Re: RE:Working Precision

*To*: mathgroup at smc.vnet.net*Subject*: [mg23976] Re: [mg23928] RE:Working Precision*From*: David Withoff <withoff at wolfram.com>*Date*: Sun, 18 Jun 2000 03:01:00 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

> Richard Fateman showed how arbitrary precision arithmetic can > produce strange results. I have some more strange behavior. These examples are really just different ways of disguising some basic issues in numerical analysis, most of which are not specific to Mathematica. All of these behaviors are entirely reasonable. Any system of arithmetic can be made to show behaviors that might be surprising to some people, and machine arithmetic can be a lot more amusing in this regard than Mathematica arithmetic. 16-digit machine arithmetic, for example, routinely must make up digits to fill in the required 16 digits, a fact which can lead to all sorts of mathematically indefensible effects. The first example, however, is not very elaborate, and since it is just showing basic propagation of error, this behavior really shouldn't be unexpected: > Richard demonstrated that (x == x+1) returns True after > evaluating the Do loop below. Well you can also get f[x] defined > for lots of values you don't expect! > > In[1]:= > x=1.11111111111111111111; > Do[x=2*x-x, {100}]; > Clear[f]; > f[x]=29; > {f[-5.2],f[-2.1],f[4.3],f[8.2]} > > Out[5]= > {f[-5.2],29,29,f[8.2]} > > ------------------------------- > > The way to fix this problem is to change the value of $MinPrecision > as I do below. Then my definition for (f) doesn't apply to f[-2.1], > f[4.3]. This will also ensure (x==x+1) returns False. I haven't > checked but a positive value for $MinPrecision might solve the > problem Bernd Brandt had with NIntegrate. > > In[6]:= > $MinPrecision=1.0; > x=1.11111111111111111111; > Do[x=2*x-x, {100}]; > Clear[f]; > f[x]=29; > {f[-5.2],f[-2.1],f[4.3],f[8.2]} > > Out[11]= > {f[-5.2],f[-2.1],f[4.3],f[8.2]} This behavior isn't necessarily a problem, since it is all correct, and changing $MinPrecision isn't necessarily a reasonable way to "fix" it. To understand the behavior of this example it is useful to recall that inexact numbers correspond to intervals. With 3-digit decimal arithmetic, for example, the number 1.23 is the best available 3-digit decimal representation for any number between 1.225 and 1.235, so mathematically this number represents the interval between those points. For any inexact number, the numerical "roundoff error" of the number gives the width of the corresponding interval. The Do[x=2*x-x, {100}] example generates a value of x that corresponds to an interval around 1 and with a width of approximately 2. This interval arises through application of standard propagation of error in which the error in the result is estimated by accumulating upper bounds for the error in each step, with all errors assumed to be uncorrelated. In this example, which repeatedly adds and subtracts the same number, all of the errors are perfectly correlated, so this calculated upper bound to the error is significantly larger than the actual error. Most practical calculations aren't like this. In most practical calculations, the assumption of uncorrelated errors is a good assumption, or at least it is better than any realistic alternative assumption. With that value of x, the rule for f[x] will apply to any argument in the interval associated with the value of x. By setting $MinPrecision to a larger value you can prevent the estimated error from accumulating beyond a certain value, but without analyzing the calculation there is no mathematical justification for doing such a thing. It might be justified in this example, which is contrived so that the numerical errors in each step do not accumulate, but resetting $MinPrecision is not justified in general, and is not recommended. Your next example is unrelated to the previous examples. It also has a straightforward explanation. > --------------------------- > Now consider another example. Below Mathematica thinks (b1) has > much better precision than (a1). This doesn't make sense, and > Mathematica doesn't do much better using the default setting > ($MinPrecision=0.0). > > In[12]:= > a1=Exp[800.0]/Exp[800]; > b1=a1+Pi-1; > SetOptions[Precision,Round->False]; > (* Precision only has an option in Version 4. *); > {Precision[a1],Precision[b1]} > > Out[15]= > {13.0515,25.2788} With $MinPrecision set to zero, the accuracy of the zero generated by a1-1 must be quite large to get a precision bigger than zero. That large accuracy shows up in the precision of b1, which results from adding that high-accuracy zero to the exact constant Pi. Removing this minimum on precision gives the behavior in your next example: > The result above comes out much better if $MinPrecision is much less than > zero. However, I can't understand why (b1) below still has slightly better > precision than (a1). > > In[16]:= > $MinPrecision=-Infinity; > a1=Exp[800.0]/Exp[800]; > b1=a1+Pi-1; > {Precision[a1],Precision[b1]} > > Out[19]= > {13.0502,13.5473} and the reason for the precision of b1 being slightly higher than the precision of a1 should be obvious upon recalling the definition of precision. Precision in Mathematica is a measure of relative error. If you make the number bigger without adding any error, which is what happens when adding a positive exact constant (such as Pi) to this positive number, the relative error goes down, so the precision goes up. Returning then to the first example, but with $MinPrecision=Infinity: > ---------------------------------- > Well, ($MinPrecision=-Infinity) allows a better result from the > last example, but now the definitions for (f) that I considered > earlier are even more strange. > > In[20]:= > x=1.11111111111111111111; > Do[x=2*x-x, {100}] ; > Clear[f]; > f[x]=29; > {f[-534.2],f[-2.1],f[4.3],f[815.2]} > > Out[25]= > {29,29,29,29} yes, releasing the lower bound on precision widens the interval available to the value of x. And finally, yes, reading up on this subject is certainly recommended for anyone who is interested in these details, and yes, the following discussion is a good place to start: > ---------------------- > A very good discussion of arbitrary precision arithmetic can > be found in the Help Browser under: > Getting Started/Demos > Demos > Numerics Report (near the bottom) Although Mathematica has perhaps given wider exposure to the topic of reliable arithmetic, this has been an active area of research for decades, and there is extensive literature on the subject. If you are interested in this subject you might try exploring the literature in a good technical library. There are very few questions about the behavior of this aspect of Mathematica that can't be answered by a review of The Mathematica Book, study of the above mentioned Numerics Report, and a few minutes of careful thought. > -------------------- > Regards, > Ted Ersek Dave Withoff Wolfram Research

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