[Axiom-developer] [#87 solve(x + 1.1, 0.001) fails] reply and experimenting with new format

[anonymous]

++added:
{\tt [Kostas Oikonomou <address@hidden> wrote:]}


++added:
{\tt [William Sit <address@hidden> replied:]}


??changed:
-One should be careful interpreting these results.  The second one solves it to 
5 binary digit accuracy (closest binary to 0.1 (decimal) is 1/16+1/32 = 
0.09375) hence the answer is not {\tt -1.1}. A similar loss happens in the 
third one. To obtain {\tt -1.1} as the solution, one needs much higher 
accuracy, as in the fourth one (that is the minimum needed).
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-From unknown Fri Feb 11 07:25:14 -0600 2005
-From: 
-Date: Fri, 11 Feb 2005 07:25:14 -0600
-Subject: Not a bug. 
-Message-ID: <address@hidden>
-
-Severity: normal => minor 
-Status: open => closed 
-
-
-From KostasOikonomou Sat Feb 12 09:03:08 -0600 2005
-From: Kostas Oikonomou
-Date: Sat, 12 Feb 2005 09:03:08 -0600
-Subject: [#87 solve(x + 1.1, 0.001) fails]
-Message-ID: <address@hidden>
-
-
-[9 more lines...]
One should be careful interpreting these results.  The second one solves it to 
5 binary digit accuracy (closest binary to {\tt 0.1} (decimal) is {\tt 
1/16+1/32 = 0.09375}) hence the answer is not {\tt -1.1}. A similar loss 
happens in the third one. To obtain {\tt -1.1} as the solution, one needs much 
higher accuracy, as in the fourth one (that is the minimum needed).


{\tt [[Kostas Oikonomou Sat Feb 12 09:03:08 -0600 2005 replied:]]}

++added:
{\tt [[William Sit replied:]]}

Of course, yes. However, there is a dilemma: when you give Axiom an equation 
with floating point coefficients, should Axiom "solve" this algebraically, as 
if Float is just like any other domain, or numerically, giving Float a special 
treatment? Since Axiom algorithms are categorical, rather than writing two 
separate algorithms, Axiom solves, if possible, algebraically (that is, 
exactly) and gives numerical answers as options when the precision parameter is 
given. This choice does not work well with equations over Float because Float 
does not have some of the algebraic properties as Fraction Integer or Fraction 
Complex Integer (such as factorization or GCD), which is why there is a warning 
in solve(x^2-1.234). The package is numsolve.spad and you see that these 
restrictions are well documented. So the above signature is really not meant to 
be used at the moment. A similar situation occurs, for example factor(1.23) is 
legal, but is really useless. Axiom does not use a mechanis!
m to exclude specific domains from a category. It adopts an "include" 
philosophy but let things fail with warning or error. If you look into 
numsolve.spad, you will find that the innerSolve1 algorithm {\it 
implementation} is restricted. (So if later someone finds a way to implement a 
solve algorithm over Float, that would be just fine).

So a lot of Axiom failures are not bugs, but by design. One way to improve the 
user interface would seem to be to automatically lifting a polynomial over 
Float to one over Fraction Integer. A moment's reflection would convince you 
this is not always possible (for example, sqrt(2) or pi are technically both 
belong to Float (model for real numbers), but of course, in reality, every 
floating point number is a rational number. Such a lifting package would have 
to take into consideration the precision to convert some symbolic constants to 
a decimal approximation and then convert that to an exact rational number. 
However, even this would not create satisfactory results because we know the 
sensitivity of solutions of polynomial equations to small changes of its 
coefficients. Wilkinson has this example 
\begin{equation*}
f(x) = (x+1)(x+2) \cdots (x+20) = x^20 + 210 x^19 + \cdots + 20! = 0
\end{equation*}
where a change of the coefficient 210 by $2^{-23} \approx 1.2 \times 10^{-7}$ 
would turn the root $-20$ to $-20.8$ and five pairs of zeros to complex roots. 
So if we want numerically accurate solutions, we should use a robust numerical 
library. I believe this is not yet available in Axiom (the NAG version allowed 
interface with its Fortran libraries, at extra costs).

If we are really (no pun intended) only using truely floating point 
coefficients, then it can easily be converted to Fraction Integer, but one has 
to beware that the algorithm would take a very long time because exact 
arithmetic with large integer coefficients are expensive.

{\tt [[Kostas Oikonomou wrote:]]}


++added:
\begin{axiom}
solve(x^2 - 1.234)
\end{axiom}

??changed:
-                                       Kostas
-
-
-From MartinRubey Sat Feb 12 13:07:55 -0600 2005
-From: Martin Rubey
-Date: Sat, 12 Feb 2005 13:07:55 -0600
-Subject: [#87 solve(x + 1.1,   0.001) fails] [#87 solve(x + 1.1, 0.001) fails]
-Message-ID: <address@hidden>
-In-Reply-To: <address@hidden>
{\tt [[Martin Rubey Sat Feb 12 13:07:55 -0600 2005 &lt;address@hidden> wrote:]]}

??changed:
-Martin
-
{\tt [[William Sit wrote:]]}

Yes, as explained above. Algebraic methods do not work well with Float. 
                        


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