Test 1 - 2009 Solutions

# Test 1 - 2009 Solutions - UNIVERSITY OF NOTRE DAME...

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Unformatted text preview: UNIVERSITY OF NOTRE DAME Department of Civil Engineering and Geological Sciences CE 30125 Computational Methods Fall 2009 J.J. Westerink October 29, 2009 Test 1 NAME SOLUTION This exam is being conducted under the HONOR CODE. Please Sign your exam to indicate that you agree and have adhered to all the stipulations of the honor code. Signed 110\$ 1. Please highlight your answers with a box. 2. Clearly show your steps!! 3, Taylor Series: _ - df (.t—U)ld2f f(x)~ﬂaJ+(“””)Eu=u+ 31 — +(r—a)3daf i _ “d1- ?” Frza 4|(‘ a)Er:E Problem ] (20 points) Solve the following system of equations using the LU decomposition method. Note that you should re- arrange lhe system prior to solving. Indicate why you should rte—arrange the system. ’ I ~JF/i/rusT 3E Dig-Gamay Damn/AM j 4 9 II 7 (H?- ‘X H A 5 3 2 x3 7 7.3 5 X, “an a”- ' I 0 U ‘ (in qlL 4.5 I O 0 Pt" ' 2 1 P1: 1 a e 0 “in (3:5 ' : l O ‘ ' ' 'Psi P31 I o 0 C155 9/9 0.3%" g “in "5 Chi '5 9! out : (in : Lf qig 7-7 \$3 -'—' Q2. 5 [011qu i: '2-7 Pal I: 5'21 = Putin +922 = 3: (If/q )(LD‘WZL 3? C322 '3: 6'12 : SG/q C429, = Pacts 19:3 = 3=I‘f/e.)(&3+‘izs 2) gas 2 2 = “V”! Q's! ‘—’ F3! “in "-‘* 2 => P3: 2 2/7; U :31; :4 Psi (it: “5' Page: :2 3 =1 (2/000? 1331(6921>=> P32: oral-[-12 (Di/\$6 '53 ~ P2: Ch} 4' P32. ‘723 "r 933 3: 5 :— ﬁﬁxﬂ +(d-é‘i'X-leD-I- 7‘5 AiLU ""7 AX=B => LUX=B iF“UX.=Y => “[733 f‘ZIXSé EoLUE LY 5: [3 1 ss— ﬁ—o -6ﬁ 1....__~_( is.) Problem 2 (20 points) Given data In = 0 110) =fo I1 1‘ hﬂh) =fi r gm = In” “,1” +1.31; Derive (using Taylor series) the error associated with this Lagrange Linear interpolating poiynomial. 8(1) = f(x) ‘ EU) Demonstrate that your answer conforms to the fundamental precept of Lagrange basis functions. fcx -: 7g + (X'X03ﬁh + ‘X *Xo)z£[2)+ #517: 2'. 31:05:) 2' f:- j; -{‘ (Xi-"yaﬁm .9 (xihﬂ)zﬂ(a "'1" Z’. _ (I): __' a _ j; “j; _ (XIZXD)JC(Z)+ MOI-Jr, 5) X,—-Xo=h foo: j, + (kayo) ‘" '%'ﬁk‘}* M07] “*(ngdwm WW, 0’): a l—u/X'Xa} X—Xu ﬂ) —h(X—Xu) (X—Xo) f h>+j(h z 4 7. “Lil/J11 =7Xo=0 f0) :1. (in) + 95%?) -+ fJ"‘(—A'>‘< M) + My. 2 ea) :ffx)~jC><) = ﬁthfé‘JXZJﬂ/OT c) : -— [2)‘1’1559—90 : A '2) "-1100 4"} ._ 601) j]; —m( i I/ Problem 3 [25 points! Consider the following data: 2.0 . . r I ‘ Write the most accurate interpolating polynomial possible with this data. Compute an associated estimated error. 2 J' _ '5 I 909: ,5 .f (Ac—yo) + if (X—%)(X-Ml%; 4, giro—yaw Mx ninng + O,r O +m 30‘): #6.; + (X'z) 3&2 + ~_L(x—apt—10615.1)A _r___ (x-zxxvaxvé) 29% 2 Z 21 2-3 23 +0+04.,_ j“): —2a.l -+ 2x1 + 5.2x3 Fauna, THE TAELE WE ij; J 59 QCX)=O 062, AV}; Problem 4 {25 points) Given the following data: at x0 = 0 f0 is specified . . .. ....a_t x...= I]. and .ﬁ'...’ are \$P..E¢§ﬁ.¢d .. .. . .. ... .. .. .. .. . (a) Derive g(.\'). (b) Derive the form of the Hermite basis functions when the interpolation function is expressed as: gm = fuun(-")+f1[11(1)+jini31(x) (c) Sketch the three basis functions. (auﬁTﬂFrﬂTgi 0:) 3m: comm-Myra 3(o)=fu => on =31; z ’0 0 ‘q, j; 5‘0): 01. +2a1x 9013:}: =7 Qa+qih+qzh=fr l L 111] {it 5 J] 3‘(m:j.“‘=> qi+2m=ﬂ° o I ah at: f,“ an.) at” (ﬁsﬁqt %7qgﬁtn)x+é _ f, +34”; X2 ' a o 0 (you: a bljfkT‘fﬂ—zx x1 ‘h h; T; ’ h h: 0‘ 1 " ° ‘1: W)*£(%*%)+Jﬁm «+39 0“ “I Gin-:42; qliw%_ﬁ*%jfl ﬁf'm 01: if" D Z: + jam ’1‘ H T Problem 5 10 oints Consider solving a diagonally dominant sparse system of 150,000 linear equations 29,000,000 times. There is no banded structure (Le. entries occur throughout the matrix) and there are 18 entries per equation. Given that the Gauss elimination solution of the full matrix takes 342 CPU hours, estimate how long the LU decomposition solution will take on the same computer. Then estimate how long the Gauss-Seidel solution will take on the same computer assuming that 7 iterations are required for eachof the 129,000,000 N: [.5405 _ 7 M: 23x10 "a, a] 3 3 'L M N gems dFEIZATmNs N OWN) KM (N *0] ) ——— M 1* OFEMTM = Mutt/3: 1787321022 Timﬁ FEE open-77m = WmE/ﬂ ofEQATwnJS: E‘HZ (Fa/5:73?ng crematan :_ gcququ/EE—C a I 2— 1 arm ‘ — 1 .. EPU [1mg : Tm: {1m UPEQAT'MN x M X“ : atom 1753; x 2.9 Ira-’KQSWDﬁ z OFEK t: 2.23XIO-3f/g Gﬁ055~ §EiDEL 1-— A/x We “ cw 11m: ‘— Tuna 6254' opt-7:24am K ﬂ/*< 1W KM ° ' .— : ski-wriz'src >< tsww‘ x 7 K Z-Q’V‘FXI? ENTR'ES ...__— of??qu : m) x10"; ’yaﬂi l0 ...
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## This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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Test 1 - 2009 Solutions - UNIVERSITY OF NOTRE DAME...

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