EE103%20-%20Jacobsen

EE103%20-%20Jacobsen - BE 103 Fall 2007 Midterm Exam...

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Unformatted text preview: BE 103 Fall 2007 Midterm Exam 10/25/2007 Prof. S. E. Jaeobscn EE103 Applied Numerical Computing Midterm Exam October 25, 2007 (Closed-book examination, No calculators) Name: ' Student ID Number: bc’l bq b Lname Fna m!— >5 2 _’_ Page 1 / 10 BB 103 Fall 2007 Midterm Exam l0/25/2007 Prof. S. E. Jaeobsen Problem 1: The following data comprise the output from applying Newton’s method for finding an f such that f(f)=0. 1. To what value is the sequence of iterations converging? 2. What is your estimate of the rate of convergence, 0.? State why. 3. What is your estimate of the asymptotic error coefficient, 11 ? State why. 232? 4. What can you say about the function’s derivative, f‘"(3r’) , at your limiting value of the 1: sequence? Why? What about g2’(f)? HHHHHHHHHHHHHHHHHHHHHHHHHMwmmH .626923076923077e+001 .081360946745560e+000 .437414656349567e+000 .465672157837611e+000 .375162859924642e+000 .767842843999006e+000 .428888726391136e+000 .239587641978233e+000 .133843976215715e+000 .074771786275115e+000 .041771307641842e+000 .023335591519508e+000 .013036460375617e+000 .007282837243768e+000 .004068567565098e+000 .002272911177978e+000 .001269765133417e+000 .000709356138076e+000 .000396282838074e+000 .000221383983777e+000 .000123676484484e+000 .000069092047913e+000 .000038598373044e+000 .000021563037233e+000 .000012046222108e+000 .000006729639499e+000 .00000375952289Ze+000 .000002100262932e+000 .000001173314941e+000 .000000655474098e+000 PageZ/IO u' 1 \ ©W\v®c§€‘>m i:\ Q) EMQWO N m ‘3)UQACCCQ‘WR (0w “#0? N7 (*4 BB 101me Midtamfixam 10/25/2007 Prof. S.E.onbscn \/ hfimme ‘\8\ “Wm We \mow.'\°‘\0“¥ Q0.“ a mew‘ms’b P CUNQVCQQ A 3t1\fme T0 “‘3 (nth “a (“WE‘th ‘\f\ 7 E5 103 F8112007 Midtelm Exam 10/25/2007 Prof. S. E. Jacobean \ Problem 2: ac} , Assume f(x) has a fourth continuous derivative. Show that 7)? W 2 -fi (0 “f by +1 ’(0 (Fifi i170 1“ $32 3 \ 3 is “n 00") WXimflfiOfl fOT fmfi) . the second derivative of f evaluated at f. + E ’f (yo‘xsz) WT ”“T°"°"‘“°°'°““‘"°°-‘ ‘Véu {flows /6 2 _ )i ))(;)\\~< \ r”: _‘ V, AV {k7m32pw— ’(vm +£{K‘flh “Km? .2 ., MT , 492“ {RN m Maui26W ” ‘ ; k {JG—2 \ 3 ” W a _ J VK \\ «mm; iuxg_1"“‘“*~~-~-‘ A «x \f _ f... 4E? \ a“; Page4/l0 BB 103 FallZOO'I MidtermExam 10/25/2007 Prof. S. EJacobsen Problem 3: Suppose we're using Newton's method for finding a solution of the equation f (x) = 0. Suppose the method works and we have x. —) J? and we also know that f is a root of multiplicity 4. We're interested in studying the rate of convergence. To make matters specific, let's consider f (x) = x‘ (of course, we don't need an algorithm for this problem; our point is to understand rates), where we begin Newton's method with the positive value X0 =1 . (a) What is the rate of convergence of Newton's method, x <— x- f(x)/ 1' '(x), in this case? What is the value of the asymptotic error coefficient, 1. ? (b) In place of Newton's method, suppose we use 1: +- x—2 f (x)! f '(x). What is the rate of convergence of this method? What is the value of the asymptotic error coefficient, 11 ? How many iterations will it take to be within 10" of the root (An expression for the answer is sufficient). (c) In place of the method of (b), suppose we use x(—x—4f(x)lf'(x). How many iterations will it take to converge? ((1) Now suppose that f (x) = 2.0:), an n" degree polynomial with n distinct roots. As above, Newton's method produces a sequence x, 41?. Show whether or not the convergence rate is at least quadratic. Hint: Consider the facts PM = (Jr-mm). 12%) = ? P(x)P‘"(x) 2)(x)- g ".5. ' (P:"(x)’ , Q LNG: #3 {lg-AN} 3% (gum 54‘ >4‘%‘X 4* own iri'élgf’iokzw } . , ‘zfi \l \‘ZO-C ‘6 \PYFCP. v_~\Tt\/\ ’4 Page6/10 ER 103 Fall 2007 MidtamEnm 10/250007 Prof. S. E. Jacobsen Q szk‘n“ 74‘ 2 w I A eY—‘fl/k“ 26‘ L Q33 R 2 .Q C7/6 \ g == )0 00“ 10,} “WNW \ Ve fievCI‘UODS T0 C) A «00* ~ I " 2, “0 up 6 “\209 X“. NW C) Qyub52§< " / ‘4( wow\\ kafi \ _,_ O $\1‘€‘\\\\\m\\ (d\\l (O “firs? y'“(, 85103 FillZOO'l MidtamExam 10/25/2007 “of. s. E law we <3“ we 5>m>¢6 :7 6B}; (0:0 V mg S‘mce Q h‘1 5 CMWM’SF \ Long“ 27%? gbecmg 3k /$ g Wmmx) v \ m‘ \' > ” . aflg Rafflepdv a ‘9 ivwdmth ‘7»; PageB/IO BE 103 Fall 2007 MidtermExam 10/25/2007 Prof. S. E. Jacobsen -NMK Problem 4: i ‘ I We (a) Let f(x)=x+e"°" . There is a unique solution, 1?, for f(x) =0 in the interval [-1,o]. (i) The plot of f (x) is given below. By inspecting the graph, what can you say about the rate of convergence if Newton‘s method converges to the solution? Whfl (ii) Starting with x, = 0, mark the approximate locations, with x, and x,, where those values would be if we had implemented Newton‘s method. xomso x’) (iii) Now, implement Newton's method numerically, starting with x0 = 0 as the initial point. Also, assume we are implementing Newton’s method on a machine that uses the IEEE double-precision standard. Compute the first 5 numbers generated from the Newton iterations, starting with x0 = 0. (Hint: use e'” =1.9287 x 10‘” , and recall that a, a 1.1 102 x 10'“ for an IEEE double precision machine). (iv) What do you observe fiom part (iii)? Explain. W #97 - e M +51%)” -5011 VA \ 14¢} ') _ _ 100* e ___. O % k“ k \ emf Xlg ~ “) _ wre 09(03X" T—noxew XR/j M20)\W:\ __ Lg‘fii M. l—') H ar© 90 ON” W3 7K mew CoNECOSG; 'T’O Q VOAva ‘m ?q"\ Km“? Q mfihec‘flqn yCKMQ 'n A ~80 O “6% 6 cm woe—G 09 “new 39 \ Cth 0 MN memokr. Page 10/10 ...
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EE103%20-%20Jacobsen - BE 103 Fall 2007 Midterm Exam...

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