/UL aim -—-’-~ T A : T—J“ ‘ M ATA: 1 ‘2 2‘ 2 \ I. [2‘ (g AAX. A b , , <‘ O 3>(_2 O)- 13 W W: (i “53‘ Elli) =2 (:2;er A X (Li Mark5> *9 Q = -\-’*(E “Ll “2+ :: iiuz Umt~§§rarw "Li é '2 3 a Frog/Lima?“ . (b) Comimcrlin "lids. Korma/l L%waiﬁm Ag? />\<=A*lr: “M WWW ‘2 1X2 W“ m A —i O '3; “i ? ~85 Io maA%=2—22>3g __Li L (-6 O “3*? (\$1)- (aux/Rina) /\ i -l w- M k F‘, i, 5 4+; / )4: w (/l “'23 LH A A l 3 “LP” L+>*("/_+ ‘ll‘ :1 X: b” ‘ \ " . M 5 Ware/3 > 28( L (3)4!” ‘SL/r'r w O‘PPCQKmeCl‘CM (Ll’ marks) 1..) 2. You are given the data points (—2, 2), (~1, 1), (1, 0), (2, 1). (a) Find the best linear ﬁt to this data having the form 3/ = [30 + 51:13. (b) Find the best quadratic ﬁt to this data having the form y 2 50 + ,8100 + 52302. 521:: (0.) Fast hm Arum/3M Aiéf; : /‘)>°_2/gl:2 ’2 2 ion/w ‘ -. (gjyzp, / M+Aw> :1,” . ﬁe "1“ Qﬂgl:\ 32 \{54 A T /\ 1“)- NJL/XJV Oomhmcgf ALL Mrmaﬂ WQHMA A>L=A lo T __ \ -1 NW A A“. (-12% \\ ELM“ dub): (LC: Cl>c>)’\/ \ l ATE: )1), : ﬁghm / 14% ﬁg 2 T Io/rhtt «%:1‘ @)=@)AA1\$30)thQ“/ l 6. Consider the vector space 0 [0, 1] of continuous functions on the interval [0,1], together with the inner product given by < f,g >= fol f(t)g(t)dt, for f, g E C[0, 1]. Let P2 be the subspace of C]0, 1] having a basis given by B = {1, 2:, x2}. Find the best quadratic approximation to V0? on [0,1]. Eff: First am :3er amp] wa/J ‘]:or [P ]> off) CA, H11 Grm“3c]\m&dct ProongFQ+o+PxQ >438 1 g v] =\ 4 > 51 a L ‘ __ x1 = x x: (1 \>= v ~ ><~ ) \ ix,” 2 1 4m» ’ ° ’7‘ I :; X~aéz \ V :_ X2“ (xi/B] _.. 4W6“? \/\/ 3 (MW <x~li,;<~3,;> \ -— - .L 4X2.) \>:SXZC]X=§ ) (X2) X‘i‘>“§)<7(><“p~d‘]x L].- 6 l1 0 . ("‘0 \ P A“); (Vi/Vt) AVID/7) l <v3/V3> 3 \/ WM mm»: «a»: gmxﬁ 0 (VI/Vl>‘—"-‘ \ <5§V¥>= §()<-Ji)c\x= 9’13: __L_ (vlivlﬁ— HT 4EV3>=\$ r;z(>< ~x+1§3clx= 11;; O \ (Va/Vs>:3("2"x+ﬁ A”: \$3 my), ' -.-.: 2;... “A- —- (It. 1-— +‘L' proAﬁLJ—X‘) 3U SCX .2.) :?.<7< X é) —--—L5- 2 i9: is... “ix + 35X +35“ ‘/ (A QPPPOKJLMCCRW “gar R MP ‘HLL. [ZARFULE [0,11 (“gape”ch +0 +ka give/m W Prowg'c (10 marks) \/\/\/ 7. Over the past four Winter Olympics Games, the Canadian team won the following numbers of medals: Year Location Number of Medals 1994 Lillehammer, Norway 13 1998 Nagano, Japan 15 2002 Salt Lake City, USA 17 2006 Turin, Italy 24 Scaling the years, we can translate this information to the data points (0, 13), (1, 15), (a) Determine the least squares line y = ,80 + ﬁlm which best ﬁts these data points. (b) Using your line from (a), predict the total number of medals that Canada will win at the Winter Olympics in Vancouver in 2010. 