       Re: Why Mathematica does not issue a warning when the calculations

• To: mathgroup at smc.vnet.net
• Subject: [mg117653] Re: Why Mathematica does not issue a warning when the calculations
• From: John Travolta Sardus <pireddag at hotmail.com>
• Date: Tue, 29 Mar 2011 06:54:12 -0500 (EST)
• References: <imfa3k\$iks\$1@smc.vnet.net>

```On 24.03.11 12:32, Daniel Lichtblau wrote:

(several things and)

> precision tracking and fixed precision

ok, so I realize there is a deeper point which I had failed to
appreciate properly. I will take my time to read through the
documentation, although probably I won't understand it up to the lat detail.

Anyway, when writing I had something in mind, somewhat naive, but I
thought it had something to do with the general issue (that is, it would
be nice to have software that *always* warns when the calculations are
losing precision).

First a (very naive) point of view

- when a calculation is impossible for a computer, then just do not do
it ... do not subtract two very large numbers that are almost equal to
each other hoping to get their difference correct to the n-th decimal
digit (ok, almost joking here)

What I actually had in mind

- there are algorithms which, I know, can give a nice rule on the
expected usefulness of the result beforehand (the most important case
that I know of is the condition number for matrix inversion: by the way,
I have verified that Mathematica issues the usual warning when dealing
with an ill-conditioned matrix, and that stating a precision). When
reflecting about it, it seems very unlikely that there is a simple
criterion for every algorithm (quoting again ...
> Many computations cannot be done with good
> error control, or a priori estimates of needed initial precision.
So, to make my original question more precise, are the algorithms
corresponding to each Mathematica command (of which there is only a
finite set ...) classified into "algorithms with good error bounds" and
"algorithm without" and is there such a list? I guess not, and in any
wants to use Mathematica with its "arbitrary precision arithmetic".

Just to make an (extreme?) example of what can happen, I was fooling
around with ill-conditioned matrices, to see whether I could obtain
accurate results for the solution of ill-conditioned linear systems
using the second argument to the N function. After a while of playing
around I entered the following (I wanted to introduce a new vector of
known values into my calculations)

b = {1, -999999999999998}

Resulting into the output:

{1, -999997999000002}

I guess one needs first to learn how to use Mathematica, then he can use
it effectively

```

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