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From: hbaker@netcom.com (Henry Baker)
Subject: Re: arbitrary-precision real arithmetic
Message-ID: <hbaker-2808951906100001@192.0.2.1>
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References: <hbaker-2608950928260001@192.0.2.1> <DDzExB.KGv@bath.ac.uk> <hbaker-2708951704280001@192.0.2.1> <41tac3$pci@fsuj01.rz.uni-jena.de>
Date: Tue, 29 Aug 1995 03:06:10 GMT
Lines: 43
Xref: glinda.oz.cs.cmu.edu sci.math.symbolic:18910 comp.lang.functional:6294 comp.lang.lisp:18976

In article <41tac3$pci@fsuj01.rz.uni-jena.de>, prm@rz.uni-jena.de (Ralf
Muschall) wrote:

> In article <hbaker-2708951704280001@192.0.2.1> hbaker@netcom.com (Henry
Baker) writes:
> :: I forgot to say that for algebraic numbers (roughly solutions of
polynomials
> :: with integer coefficients -- e.g., sqrt(2)), there is a very
straightforward
> :: way to calculate with them -- by representing such an algebraic number by a
> :: square matrix whose characteristic polynomial is the minimal
polynomial which
> :: has the given algebraic number as a root.
> 
> :: The Cayley-Hamilton theorem tells you that one of these matrices satisfies
> :: its own characteristic polynomial, so you can now calculate with the set
> :: of square matrices generated by one of these particular matrices.
> 
> A layman's question:
> 
> Wouldn't this approach map all zeros of the polynomial to the same
> [set of] matri[x|ces]?
> 
> I.e., sqrt(2) and (-sqrt(2)) would be indistinguishable.
> This might not be a problem in pure algebra, where they
> *are* indistinguishable, but in other sciences, the users might
> want to know whether a number is +1.414 or -1.414.

In the mapping a+bi => 
[a -b]
[b  a]

the sign of b is remembered.  I believe that similar conventions are possible
with other matrices to remember which zero a particular matrix indicates.

The general problem of algebraic numbers is that the roots _are_ pretty
interchangeable--that's one of the reasons why polynomials are relatively
difficult to solve, so you have to go to some effort outside the realm of
the usual algebraic operations in order to distinguish them.

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