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Unformatted text preview: Exam 1: Numerical Computing and Solution of Linear Systems MATH 3315 / CSE 3365 : 801C – Spring Semester 2007 Thursday 15 Februray The SMU honor code applies. An answer should include necessary steps, unless it requires only one step. Otherwise you will not get full points even though your final result is correct. Question 1 (3 points) A normalized double precision (DP) number y belongs to binade k (where k is an integer) if 2 k ≤  y  < 2 k +1 . Which binade does the DP number 0 . 375(= 3 8 ) belong to? Answer: 2 2= 2 8 < 3 8 < 4 8 = 2 1 . So 3 8 is in binade 2. Question 2 (8 points) The distance between two neighboring DP numbers in binade k is ² DP 2 k , where ² DP = 2 52 is the DP machine epsilon. (1) What is the distance between two neighboring DP numbers in binade 100? Answer: ² DP 2 100 = 2 52 · 2 100 = 2 152 . (2) What is the distance between two neighboring DP numbers in binade 0? Answer: ² DP 2 = 2 52 · 2 = 2 52 . (3) What is the distance between two neighboring DP numbers in binade 52? Answer: ² DP 2 52 = 2 52 · 2 52 = 2 = 1. (4) Are all DP numbers equally spaced on the real number line? Answer: No, as seen from the results in (1), (2) and (3) above. Question 3 (6 points) Only some real numbers can be directly represented as DP numbers. In general, a real number z is approximated by a nearest DP number, which is denoted as fl DP ( z ). We know 2 52 and 2 52 + 1 are two neighboring DP numbers in binade 52, so (1) fl DP (2 52 ) = 2 52 , because 2 5 2 is a DP number. (2) fl DP (2 52 + 0 . 1) = 2 52 , because it is closer to the DP nubmer 2 52 than the DP number 2 52 + 1 for 0 . 1 < . 5. (3) fl DP (2 52 + 0 . 8) = 2 52 + 1, because it is closer to the DP nubmer 2 52 + 1 than the DP number 2 52 for 0 . 8 > . 5. Question 4 (6 points) In normalized decimal scientific notation, a nonzero real number T is represented as T = ± m ( T ) · 10 d ( T ) , 1 where the decimal significand 1 ≤ m ( T ) < 10 is a real number, and the decimal exponent d ( T ) is an integer. Write the following numbers in normalized decimal scientific notation and round them to nearest 4 significant digit numbers. (1) 3 . 1416 = 3 . 1416 · 10 ≈ 3 . 142 · 10 (2) 39998 = 3 . 9998 · 10 4 ≈ 4 . 000 · 10 4 (3) 0 . 0013331 = 1 . 3331 · 10 3 ≈ 1 . 333 · 10 3 Question 5 (6 points) Let A be an approximation to a nonzero true value T . The abso lute error is defined as  A T  , and the absolute relative error is defined as A T T . If the absolute error A T T ≤ 1 2 10 q , where q is a nonnegative integer, then the approximation A to T is qdigits accurate....
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This test prep was uploaded on 04/07/2008 for the course MATH 3337 taught by Professor Xu during the Fall '07 term at Southern Methodist.
 Fall '07
 Xu
 Linear Systems

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