M294F09WS_LeastSquaresSol

# M294F09WS_LeastSquaresSol - Math 2940 Worksheet Ch 5...

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Unformatted text preview: Math 2940 Worksheet: Ch. 5 Orthogonality and Least Squares October 22, 2009 1. Let S (t) be the number of daylight hours on the tth day of the year 2008 in Rome, Italy. We are given the following data for 8(1): M February 1 32 10 March 17 77 12 April 30 121 14 May 31 152 15 We wish to ﬁt a trigonometric function of the form 27f 27r N) " Mbsm (366 ) +0608 (366) to these data Find the best approximation of this form, using least squares. (Only set up your matrices and the equation, do not actually perform the calculations) (ﬂI/AA WWW? m 7%,! a/A’Mlm 14/! W M an AWM 710 MM W [2 AA ‘ M AHAI‘m gfzIFZII+CW wlbf/ﬂ/ I S’fn/ gill/7‘ Wig) ID I I U: 01 As‘méjzr/WpALCW/ﬂéfﬁ) l4: a AAyiAélzﬂz/JN Cm 9275/13?) WA (”TV 9AA) WWW) 1F 01+ Arz‘rv/ [gamma car/n— (/9!) AAA/5' WA ﬂ/V’C A MAM/l7?” _ ”76%;??ng Mfg/117’ r4?“ {/1 MWWV/A M How many daylight hours does your model predict for the longest day of the year 2008? (The actual value 1s 15 hours, 13 minutes, 39 seconds. Again, just set up the problem) a * 12. 26 Use the fact that b R: 0431 c —2.899 A2 A #7: /,2 AA + Mr/ AA [77-515) NW w/ﬁé} ,9“ AAA Af/m/WAAA %W. A" M1 er 'ZIWI 9 FM/AAW/ﬂy Mﬂﬂﬂﬂ'mﬁ’e I ~' 1;; 14M?/ “FAA'W/Wéﬁé" 72} ”[AA/ W A AA AAA/23% W "W7!” "lg/WM W7 - If 91 A A/MMZA] ”W41 £0 Cheﬂé' ZWé/Jéé 3-“ 219969175 él/fé [#ﬂ/‘f’wkﬂ 3%Zg4012’7Ii I I )777 a 1 VAk/WZAA WA A/zAAA) AAAAWZZ , Amide/212A AME/mm/QA) A 2. Consider the space, X, of all piecewise continuous functions on the interval [—7r,7r] (having at most ﬁnitely many discontinuities) with the inner product deﬁned as < ﬁg >= % /:f(t)g<t)dt- - Compute mu, usinmn, Hoosoem, ”small, and Hcos(2t)H-' - a) HM” = (W: gag/2A: #4: WWW / 5w /////: 1/2. “‘7 /""""”’” [‘7'V“"’“‘>: if? WW #15 “em as /%— W; = Mam»: 7, mtg/=1 W ”WP: %’ ”WM: #7,” aim/M “7 '75 H '* @ﬁifz/Q—MmV—f w Mia/i=7 M //W//“: sew/2% a; PM W .%/;-- —~///~/M=/, it ”WA 2 V) x/wzw//’= a; “1W; {ft/4’ Kiwi/1W as -Q§@”/Z}%/%’+W/°¥W/)’f( a M/zw/M I ' 0 Let T2 be the set of all functions of the form f (t) = a + ()1 sin(t) + c1 cos(t) + b2 sin (2t) + C2 cos (2t). Show T2 is a subspace of the space X. /: ﬂw‘ ﬂ, £01"; Y’d‘ Mll'fﬂ'dﬁhzzﬂl‘ ﬂ‘ elf/Z”) #2? 