tag:blogger.com,1999:blog-11295132.post742612324236798788..comments2017-01-21T08:58:46.751-08:00Comments on A Neighborhood of Infinity: Computing errors with square roots of infinitesimals.Dan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-11295132.post-40567616071871541162013-07-23T20:44:00.619-07:002013-07-23T20:44:00.619-07:00Abraham Robinson's Non-Standard Analysis intro...Abraham Robinson's Non-Standard Analysis introduces infinitesimals into the number-line. It would be interesting (and challenging) to construct an algorithmic 'library' implementing this rigorous, but relatively simple theory. I expect that implementation of your ideas in NS Analysis would be result in very simple programs.<br />The library would have application to many areas of mathematics. That said, I am not volunteering to map the theory into any existing language!Polymathhttp://www.blogger.com/profile/14848236005210537884noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-64579004743375467982011-12-03T10:24:38.070-08:002011-12-03T10:24:38.070-08:00It is fair to point out that rounding errors in co...It is fair to point out that rounding errors in computation do not match errors of estimate in physics, which may run to three decimal places. Conversion to binary makes the problem worse: to think otherwise is just structuralist myth-making. <br /><br />Only in the ancient continued fractions are rational and irrational numbers comprehensively distinguished by termination. But an error of estimate encompasses noise as well as finite accuracy, so the problem is generic, and comparable rather to heat of computation!Orwinhttp://www.seri-worldwide.org/id591.htmlnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-42039284711179619902011-08-13T18:34:19.092-07:002011-08-13T18:34:19.092-07:00Stochastic calculus comes in two forms, Itō as you...Stochastic calculus comes in two forms, Itō as you noticed, and Stratonovich. They're both defined with a limiting process similar to the Riemann integral. They basically differ in that the Itō formulation is explicitly "causal" and function evaluation happens at the beginning of a segment. The Stratonovich formulation uses a balanced "midpoint" evaluation strategy. Unlike the Riemann integral case, these two converge differently. The big draw for the Itō formulation is that the calculations are much easier, as many expectation values are easily seen to be zero. The big draw for the Stratonovich formulation is that corresponds more directly to physical differential equations, and is the only one that has the right transformation properties to be used on manifolds.Aaron Denneyhttp://www.blogger.com/profile/15613957348593645695noreply@blogger.com