Anonymous user
Talk:Numerical integration/Gauss-Legendre Quadrature: Difference between revisions
Talk:Numerical integration/Gauss-Legendre Quadrature (view source)
Revision as of 16:40, 28 June 2011
, 12 years agohard to say...
(Utility in cases of limited necessary precision, and aims of irrational representation.) |
(hard to say...) |
||
Line 12:
::::: 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)
:::::: 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)
::::(unindent) It's difficult to talk about feasibility abstractly. You could represent var x simply as "root of that function" without fully evaluating it, and later logic may simply eliminate that function, hence the root, altogether, then good, lots of work saved; but if you are forced to take a value at some point, it then all depends on how you are going to use that value: you need to make decision now on precision desired, which can be [[wp:Wilkinson's_polynomial|surprisingly non-trivial]]. Stuff may be theoretically possible, but it's quite different from being practical, alas. --[[User:Ledrug|Ledrug]] 16:40, 28 June 2011 (UTC)
== Tables and layout ==
| |||