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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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