MATH 251 Exam 1, Fall 2007

AMERICAN UNIVERSITY OF BEIRUT Faculty of Arts and Sciences Mathematics Department MATH-CMPS 251 QUIZ I FALL 2007-2008 Closed Book, One hour 15 minutes SUBMIT THE QUESTION SHEET WITH BOOKLET (ONLY NON-PROGRAMMABLE AND NON-GRAPHIC CALCULATORS ARE ALLOWED) STUDENT NAME ID NUMBER Problem Out of Grade 1 10 2 8 3 8 4 8 5 16 TOTAL 50 1 1. (10 points) Answer the following: (a) Let x = 7.477 and y = 3.789 be 2 floating points in F = F(10, 4, −2, +2). What is the relative Error in the computation of x ⊕ y in F? (b) Let x = 0.3721448693 and y = 0.3720214371 What is the relative error in the computation of (flp(x) ⊖ flp(y)) in F? 2 2. (8 points) Answer the following: (a) Let x = [94F96A0]16. Is this a hexadecimal representation of some element in Fs? Justify your answer. (b) In case x ∈ Fs, determine the decimal number that has this hex- adecimal representation in Fs. Otherwise, modify x first, then find the required decimal number. 3 3. (8 points) Loss of significant figures may result in the computation of the following functions of the variable x for certain values of x. Specify these values then propose alternative functions that would remedy the loss of significant figures. (If necessary you may use Taylors series). (a) f(x) = x − √ x2 + 1 (b) g(x) = 1 − cos(x/2) 4 4. (8 points) Consider the polynomial p(x) = 4 + x2 + 3x4 + 2x8 − 5x16. (a) (4 points)) Put p(x) in nested form. (b) (4 points) Find consequently the minimum number of floating- point operations to compute p(x). 5 5. (16 points) Let f(x) = ln(1 + x) − 1 x+1 (a) (7 points) Prove that the function f(x) has a unique positive root r in the interval (0, 1): • (3 points) By plotting both functions ln(1 + x) and 1 x+1 • (4 points) By studying the behaviour of f(x) and f ′(x) on [0, ∞) 6 (b) (3 points) Apply 2 iterations of the Bisection method • First Iteration: r1 = • Second Iteration: r2 = (c) (3 points) Write Newton’s method iteration formula and apply 1 iteration of the formula using r1 (found above) as initial choice. (d) (3 points) Write the Secant method iteration formula and apply 1 iteration of the formula using r1 and r2 (found above) as initial choice. 7