Scientific Computing

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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 floating-point 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 ) k From this bound, we can see that there must exist some * that satisfies: (1 + * ) k = (1 + 1 ) · · · (1 + k ) 1
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1.3 Multiplication Series Example
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