CMPS 251 Final, Fall 2000

AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH 251 FALL 2000 FINAL EXAM Wednesday Jan. 31st, 2001; 2:00-4:00 pm Closed Book, Two hours Calculators are allowed 1. Consider a set of data {rey = 1... np. with o) # Tj. i#xj- e State and prove the theorem on the existence and uniqueness of the Lagrange interpolation polynomial Pn-1(2) Pn-U@i) = Yas P= Levee (10 points) e Write the MATLAB function Y=DivDiff (x,y) that constructs the divided difference table associated with the data {(2iys)li = 11}. The input data are the column vectors of length n x, J. The output nx 7 matrix Y is lower-triangular, such that the first column of ¥ is the vector column y. (10 points) 2. Consider a uniform partition of the interval [a.}] into n equidistant intervals of length h = boa : as7r< 22 =ath<..<ri= at(i-l)h <.. <2nti = atnh = b. Let f(z) be a function defined on [a,b], with f. fly wes f contin- uous on [a,d}. (f € C*([a,d])). We adopt the following notations : {fi = fled.t = Lean + 1 and fig = f(z.4) where = wttit1 is the middle point of the interval (ri. vival z; 2 L 2 jalan. Let D= ff f(x)de. e Derive the composite trapezoidal rule T(h) to approximate f. Give (without proof} the expression of I — Th) (5 points). 1 Derive the composite Simpson’s rule S(h) to approximate I. Give (without proof} the expression of I — S(h). (5 points) Show that 35(h) = T(h) + 2k fio. t=] . (5 points) By writing T(h) = 8(f1 + fati) + De fi. show that : } nm 2T(5) = Ta) +h figs - i=l (5 points) Derive the relationships between T(h). T( 4) and S(h). (5 points) Consider the following table of data, associated with f(x) on the interval [0,2]. 2 fl) 0.00 1.000 0.25 1.284 0.50 1.649 0.75 2.117 1.00 2.718 1.25 3.490 » 1.50 4.482 1.75 5.755 2.00 7.389 | Find T(1). T(0.5). T(0.25), $(0.25). (10 points) 3. Assume that f € C?({a,d]) and that c is a unique simple root of the function f(z), in the interval [a, d], i.e. f(c) = 0, f'(c) #0. Give Newton’s (Newton-Raphson) iterative procedure to find c and prove that, if {rg].4 = 0.1....} is the sequence of iterants of this method then : tim Matic yy, kx |rR _ e|? where M is a constant depending on the function f. Find M. (10 points)