Scientific Computing

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CS205 Homework #1 Problem 1 Arithmetic operations are subject to roundoff error when performed on a finite precision computer. In order to perform an operation x op y on the real numbers x and y we deviate from the analytic result when discretizing those values to machine precision as well as when we store the resulting value. Let ¯ x denote the discretized, floating point version of x that is stored on the computer. You may assume that ¯ x = (1 + ) x where is bounded as 0 ≤ | | < max where max 1 is the machine roundoff precision. Assume that the result of the arithmetic operation between two floating point numbers ¯ x and ¯ y is computed exactly, but when stored on the computer it is once again subject to roundoff error as x op y = (1 + )( x op y ) where the roundoff error obeys the same bounds 0 ≤ | | < max . The relative error of a computation is defined as E = Computed Result - Analytic Result Analytic Result Provide a bound (in terms of max ) for the relative error induced by the following arith- metic operations, or prove that the relative error is unbounded.
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