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If $ x $ and $ y $ have $ n $ significant places, how many significant places do $ x + y $, $ x - y $, $ x \times y $, $ x / y $, $ \sqrt{x} $ have?

I want to evaluate expressions like $ \frac{ \sqrt{ \left( a - b \right) + c } - \sqrt{ c } }{ a - b } $ to $ n $ significant places, where $ a $, $ b $, $ c $ are nonnegative integers. I was thinking about doing it recursively, i.e., if want to evaluate $ x / y $ to $ n $ places, I need to evaluate $ x $, $ y $ to $ m $ places, if want to evaluate $ x - y $ to $ n $ places, I need to evaluate $ x $, $ y $ to $ m $ places...

What book should I be reading?

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    http://en.wikipedia.org/wiki/Interval_arithmetic might be useful.2011-08-11
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    The book I have on hand is the one by Stoer and Bulirsch; the first chapter has things you might want to see.2011-08-11
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    Qiaochu Yuan, that looks very promising, thanks. J. M., do you mean Introduction to Numerical Analysis? I'll try to grab a copy.2011-08-11
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    That is indeed the book I am talking about.2011-08-11
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    In the case of $x/y$, it depends on how close $y$ is to $0$.2011-08-11

3 Answers 3