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Unformatted text preview: LECTURE 1. Start with Strangs elegant description of errors and blunders. Errors are unavoidable aspects of any computation, whether because of a computers inherent limitations or our own human capacity for mistakes. Blunders are avoidable mistakes that come from careless interpretations of data. I expect you not to make blundersit would be too much for me to expect you to make no errors! Main example: Suppose we have a system weve analyzedamount of two chemicals, for instance. So suppose we have ~x t be the state of the system at time t , and suppose that we find ~x 9 = 1 1 1 1 . 00001 ~x . Note that we seem to see that, at time 9, we expect to have approximately equal amounts of each chemical. Write the matrix above as A . Now suppose we know how much chemical there is at time 9, and want to work backwardsgiven ~x 9 , how do we find ~x ? Well, thats easywe just have ~x = A 1 ~x 9 . So suppose you find ~x 9 = 100 100 . Then you can easily compute A 1 = 100000 100000 100001 100000 and A 1 100 100 = 100 Great! But maybe not so great. After all, theres naturally some exper imental error. But your instruments are really good, and you know that youve measured the chemicals correctly with an error of at most 0 . 0004 grams. So you say, well, OK, maybe ~x 9 = 100 100 . 0004 . Thats not a very big difference. But when you work backwards, you find ~x = A 1 100 100 . 0004 = 40 60 1 In other words, a really tiny uncertainty in measurementless than one one...
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 Winter '09
 IgorDolgachev
 Linear Algebra, Algebra, Numerical Analysis, Matrices, condition number, Δb

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