Metallic MOS: Difference between revisions
Dave Keenan (talk | contribs) Changed |
Cmloegcmluin (talk | contribs) →As interval finder: great suggestion from Dave, to include the direct formula for these intervals |
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We’ve left still another part about our instructions for finding metallic generators vague: the “an interval” part. It turns out that not just ''any'' interval can yield a metallic generator — only those whose bounding ratios are sequential convergents (or semiconvergents) of a [[wikipedia:Continued_fraction|continued fraction]]. | We’ve left still another part about our instructions for finding metallic generators vague: the “an interval” part. It turns out that not just ''any'' interval can yield a metallic generator — only those whose bounding ratios are sequential convergents (or semiconvergents) of a [[wikipedia:Continued_fraction|continued fraction]]. | ||
Now that’s a mouthful, to be sure — but we don’t need to understand what that entails at this point, because fortunately for us, an algorithm called the [[wikipedia:Stern–Brocot_tree|Stern-Brocot tree]] graphs all of these intervals for us: | Now that’s a mouthful, to be sure — but we don’t need to understand what that entails at this point, because it can easily be tested by checking that <math>a_1b_2 - a_2b_1 = ±1</math>. And fortunately for us, an algorithm called the [[wikipedia:Stern–Brocot_tree|Stern-Brocot tree]] graphs all of these intervals for us: | ||
[[File:1200px-SternBrocotTree.svg.png|618x618px]] | [[File:1200px-SternBrocotTree.svg.png|618x618px]] | ||