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Unformatted text preview: [ML 3034: Modeling Methods in MMAE Name: Kt (' /\ (3 #gm EXAM 1  Spring 2012
Open books and open notes and put your answers on bubble sheet mmm #1: (60 points) Answer all 15 short answer questions: bubble in g choices that are m (there may be
more than one correct choice). 1. Direct methods for the solution ofan N x N system ofsimultaneous equations
fsolve the equations in M x 0(N’) ﬂoating point operations, where M = number of iterations to convergence.
% solve the equations in M x 0m") ﬂoating point operations, where M = number of iterations to convergence.
c) are characterized as schemes for which the dominant source of numerical error is truncation error.
.? are characterized as schemes for which the dominant source of numerical error is roundoff error.  solve the problem exactly if there was no round off error. 2. The solution ofa linear system ofequations by Gauss elimination consists of using ERO’s on the system to
(a) triangulate the coeﬂicient matrix into aw matrix and solving by backsubstitution. (b) W the coeﬁicient matrix and solving by backsubstiurtion. (c) triangulate the coeﬁicient matrix into an upper triangular matrix and solving byw. (d) triangulate the coefﬁcient matrix into a W and solving by backsubstitution. . triangulate the coefﬁcient matrix into an upper triangular matrix and solving by backsubstitution. 3. For ﬁxed point iterations, 3,,“ = g(:r:,.), for g(z) = ﬁe: and with an initial guess so = 1.0, we have a sufficient
condition for the iteration will converge? I. _
. yes, because g’(zo)l = 0.544 and all subsequent derivatives are smaller than one. $&\\~\\R\ ._ ’ _ (b) W " .4‘
(c) W (d) no, becasue Lq’(£9)l > 1 for all 2:,I and all subsequent derivatives are greaterthan one.
(e) none of the above ‘ 7’ 4. Machine precision is deﬁned as
the smallest number you can add to one on the computer and still recognize the addition.
the smallest number you can substract ﬁ'om one on the computer and still recognize the substraction.
(c the range over which integers that can be expressed exactly on the computer.
((1) the range over which real numbers that can be expressed exactly on the computer.
(c) none of the above. 5. The elementary row operations (EROs) that are used to solve simultaneous linear equations by direct
methods are
row switches.
) switching diagonal coefﬁcients.
. adding a multiple of a row to another.
. multiplying a now by a nonzero constant.
(e) none of the above. . 6. Direct methods for solution of simultaneous equations include:
Gauss elimination. _Q~ LU decomposition.
(e) none of the above. For the ﬁrnction f (z) = 5:1: — 2ez with an initial bracket [a,b] = [0, 1.5], the residual at the lst estimate of the root
.18ng bisection is to 3 decimals places: (a) 0.378 (b) 1.633 (c) 0.013 0. H7, ll
(d) 2.I85 @lnone of the above 8. In order to evaluate the ﬁrst derivative of f (1:) at 1., using a backward ﬁrst order ﬁnite difference, “30) = W). +0031.) on the computer
(a) we can always increase the accuracy by taking A2: to be increasingly “MW!! ‘ﬂwe can always increase the accuracy by taking A1: to be increasingly small within the limitation posed by
mtmdoﬁ‘ error ampliﬁcation.
(c) we can always increase the accuracy by taking A2: to be increasingly small within the limitation posed by
Mention. is. mrmd oﬁ‘andmr competition places an optimal limit on how small we can take A1: and beyond
which decreasing A2: causes e error to stop decreasing and to grow.
(e) none of ﬁre above. 9. With an initial bracket of [2.5,3.75] for the root of the function f (z) = 1:3  192: + 30, using bisection, after 10
bisections the error is estimated to be:
(a) 2.232 x 10‘5 a 1 .x ‘
. _ _ \ ' ‘ '_
8.1.2233: 502...... 5‘an HM \. Dike; (d) 6.104 x 10’4 —‘_ . cannot be estimated. ‘ I ' ' /
g) P ’ Pn \ é I M 10. The secant method to ﬁnd the root(s) of a nonlinear scalar equation, f (z) = 0, Q A (a) requires oneinitialguess.  '~ 31¢ A (b) requires function evaluations. ’ m  (J 00 1:2, (c)requires both function and ﬁrst derivative evaluations. \ r (d) requires both function and second derivative evaluations. . requires two initial guesses. 1]. In solving for the root(s) of a nonlinear scalar equation, f(1:) = 0, which method converges fastest to the
solution (assuming all methods converge to the root from the given initial guess), (a) The Thomas algorithm (b) secant method. (c) Hilbert's method. (d) bisection method. . Newton~Raphson method. l2. Due to the ﬁnite accuracy of the computer, zero is expressed in double precision as (pick closest value)
(a) 1038 61"“ (b) 10'108 (c) 10"2 Q 10‘308 (e) none of the above, zero is written exactly on the computer. 13. The main sources of error in all numerical computations are
’round 011'. (b) segmentation. . truncation. (1:!) phase error. (e) none of the above. (8) 0.010111 0 0.110010 
(c)0.001101 (d) 10.001101 / .