5(3le (a) For raving :1: +0 I o \3 3° :‘3 I 1 /5° _ 15 await-as r 2 (M— n / ﬂ°+2ﬁ\:r¥ l \/’\:/ erg/32:“ T >< r» l‘ T /\ ’7’“ \ \ Cm Tumo A Axe-A lo WM \/ COMT‘Fucl’ mrmllﬁvﬁ L+ é yr» 1 l ‘ \) IE T=llll =< >2Al’f‘113l3L AA<Ol73\‘\Z 4H 0 12% l 3 2 SCI <m> : Bali Elf)??? (Lt; '0 (amaawaarmasvmatatu%:a+;LJV US CA3 lb Scsuod‘ao llrmnrx (a) ) :1 +2 4% ><=Ll~/('rla arlolo Lj l 7~7< w Cdei/QPOHO‘A'l'OXf-Ll) we. Otto.“ if; \l—HL’r =24, M 711%, All» {/3 26 . ‘3 (zmwey _ MA6 Question 8: So1ution First set_up the system of equations corresponding to this prob1em. The coeff1c1ent matrix, A, is: A = [1.0 0.5 0.25; 1.0 1.0 1.0; 1.0 1.5 2.25; 1.0 2.0 4.0] A = 1.0000 0.5000 0.2500 1.0000 1.0000 1.0000 1.0000 1.5000 2.2500 1.0000 2.0000 4.0000 The right—hand—side vector, b, is given by: b = [23.7; 45.1; 64.0; 80.4] 23.7000 45.1000 64.0000 80.4000 Form the norma1 equations, (AAT)*AX = (AAT)*b: B = A'*A B = 4.0000 5.0000 7.5000 5.0000 7.5000 12.5000 7.5000 12.5000 22.1250 c = A'*b C = 213.2000 313.7500 516.6250 So1ving for the so1ution vector, x, yie1ds: X = B\c X = —0.2000 itj: 50.3000 ~5.0000 / Note that in MATLAB the simp1er command: x2 = A\b x2 = —0.2000 50.3000 -5.0000 Page £5 MA6 produees the same resgit since MATLAB wiH a1ways find a 1east squares so1ution to an inconSistent system. Thus, th) =_—O.2 + 50.3*t — 5.0*t/\2 is the best quadratic fit. Comparing this with the known equation th) = v*t — g*1:l\2/2 we obtain: v = {Of and —g/2 = —5.0 => 9 = 10.0.\/ (5 marks) 04E“ MCU‘kS) Op‘tloncd Tog/go'- CoJouJus Approacdk‘h Qw‘tfmﬁié O'fL x453 Cami ’H‘é In, m Mam W. W): +0 1% 44¢ not @mecﬁim JYO )7? 043% “Rnrm Oc’r EX+C><1. Th fromaﬁm Co #4: m #051; 1A9. ocbmrc o'M‘LL CU. Fencgh 01mg~ a+kx+c4<3ﬂ SCHQQ Nev are, not ga‘varx AWLQC. Pun-LN}le we, #0 oLi'M—Uencx weak mks“; @41an [OI [l M, We. +0 4’ka \$W~H~j E Wag: l E = g [W‘meX'TCXZJICi-X o z 5‘ : S [ X ‘20 I}; *ZIDXWZ «gag/7‘ +2ak>< +2acxl+25c><3+ 53+ ﬁx: ciﬁkﬁx Injftgf‘oé‘iné ‘kﬁ‘m' Lam. CM : E: ﬁf-%Q*L§B*%C+C~\o+%ac+ékc+a1+éﬁf+égcg ’6 4h we, AL'EEE—zagtb—ExQ 30, 9b 3‘3» 713.0 +0 a Mr“ 960% O‘p Eiuoﬁim £2»? 035/0... givamkg M, 3 2\/O~\ /’+\ - \$3.. 192%,?— cz—ni 35” ‘ Thu)! @mea‘ﬁmﬁ )3? Lo; ‘%X1+ﬂ>¢+% 3:; LA JLXQCj: mgfeamqr‘ff: 4% Max!" agglzf‘o‘ 4O Slum) mrfmponk4o (1, Wm. _ Raving REL Sohﬁm 0»: ...
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## This note was uploaded on 01/05/2012 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

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M235A6Soln - MATH 235 Assignment 6 Solutions Fall 2006 1....

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