15-Errors-in-Computation-4UP

15-Errors-in-Computation-4UP - Errors Experiments Errors in...

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Unformatted text preview: Errors Experiments Errors in Computation Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Errors in Computation MATH 2070U 1 / 22 Errors Experiments Errors in Computation 1 Quantification of errors 2 Computational experiments Polynomial evaluation Summation Cancellation Overflow/underflow c D. Aruliah (UOIT) Errors in Computation MATH 2070U 2 / 22 Errors Experiments Absolute and relative errors Definition Let a ∈ R be an exact value and let e a ∈ R be an approximation of a Absolute error in e a 4 a : = | a- e a | Relative error in e a 4 a a , ( a 6 = ) Example ( a = √ 2, e a = 1.414) 4 a = √ 2- 1.414 = 0.0002135 . . . 4 a a = 0.0002135 . . . √ 2 = 0.0001510 . . . c D. Aruliah (UOIT) Errors in Computation MATH 2070U 4 / 22 Errors Experiments Example ( a = √ 2, e a = 1.414) 4 a = 0.0002135 . . . , 4 a a = 0.0001510 . . . |4 a | ≤ 0.00022 4 a a ≤ 0.00016 |4 a | ≤ 0.0003 4 a a ≤ 0.0002 The following are equivalent: a = √ 2, e a = 1.414, |4 a | ≤ 0.22 × 10- 3 a = 1.414 ± 0.22 × 10- 3 1.41378 ≤ a ≤ 1.41422 c D. Aruliah (UOIT) Errors in Computation MATH 2070U 5 / 22 Errors Experiments Rounding versus chopping Rounding to three decimals 0.123767 is rounded to 0.12 4 0.774501 is rounded to 0.77 5 0.6325 is rounded to 0.63 2 0.6335 is rounded to 0.63 4 Chopping to three decimals 0.123767 is chopped to 0.12 3 0.774501 is chopped to 0.77 4 0.6325 is chopped to 0.63 2 0.6335 is chopped to 0.63 3 0.69999 is chopped to 0.69 9 c D. Aruliah (UOIT) Errors in Computation MATH 2070U 6 / 22 Errors Experiments Significant digits Significant digits: leading nonzero digits in a number, e.g., 1.7320 has 5 significant digits, 0.0491 has 3 significant digits “Correct” significant digits problematic concept, e.g., a = 1.00000, e a = 1.00499, 4 a a = 4.99 × 10- 3 b = 9.00000, e b = 8.99899, 4 b b = 1.12 × 10- 4 e a agrees with a to 3 or 4 significant digits (by most definitions)...
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This note was uploaded on 02/23/2010 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Spring '10 term at UOIT.

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15-Errors-in-Computation-4UP - Errors Experiments Errors in...

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