4+ mm mm! + , @1ng H (415! [(064, 4-! Riv/‘1 éh/ (174! fix/bl) "W WWO: ﬂw) ~+/A«2);~M+/zy)m ~r 0 Find an orthonormal basis for T2. (Check it) awe/24 [2%) % fedﬁaf [2/] A“ in Z3” M moi/yd — < 0 Compute ProjT2( f (t)) where f(t) = { (2’ 07T<_tt<<w0 . m W K76 W WMWA w WVVVMVVVVV/ mm/zﬂlwm/w 7 W T . 7” M .ﬁ’szmIai‘VA/Am ‘ MM 6F6fm-«Q%w%71 ’49 {44W WMyMW/ém; I7 ”,3 M/ﬂl/m i: //147: J k / V04; , M, w, WUVV‘. W (7) ”2’ Vii/M2!” W ’{JL M/ ,3) g ymt WE") : 3%: Wi S‘I‘Mf’ﬁw'zo S’IW gmé ’15) £0 V24 rfo .29 ML: Vil’ﬂygy 5 M 2‘1‘1‘) M3: VS'VL/AQH/ WW l/gJ'r .1/33»(l§l”/>lll ”1146(2)”; ”M131 ’lm'lrI/l/DI/ﬁ n a (W. Va» H7, WM» = 1m W21?!) < MA (W7: 7%” Caéhmwf' :0 KW WilKWM/VHLVVW 5’0 MM 1/» A?" i‘ .L 30 V31: W —~> a]: “V ML” -= 504%, «mm-MMW W W: Mr 1 Vintnﬂl, *(Vgﬁ amt/2,, 04%;”; V’ZVmb/D : \$74!; dagfnlvlﬁa'z 0 ((072,: [RV-Al) MIT) 41: SWIM (”WA WI 92,4214 sv‘rMF #LW/zrmimlk‘wfﬂ \$74 Zulaléo vﬂfzwla \$9. .i ‘ ‘ .lwlr) , . <qu3) 7r [I FmZthai-Loa: 1; [ yawwﬁm u Fir “33W ,ﬁzerW/J: a u. "‘ 44\$ (41-?) 9,; 14‘: may ~42 m4: wax—u : Main 4V _ L/ «(M («N/VI WW“: VfL/IVHII W Vs" 5 Vy—*Zu,lvf>u,~ (MuVs'7u1 (“M/W 4’ ‘“ ~32 z Vm “r>=# "w: WV = m max/21]}; ,. 0 -qu \ Ir ' w / trWﬂa-oé‘m) [WV “2>=%r{_f{;’sm+mzwr V/Wﬂ, ﬂf/ZmzéV-JM/ '0 ,VVWV ‘ W m M" I, , 1r- 1 Vsr: “3% I/Vﬁ,;rm*wlfﬂ my; cm [I~Zsin1%#'"' 7% AW 1372 WWW”, S 0 5 yr, “0* [ﬂrﬁﬁm Merl/V4 V l/yr (1,?“ g m (45% 2 B (m WI} ”M ,L 3» VI": mm) a Mr: W //'Vr“”” ”ﬁg—3A { ’/dé, m. A m4, KIM/2f); W/Wj N M W‘WWMM {959W ﬂy 7;; (,1 , m Inkjm) / In WM M A” W7/ A Wu WI: Kim /n~f)y;7«(h¢hé/’rﬂ M144 f ﬁrm {War/WW r” ”H” , ‘ ﬂy.” yap/mt) w/mwﬂ 5‘5 M V?!” WW WWW/W78” WM Wm WM M47; yin .eM/L 0W 7 , Qm/M/é /m(/‘E MM) W //¢).: 57/ *~7/‘égz& -// #4 75¢?“ (My: fWﬂm My): “mm + 4 ”2M: +Zg%>@f(n/%>//¢+///W')Z/r WA“ 255% v ”If W W MWM/m’o/ é{If/3'" ﬂaw. dam/g. (f, m}: 774/0) 124th)" Izrﬂﬂtﬂ‘mgﬂ/Aé 5' yy ﬁk’f’f/jfr 2/27” 45%,),— MWW‘ jﬂnW/j; : o (Mk/dWF/Wﬁ: ’AV/'%2£”/j=«%/' "121% ~ afﬂﬂgt/a <£ur>>7f—ﬁ”'wzwr%(‘ Jigﬂjjj p a "7” ,1/:_:/. VWM" W2. /VZ ...
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