lectures1_2notes - Fall 2011, CMPSC/MATH 451 Lecture 1: Aug...

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Unformatted text preview: Fall 2011, CMPSC/MATH 451 Lecture 1: Aug 23, 2011 Lecture 2: Aug 25, 2011 These notes are meant to cover whats discussed on the blackboard, and are almost entirely derived from the textbook. 1 Error Analysis: some definitions absolute error = approximate value- true value (1) relative error = absolute error true value (2) Consider a one-dimensional problem f : R R , where x : true input f ( x ) : desired true result x : inexact input f ( x ) : approximation to the function total error = f ( x )- f ( x ) (3) = f ( x )- f ( x ) | {z } computational error + f ( x )- f ( x ) | {z } propagated data error (4) (5) Note that the choice of algorithm has no effect on the propagated data error. Example. Suppose we need to approximate the value of sin( / 8). Let us first assume 3. Then, / 8 would be 3 / 8 = 0 . 3750. Further, let us approximate sin ( x ) to x , considering just the first term in its Taylor series expansion. The total/absolute error is f ( x )- f ( x ) = 0 . 3750- . 3827 =- . 0077. This is the sum of the propagated data error, which is f ( x )- f ( x ) = sin(3 / 8)- sin( / 8) . 3663- . 3827 =- . 0164, and the computational error f ( x )- f ( x ) = 3 / 8- sin(3 / 8) . 3750- . 3663 = 0 . 0087. Note that the errors partially offset each other in this case. Computational error can be further split up into truncation and rounding errors....
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This note was uploaded on 01/19/2012 for the course CMPSC 451 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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lectures1_2notes - Fall 2011, CMPSC/MATH 451 Lecture 1: Aug...

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