Re: Why?

*To*: mathgroup at smc.vnet.net*Subject*: [mg110598] Re: Why?*From*: Richard Fateman <fateman at cs.berkeley.edu>*Date*: Sun, 27 Jun 2010 04:56:17 -0400 (EDT)*References*: <hto93n$kt8$1@smc.vnet.net> <hviapu$h6s$1@smc.vnet.net> <i049do$5sq$1@smc.vnet.net>

Noqsi wrote: > On Jun 21, 12:11 am, Richard Fateman <fate... at cs.berkeley.edu> wrote: >> Noqsi wrote: > >>> Approximation is often puzzling. The ideological war you wage against >>> WRI is unhelpful here. The wise person understands that there are >>> multiple points of view. >> If it is required to be wise in the ways of WRI's (unusual) arithmetic >> to use WRI software, then that is more than a user interface problem. > > You must be wise in the ways of the tool to use it. Very Zen. Or maybe Karate Kid? That's always > true. There's nothing wrong with being "unusual": Mathematica is > unusual in quite a few ways, and that's key to its ability to quickly > dispose of problems that are more difficult with other tools. When a > different tool is better, just use that and quit carping. I suppose you count as "carping", a complaint when the tool disposes of problems with incorrect or misleading answers. > >> Your IEEE754-based ideology has its own >> >>> weaknesses: abuse of the concepts of "rational" and "finite", and weak >>> connection to the real number system. Matthew 7:3 applies. >> I'm not sure what you mean by IEEE754-based ideology. > > The ideologue never understands his own ideology. And it appears that you are unable to explain your words. The IEEE-754 binary standard embodies a good deal of the well-tested wisdom of numerical analysis from the beginning of serious digital computing through the next 40 years or so. There were debates about a few items that are peripheral to these decisions, such as dealing with over/underflow, error handling, traps, signed zeros. >> Probably the >> whole community of numerical error analysts agrees on a model that is >> different from WRI's. > > In other words, your fellow ideologues agree with you. Unanimity in an > area as tricky as this is a sure symptom of groupthink: difficult > problems demand *multiple* points of view for truly effective > understanding. Ah, the Bozo the Clown theory. They laughed at Columbus, but he was right [so it goes... did they really? but anyway...] Hence: They laughed at Me, so I am right. Neglecting the other set of examples, such as They laughed at Bozo the clown ..... Even in physics, where we think we are all working with > a common reality, we have multiple ways of looking at it. Yes, I especially like the flat earth society http://theflatearthsociety.org/cms/ > Numerical > analysis lacks that common reality: it serves a diversity of > applications, with a diversity of requirements. Computing can model mathematics, mathematics can model reality. At least that is the commonly accepted reason we still run computer programs in applications. Good numerical computing tools allows one to build specific applications. For example, one would hope that the computing tools would allow an efficient implementation of (say) interval arithmetic. This is fairly easy with IEEE-754 arithmetic, but much much harder on earlier hardware designs. The basic tool that Mathematica (and many other systems) provides that might be considered a major extension in a particular direction, is the arbitrary precision software. Mathematica has a different take on this though, trying to maintain an indication of precision. None of the other libraries or systems that do arbitrary precision arithmetic have adopted this, so if it is such a good idea, oddly no one else has taken the effort to mimic it. And it is not hard to mimic, so if anyone were to care, it could be done easily. Apparently people do not want some heuristically determined "fuzz" to be mysteriously added to their arithmetic. I do not know how much of the code for internal arithmetic for evaluation of functions in Mathematica is devoted to bypassing these arithmetic features, but based on some examples provided in this newsgroup, I suspect this extra code, which is an attempt to mimic arithmetic without the fuzz inserted by Mathematica, becomes a substantial computational burden, and an intellectual burden on the programmer to undo the significance arithmetic fuzz. > > Where I fault Mathematica's design here is that "Real" wraps two > rather different kinds of objects: fixed-precision machine numbers, > and Mathematica's approximate reals. Both are useful, but > understanding and controlling which kind you're using is a bit subtle. > "Complex" is even more troublesome. I know of 4 systems which provide arbitrary precision numbers that mimic the IEEE-754 arithmetic but with longer fraction and exponent fields. Perhaps that would provide the unity of design concept that you would prefer. One just increases by a factor of 4 (quad double), the other are arbitrary precision. RJF >