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Re: Zero does not equal zero et al.
> Hello, > I would like to comment on some of the posts in the thread "zero does not > equal zero". I agree completely with Richard Fateman and am disappointed > that Dan Lichtblau "passed" on addressing the question of whether > Mathematica's fuzzy handling of floating point makes such programming > mainstays as x==y and x===y untenable. This is not a minor point but a > fundamental design issue. The issues that have been raised in this thread have been addressed many times, in this group and elsewhere. The basic principles are also discussed in the formidable open literature on this topic. This thread has expanded to include so much of numerical analysis that it probably isn't realistic to respond to everything. If you don't see the responses to all of your questions in this particular thread, the answers are available elsewhere. > I continue to think that 1.0 == 1 returning True > is an outright error, because it trashes the pattern matcher The pattern matcher does not use Equal (==), so the design of Equal has no effect on the pattern matcher one way or the other. > and offers no > defense against propagation of errors -- how fuzzy does something have to > get before it becomes intolerable? The notion of "defense against propagation of errors" in numerical analysis generally means designing algorithms to reduce accumulated error. It isn't clear what equality testing might offer that is relevant to this. > At the very least, why not introduce a new symbol, such as =N=, to test > for floating point "equality". You can use SameQ[Order[e1, e2], 0] if you want that operation. > The meaning > of == is sacrosanct and should not be sullied by contact with the dirty > linen of floating point arithmetic, vagaries of hardware representations, > etc. Equality testing certainly must deal with those "vagaries" at some level. If anything, functions like Equal and SameQ are less sensitive to those vagaries than corresponding functions in other systems. > I also think Mathematica's adoption of a "least common denominator" > approach to comparing floats with other floats or even with what are > supposed to be exact numbers (like 1 as opposed to 1.0) is objectionable. > Thinking probabilistically, if 1.0 is the result of an experimental > measurement, there is zero chance that 1.0 can exactly equal 1, so why > confound them? You can use SameQ if you don't want a comparison of 1 and 1.0 to return True. It doesn't seem that it is bad to provide several functions that give different common ways of handling this comparison. > Concerning Fateman's comment (in Re Re Zero does not equal Zero): > > Another choice is to test, and reset the > > accuracy of numbers at critical points. > > Realize, for example, that certain convergent > > iterations will not produce more accurate > > answers (as is normal), but will produce > > vaguer answers at each step because of Mathematica's > > arithmetic. They will terminate not when answers > > are close, but when they are essentially unknown. > > > This is a major hazard, but at least a way to defeat this behavior-- by > setting $MinPrecision=16 (say)-- is discussed in the Mathematica Book. This > device obviates the need to reset accuracy at various critical points (how > tedious!) and is crucial, as otherwise with root-finding algorithms (e.g.) > Mathematica will return a result with zero precision. This is because those > algorithms rely on the fact that although approximate roots of F[x] = 0 are > accurate only to the square root of the working precision, when you feed the > approximation back to F in the iterative loop you regain the lost precision; > there is no way for Mathematica to know this, so it assumes the worst-case > scenario of progressive degradation of accuracy. I have found Mathematica's > high-precision arithmetic, fuzzy or not, to be useful for empirical error > analysis. This (to me anyway) is the only somewhat interesting concern about Mathematica arithmetic. This too has been addressed many times, but at least it is interesting. While it is true that numerical algorithms often "regain lost precision", this is certainly not always the case, or even usually the case. In most practical calculations numerical errors are uncorrelated, and treating them as if they somehow cancel out is just plain wrong. The default behavior of variable-precision arithmetic in Mathematica reflects this reality, and assumes by default that numerical errors are uncorrelated. A simple counter-example is a program that adds and subtracts the same number many times. In such a program, the error introduced in each addition exactly cancels with the error introduced in each subtraction, so the errors do not accumulate. Treating the errors as runcorrelated, as is done by default for variable-precision arithmetic in Mathematica, lead to an over-estimate of the error in the result. The problem with that counter-example is that most numerical calculations involve a lot more than just adding and subtracting the same number many times. In the overwhelming majority of practical numerical calculations the algorithms have not been meticulously contrived so that the errors cancel out, and the only mathematically justifiable assumption is that all of the errors are uncorrelated. Programs that have been so contrived can be run in Mathematica either by using machine arithmetic, or by suitable use of SetPrecision and SetAccuracy. Since machine arithmetic (where all errors are effectively assumed to be perfectly correlated) is fully available in Mathematica, concerns about variable-precision arithmetic in a sense amount to arguing that less is more: that providing an alternative to machine arithmetic is somehow worse than providing no alternative. And for those examples where high-precision is needed and the algorithms have been designed so that errors tend to be correlated, inserting SetPrecision and SetAccuracy in appropriate places is hardly much of a hardship, especially in comparison to the vastly more difficult task of doing the basic numerical analysis to arrange for these functions to be appropriate in the first place. In any case, the assumption that precision is fixed throughout a calculation, as is done in machine arithmetic, or by corresponding use of SetPrecision, SetAccuracy, $MinPrecision, and $MaxPrecision, amounts to telling the computer to make up extra digits to pad out all results to a fixed number of digits. The absurdity of doing numerical analysis that way should be obvious. Machine arithmetic is popular because it is fast and compact, not because there is anything fundamentally good about it. > But even leaving aside the question of whether Mathematica's treatment of > arbitrary precision floats is sound numerically, and serious doubts have > been raised, it's far from obvious (to me) that floating point arithmetic as > currently implemented in Mathematica is compatible with Mathematica's > essential purpose as a symbolics package. All of the doubts that I know about have been addressed many times. If anyone has any remaining questions about these issues, my best suggestion would be to review the archives of this newsgroup, to look at some old tutorials on Mathematica arithmetic (available on MathSource), to prepare specific examples to illustrate your concerns, or to review the basic issues in the open literature. The book that I happen to have on my desk right now, "Numerical Analysis" by Burden and Faires, for example, has in the first few pages a decent introduction to arithmetic. In closing I could mention that the "zero does not equal zero" example that introduced this thread was largely just an illustration of the distinction between the way that numbers are stored and the way that numbers are displayed. The precision of a number can be any real number, but the display can obviously only use a discrete number of digits. A number with a precision of 3.4, for example, might display with three digits, a number with a precision of 3.7 might display with four digits, and so forth, and a number with precision less than 1 might display as zero. Although one might disagree with the way that fractional precision is rounded to a discrete number of digits for the purpose of display, this has nothing to do with basic arithmetic. Dave Withoff Wolfram Research