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lectures1_2notes

# lectures1_2notes - Fall 2011 CMPSC/MATH 451 Lecture 1...

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Fall 2011, CMPSC/MATH 451 Lecture 1: Aug 23, 2011 Lecture 2: Aug 25, 2011 These notes are meant to cover what’s 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 ˆ x : inexact input ˆ f x ) : approximation to the function total error = ˆ f x ) - f ( x ) (3) = ˆ 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( π/ 8). Let us first assume π 3. Then, π/ 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 ˆ f x ) - f ( x ) = 0 . 3750 - 0 . 3827 = - 0 . 0077. This is the sum of the propagated data error, which is f x ) - f ( x ) = sin(3 / 8) - sin( π/ 8) 0 . 3663 - 0 . 3827 = - 0 . 0164, and the computational error ˆ f x ) - f x ) = 3 / 8 - sin(3 / 8) 0 . 3750 - 0 . 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. Truncation error is the difference between true result (for actual input) and the result produced by a given algorithm using exact arithmetic. It usually occurs due to approximations such as truncating infinite series or terminating iterative sequences before convergence.

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lectures1_2notes - Fall 2011 CMPSC/MATH 451 Lecture 1...

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