HIST 251 Final, Fall 2007

AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH 251 FINAL EXAM FALL 2007-2008 Closed Book, 2 HOURS WRITE YOUR ANSWERS ON THE QUESTION SHEET STUDENT NAME ID NUMBER Problem Out of Grade 1 20 2 10 3 20 4 15 5 25 6 10 TOTAL 100 1 The next 2 questions deal with the natural cubic spline S(x) on the following table of 5 data points for a function f(x): i 0 1 2 3 4 xi 1 2 3 4 5 f(xi) 1 4 11 17 23 1. (20 points) Let wi = S ′′(xi) be the values of the moments of the cubic spline. (a) (10 points) Specify the system Aw = r. 2 (b) (10 points) Solve Aw = r using Naive Gauss Elimination followed by backward substitution. 3 2. (10 points) On the basis of w, give the formulae only of: (a) (5 points) The spline derivatives: z0 followed by zi = S ′(xi), i = 1 · · · 4. (b) (5 points) The expression of Si(x), the spline S(x) restricted for x ∈ [xi, xi+1]. 4 3. (20 points) Consider the problem of finding r = 1 √ R. (a) (5 points) Give the function f(x) for which r is the unique positive solution, such that the computation of f(x) does not require any division. (b) (7 points) Write Newton’s iteration formula that gives the se- quence {xn} that would converge to r. Give a graphic justification into why the the sequence {xn} converges for every x0 > 0. 5 (c) (8 points) Prove that: xn+1 − r = (xn − r)2 2xn 6 4. (15 points) Based on a set of data: {(xi, yi = f(xi))|i = 0, 1, ...n} where nh = 1 and xi = ih, ∀i. Consider the composite trapezoid rule, T(h) to approximate the integral I = � 1 0 f(x)dx. (a) (5 points) Give the formula for T(h) and the expression of I−T(h) in terms of powers of h. (b) (5 points) Prove the following formula: (1) T(h/2) = (T(h) + hΣn−1 i=0 f(xi + h/2))/2 7 (c) (5 points ) Complete the following MATLAB program which takes as input two vectors x and y and approximates I = � 1 0 f(x)dx using the composite trapezoidal rule. function T = trapezoid(x,y) % x = (x(1), x(2),........, x(n)) % y = (y(1), y(2),........, y(n)) 8 The next 2 questions deal with the following table of values for a func- tion f(x), arranged in a table as follows: i xi f(xi) 0 0.000 1.0000000 1 0.125 1.1108220 2 0.250 1.1979232 3 0.375 1.2663800 4 0.500 1.3196170 5 0.625 1.3600599 6 0.750 1.3895079 7 0.875 1.4093565 8 1.000 1.4207355 5. (25 points) We intend to approximate I = � 1 0 f(x)dx, using Romberg integration based on the composite trapezoid rule T(h). (a) (10 points) Derive subsequent, Romberg integration formulae, R1(h), R2(h), R3(h) by giving expressions of the errors: I−R1(h), I−R2(h), I−R3(h). 9 (b) (15 points) Based on the above f(xi) data, approximate the in- tegral I = � 1 0 f(x)dx, by filling the empty slots in the following table: (You may use formula (1) of 4.b, to fill the column of T(h).) h T(h) R1(h) R2(h) R3(h) h0 = 1 × × × h0/2 = 0.5 × × h0 4 = 0.25 × h0 8 = 0.125 10 6. (10 points) Using the forward Difference formula to approximate f ′(0), followed by Richardson extrapolation, find the best approximation to f ′(0) starting with h0 = 0.5. For that purpose, derive the formulae of the following table and fill the empty slots in its entries: h φ(h) φ(1)(h) φ(2)(h) h0 = 0.5 × × × 11