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CSC_349A_EXAM_QUESTIONS - COMPUTER SCIENCE 349A SAMPLE EXAM...

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COMPUTER SCIENCE 349A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PARTS 1, 2 PART 1. 1.1 (a) Define the term “ill-conditioned problem”. (b) Give an example of a polynomial that has ill-conditioned zeros. 1.2 Consider evaluation of ) tanh( 1 1 ) ( x x f = , where x x x x e e e e x + = ) tanh( . If ) ( x f is to be evaluated in floating-point arithmetic (e.g., k = 4 decimal digit, idealized, rounding floating-point), for each of the following ranges of values of x , specify whether the computed floating-point result will be accurate or inaccurate. (a) x is large and positive (for example, 4 > x if k = 4) (b) x is close to 0 (for example, 001 . 0 x if k = 4) (c) x is large and negative (for example, 4 < x if k = 4) 1.3 Consider 0 , ) 1 sin( ) 1 sin( ) ( + = h h h h g where the arguments for sin are in radians . When h is close to 0, evaluation of ) ( h g is inaccurate in floating-point arithmetic. In (a) and (d) below, use 4 decimal digit, idealized, rounding floating-point arithmetic. If x is a floating-point number, assume that ) (sin x f l is determined by rounding the exact value of x sin to 4 significant digits. (a) Evaluate ) ) ( ( h g f l for 00351 . 0 = h . Note that L 843088 . 0 ) 003 . 1 sin( = , L 843625 . 0 ) 004 . 1 sin( = and L 841470 . 0 ) 1 sin( = . (b) Taylor's Theorem can be expressed in two equivalent forms: given any fixed value 0 x ,
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