(e) 1.011101 (V10 13411 M 15. The number of bits used to store a real number in double precision is
(a) 16 (b) 32 . 64 (d) 128 (e) 256 Prom #2: (20 points) Find the roots of the function f (2:) = 736053 — 3. You are to solve for the roots of this equationbytheMMMethod usingtheminal guecsx, = 0.10mdcanyouton1ythemmof ml 16. The Newton Raphson method is a ﬁxed point iteration of the form :1: = gNR(z) for the root, and from problem 2
the 9013(3) is
(a) 9NR($)=73€—°5‘ — 377
‘gng(z)=a: — (he0'5“ — 3) / (7e‘°'5’  3.5134“)
(c) gNR(:1:)=z — 3/(7e‘°‘5‘ — 3.5:ce‘0'5‘)
(d) 9NR(3)=3 ‘ 7wed/(76“ — 3e" ‘ 3)
(e) 91.”;(:1:)=3ez  2 + 336“ 17.Withthe intialgueesz, = 0.10,ﬁrstandsecond iterationeetimatesfortherootaretwo3decimal places, \. 0.469 and 0.564 respectively.
(b) 1.350 and 3.7] 7respectively.
(c) 4.514 and 0.467 respectively.
(d) 0.7023 and 0.357 respectively.
(e) none of the above. 8. The iterative convergence criteria after two iterations is
(at)2.616x103
(b) 4.154x 102
(c) 3.186 x10‘3 d“ 9.50 x 10‘2
(e) none of the above 19. The residual alter two iterations is
w“ 2.158 x 10‘2 (b) 4.154 x 10" (c) 3.186 x10'3 (d) 4.132 x 10‘5 (e) none of the above. 20.AﬁerZ iterationsstratingﬁ'omtheintinlguess,:r:o = 0.10,weexpect (a) the iteration to diverge because I gm; (2:1)I = 0.4052 < 1 and indeed the iteration is converging.
.the iteration to diverge because lg’NR(zz) = 00068 < 1 and indeed the iteration is converging. (c) the iteration to converge because 9N1r(12)l = 1.4563 > 1 and indeed the iteration is diverging. (d) the iteration to converge because I g‘NR(zz) = 1.4563 > 1 and indeed the iteration is converging. (e) none of the above. w: (20points)Giventhematrhr—vectorsystem4g =15 as: 10 3 5 2:1 1
S — 9 l 172 = 3
3 7 ‘15 1:3 8 21. Is it necessary to use pivoting to solve this set ofequations?
(a) no, because the matrix is symmetric positive deﬁnite.
*no, because the matrix is diagonally dominant.
(c) yes, because the matrix is skew Hermetian.
(d) yes, because the matrix is not diagonally dominant.
(e) no, because pivoting is too complicated and takes up computer time. 23. Upon completion of the triangulation procedure, the element of the right hand side vector b3 is ‘9152
m—mn (c) 3.531
(d) 3.063
(e) none of the above la)  3 and the system of equations cannot be solved because it is negative.
9:1, 824 and the system of equationsean be solved. (c) 345 and the system of equations cannot be solved because it is too large.
(d) zero and the system of equations can be solved. (e)cannot compute the dotenninant as it is imaginary. 25. The solution vector is 3(zl,z2,z3) = (0.412, —o.163, o.527)
(b (zl,z2,zs) = (2.00, 11.00,  0.189) (c (31,22,23) = ( — 1.33, — 11.67, — 16.3)
(d (31,32,233) = ( — 1.33, 11.67, — 16.33)
(e (21,22,33) = (1.00, 2.00, 3.00) ...
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 Fall '08
 double precision, Upper Triangular Matrix, nonlinear scalar equation

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