[Axiom-developer] [#87 solve(x + 1.1, 0.001) fails] LaTeX not working

[anonymous]

??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^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 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^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 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 
&times; 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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