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Unformatted text preview: CS205 Review Session #1 Notes 1 HW 1.1 Hints 1.1 Addition Example Recall that the relative error is defined as: relative error = computed result analytic result analytic result Consider the relative error induced by the addition of two positive numbers x and y . The simple act of storing each number in floatingpoint introduces some error yielding ¯ x = (1 + 1 ) x and ¯ y = (1 + 2 ) y . Performing the addition induces some roundoff error both for the operation itself as well as the storage of the result, which we will model as (1+ 3 )(¯ x + ¯ y ). So: R.E. = (1 + 3 )((1 + 1 ) x + (1 + 2 ) y ) ( x + y ) x + y ≤ (1 + 3 )(1 + 4 )( x + y ) ( x + y ) x + y =  (1 + 3 )(1 + 4 ) 1  =  (1 + 5 ) 2 1  ≤ (1 + max ) 2 1 = 2 max + O ( 2 max ) 1.2 Aside Consider a sequence of multiplied rounding error factors: (1 + 1 )(1 + 2 ) ··· (1 + k ) It’s clear that the cumulative error can be bounded both above and below by: (1 max ) k ≤ (1 + 1 )(1 + 2 ) ··· (1 + k ) ≤ (1 + max )...
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
 Fall '07
 Fedkiw

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