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[Axiomdeveloper] [#87 solve(x + 1.1, 0.001) fails] LaTeX not working
From: 
anonymous 
Subject: 
[Axiomdeveloper] [#87 solve(x + 1.1, 0.001) fails] LaTeX not working 
Date: 
Sun, 13 Feb 2005 02:52:46 0600 
??changed:
solve: Polynomial Float>List Equation Fraction Polynomial Float
'solve: Polynomial Float>List Equation Fraction Polynomial Float'
??changed:
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^21.234). 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 mechani!
sm 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).
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^21.234)'. 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 no!
t use a mechanism 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
<center>
f(x) = (x+1)(x+2) ... (x+20) = x<SUP>20</SUP> + 210 x<SUP>19</SUP> + ... + 20!
= 0
</center>
where a change of the coefficient 210 by 2<SUP>23</SUP> (approximately 1.2
× 10<SUP>7</SUP>) would turn the root 20 to 20.8 and five pairs of
zeros to complex roots. (This perturbed equation will take a *very long* time
in Axiom, will not be solved exactly by Mathematica, but is trivially solved
*numerically* in Mathematica).
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).

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 [Axiomdeveloper] [#87 solve(x + 1.1, 0.001) fails] LaTeX not working,
anonymous <=