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_posts/2012-03-07-Cancellation-error.html
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_posts/2012-03-07-Cancellation-error.html
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---
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layout: default
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title: "Cancellation error"
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redirect_from:
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---
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<html>
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<head>
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<title>Cancellation error</title>
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<meta name="description" content="I was recently reminded of the importance of choosing numeric algorithms that
don't behave catastrophically for certain inputs. One example is the
calculation of 1-cos(θ) for small θ. In this case…">
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</head>
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<body>
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I was recently reminded of the importance of choosing numeric algorithms that
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don't behave catastrophically for certain inputs. One example is the
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calculation of 1-cos(θ) for small θ. In this case, cos(θ) is very close to 1,
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leading to a <a href="http://en.wikipedia.org/wiki/Loss_of_significance">large
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cancellation error</a> in the subtraction step.
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<p>What is to be done about this? Seek out an equivalent expression that does
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not suffer from cancellation error. In this case, employ the double-angle
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formula cos(2u) = 1-2sin²(u). This yields the equivalent expression
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2sin(θ/2)², which gives good numeric results for θ close to zero.
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<p>This plot shows how the accuracy of the original formula starts to visibly
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degrade at around 1e-7 and gets steadily worse until, at about 1e-8, it
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actually drops to 0.
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<p><img src="https://media.unpythonic.net/emergent-files/01331134959/cancelerr.png">
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<p>I wonder whether it would be useful to have a valgrind-like tool warn when
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disastrous cancellation occurs, or whether typical programs would have way
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too many such errors to ever contemplate fixing them all…
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<br><br><font size=-2>Entry first conceived on 7 March 2012, 15:42 UTC, last modified on 16 October 2013, 14:35 UTC</font>
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</body>
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</html>
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