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Unformatted text preview: Page 1 of 4 CEE 305: Civil & Environmental Engineering Computation (Fall 2011}
Professor Duc T. Nguyen; 135 Kaufman (Ofﬁte #6833761) ' Wed.I October 12, 2011; 3:002m — 4:15am; 10.12 BAL
Closed Books/Notes; calculator is allowed _ Computer 8: internet are "NOT” allowed _ _
1page (both sides 85” x 11”} formulas/examples etc...is allowed Matrix Operations, Simultaneous Linear Eguations [SELCholesky Method. Fill—inTerms
 '5 Problem 01 I . , " .
1‘3"sz I:— —' } .rﬁq: nvr ' Given the following3 x 3 SLE [A1 * {x}={b}, where: ﬁg ('71) _' 3 4 ll = ‘ L! 2 —1 1 —3_ _
[A]: —1 4 —2 [b]: 9 y .3
2 1 8 _  11 _ l _ l' i
‘ " I' ' tion method M1 0 a 5 
U N 6 Emma , _ 5
5mg aive auss /X 0 0 C} ‘21; Li a) Find the upper triangular matrix [U]?? b) Find the backward solution vector {x}?? 1" ’2 ._ [  l 1 Problem 02 Given the foilowing 3 x 3 SLE [A] * {x} = {b}, where: _
3 @ LC = 7 2 —1 .1 — 0 3( .433“
[A]: —1 4 —2 [b]: 9 r;
2 1 8 11 , D 0
Using L U Decomposition method, I 0;) v \ O O
a) Find the upper triangular matrix [U]?? L: vol? 1 O
l
l b b) Find the lower triangular matrix [L1?? _  'g , +3 '
c) Find the forward solution vector {y}, from [L] * {y} = {b}? a j ’ 1 .y d) Find the backward solution vector {x}, from [U] * {x} = {y}?? . _ [.414 O I'D Page20f4
: (AT  M95
D: 1 I . ] —o;7o’] my” 0 (To SatVIE a) iLiiLl, 0 2mm ProblemOS
GiventhefollowinngSSLE[A]*{x}={b},where: ®_X "4‘4 4030.7 .lr‘fu‘i SF
2 —1 1 —3 [UL o from 0 [To SowE D) [A]= '1 4 “2 [b]: D o 2”‘4’1
II is the given matrix [A] symmetric positive deﬁnite (SPD), explain your reasons?? #25 W 5 Me WIMAIJT ‘5 I
b) Use the Cholesky method; find the upper triangular (or factorized) matrix IU]??¥ P05 "TV E c) Find the forward solution vector {y}?? @2 '23 751'; #556 kW d) Find the backward solution vector {x}?? e} Find the product of [P]: [A]* {x} =?? 2
2x75: [P1 “3““ «“9 5f! ‘5 "1  isﬁ r
Problem '04 q Pl“: lo I” Given the following "symmetrica matrix: '  O D a) Without any numerical calcUlation (just based on careful observation about the nonzero locations},
identify the "fill—interm” ocation(s} in the UPPER triangular matrix potions only, assuming ”Cholesky
factorization method” is used?? Speciﬁcally, how many "upper triangular" ﬁllin—terms wiil you have and where arethese ”ﬁllinterms” located? 2 FILL1p T618115 @ M17, 4' ng b) Assuming METiS reordering algorithms have already been used to minimize the total number of fillin
terms, and output from METiS reordering algorithms can be given as iPERM (new#) = {oldii}; .SuchasiPERM(1)={4} I 8 O O "D O iPERM(Z)={3} 4c , o o o ."9 o
iPERM(3)={1} {A 1 '20 {go 3 ’D '9 iPERM(4) = {2} 2
. > O 0
PERM”=5” E; 3 lb Liz :50 Based on the given information/data,_ the original nonzero value for'A(2,4) = 5 (at the "old” location: row 2,
column 4) will move to which ”new” iocation??
 A L bl , l ) Problem 05 toe rm 3cm  3 . Page 3 of 4
05%) : 4N3 __ _,/ =00.30.15.
X,  Y,  4“ I 3
Given the following (n=4) data points:
X; = 1.0 4.00 6.00 
vi: 3.0 4.00 8.00  bl’f(x¢)*jm> .. finD’Jcl‘J
,  . x?— '3‘: )ioax ;
Where =0, 1, 2 and (X0, X1, X2) = (1.0, 4.0, 6.0) x2 4‘: (Y0, Y1, Y2) = (3.0, 4.0, 8.0)
0 grid” , (“U3 002.55 The Newton interpolation (2nd order polynomial) function can be expressed as:  [p ' L/ L; ., l 5 y=b0+b1(x—xo)+b2(xuxo)(x—xl) ‘6" $21. (Job? Find the unknown constants bu, b1, and b2?? L61; ) b): “0.337; )‘ 5'2“: lubleq’ Problem ’06 Given the following (n=3) data points: xi = 1.0 7 4.00 6.00 _ vi: 3.0 4.00 8.00 thuul 5’) ((0064—69)
 (x~i‘)ixV)(‘3) Where 1:0, 1, 2 and (Xe, X1, X2) = {1.0, 4.0, 6.0) ("(9 ’  l C b , 4 ) (Yo. Y1, Y2) = (3.0, 74.0, 8.0) Using the Lagrange interpolation (2nd order polynomial), write the equation for y=f(x) that passes through the
above 3 data points?? Regression...
Problem 07 5 " H” E Given the following (n=4) data points: x, = 1.0 4.00 5.00 ' 6.00 5
Yi= 3.0 —4.00 8.00 2.00
Where I=0, 1, 2 and (x0, x1, x2, x3) = (1.0, 4.0, 5.0, 6.0) Ll (b .7 8 ‘ 0,0 ‘3
(Y0, Y1, Y2, Y3) = (3.0, —4.0, 8.0, 2.0) __._ )5
. no '70 4 Uta 0“
The nonlinear (2m1 order polynomial) function can be expressed as: . a £97 '75 LlDb 2:72; 2, _ 2
y—a0+a1x+a2x Setup (you do ”NOT” have to numerically solve) the appropriated equations 50 that the nonlinear regression  "n coefﬁcuents a0, a1, and a; can be solved. . a0 : .2, 55 2;
ﬁg *‘llC‘Hi
“a = 0.!!8‘18 Page 4 of 4 Problem 08 [Al/~35: DATA Fag M Pia Woks Pace» am 1
Given the following (n=4) data points:
'5
xi = 1.0 4.00 5.00 6.00 4 W Q " _
Yi=' 3.0 4.00 8.00 72.00 _ .1 g
_ . Ila '78 0;
Where =0, 1, 2 and (X0, X1, X2, X3) = (1.0, 4.0, 5.0, 6.0) _ ' (Y0. Y1. Y2, v3) = (3.0, 4.0, 8.0, —2.0) The linear (1St order polynomial) REGRESSION function can be expressed as: 053 y=.a0+a1x Find the linear REGRESSION coefficients a0 and a1??' 40 = 2. M78 6
at ‘ 'Déé'll ...
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