MidtermTestandAnswers - NATIONAL UNIVERSITY OF SINGAPORE...

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NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE ANSWERS to MID-SEMESTER 1 TEST 2007 MA2213 Numerical Analysis I 5 October, 2007 Time allowed : 1 hour and 30 minutes INSTRUCTIONS TO CANDIDATES 1. This test consists of TWO (2) sections. It contains FOUR (4) questions and comprises TWO (2) printed pages. 2. Answer BOTH questions in Section A. 3. Answer not more than ONE (1) question from Section B. 3. Candidates may use calculators. However, they should lay out systematically the various steps in the calculations.
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PAGE 2 MA2213 Part A. Answer both questions. 1. (30 Marks) Consider the function F ( s ) = s 3 - 5 on the interval [1 , 2] . (i) (10 Marks) Show that F has exactly one root r [1 , 2] . (ii) (10 Marks) Starting with the midpoint r 0 = 1 . 5 of the interval [1 , 2] , use two steps of the bisection method to compute approximations r 1 and r 2 of r. What is an upper bound for | r - r 2 | ? (ii) (10 Marks) Use one step of Newton’s method starting from the point r 0 = 1 . 5 to obtain a more accurate approximation r 1 of r. Answer (i) F (1) = - 4 and F (2) = 3 therefore F changes signs on the interval [1 , 2] and it must have at least one root r [1 , 2] . Since F 0 ( s ) = 3 s 2 > 0 for all s [1 , 2] , F is strictly increasing on [1 , 2] . If t [1 , 2] and t 6 = r then either t < r and F ( t ) < F ( r
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