Talk:Numerical integration/Gauss-Legendre Quadrature: Difference between revisions
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(Utility in cases of limited necessary precision, and aims of irrational representation.) |
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:::: Idle musing...say you've got a scenario where your root may not be rational. If greater precision in the results may be needed later, but not now, one might save off the polynominal factoring routine's state for resumption later, no? I'm not saying it's necessarily useful for this task, just thinking about scenarios where irrational numbers in intermediate calculations might have their precision preserved. --[[User:Short Circuit|Michael Mol]] 15:35, 28 June 2011 (UTC) |
:::: Idle musing...say you've got a scenario where your root may not be rational. If greater precision in the results may be needed later, but not now, one might save off the polynominal factoring routine's state for resumption later, no? I'm not saying it's necessarily useful for this task, just thinking about scenarios where irrational numbers in intermediate calculations might have their precision preserved. --[[User:Short Circuit|Michael Mol]] 15:35, 28 June 2011 (UTC) |
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::::: Hmm, not necessarily. For polys higher than degree 5, there are no general analytical way to get roots, so all you have left is numerical solutions. You could try to use rationals during <i>that</i>, for example while using Newton's method, but often there are so many iterations, your rational representation will grow too fast to be of much realistic use. --[[User:Ledrug|Ledrug]] 15:44, 28 June 2011 (UTC) |
::::: Hmm, not necessarily. For polys higher than degree 5, there are no general analytical way to get roots, so all you have left is numerical solutions. You could try to use rationals during <i>that</i>, for example while using Newton's method, but often there are so many iterations, your rational representation will grow too fast to be of much realistic use. --[[User:Ledrug|Ledrug]] 15:44, 28 June 2011 (UTC) |
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:::::: Two things. (1) a numerical solution is of limited precision, and in practical use, often only a limited precision is necessary. In a particular use case, some signaling of a desire for greater precision would allow for further calculation as-needed (a lazy evaluation, I guess). (2) Would it be possible for those intermediate routine states to still be useful for comparison? ("is {routine state} equal to {routine state}? Does a prior or projectable state indicate equality?") Admittedly, my line of thinking is towards usefully representing irrational numbers internally and exactly by way of incomplete calculation, and using the state of that calculation's routine as a value in other operations. --[[User:Short Circuit|Michael Mol]] 16:21, 28 June 2011 (UTC) |
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