Re: numeric alg. inverting an array

From: CBFalconer (cbfalconer_at_yahoo.com)
Date: 01/12/04 

Date: Mon, 12 Jan 2004 02:09:10 GMT

Willem wrote:
>
... snip ...
>
> For those who don't know what a 'field' is: (I must admit I don't know
> exactly what it is myself, just that it has some interesting properties)
>
> The idea is that you have a finite set of numbers (in this case 2^16
> numbers), with the operations 'addition', 'subtraction', 'multiplication'
> and 'division' defined in such a way that the following hold:
>
> For *all* a and b in the set, there is a value c in the set so that a+b=c
> (Same goes for subtraction, multiplication and division, except divide by
> zero.)
> Also, a+b=c implies c-b=a, and a*b=c implies c/b=a.
> That is, you can multiply and divide with complete accuracy.
>
> (These 2^16 elements of the set are mapped 16-bit words, by the way.)
>
> This means that numerical stability doesn't enter the picture for this
> application.

IIRC a field is a set that has some operations and exhibits
closure over those operations. I am not sure whether it needs the
inverse operations. Some breeds of things have identity values,
so that "thing op identity --> thing". The identity value for
addition is zero, for multiplication is 1, in the natural number
fields. Or something like that. I recently found out that the
young professor who taught me all those things is dead after a
productive life, so there may be lots of dropped bits.

Once you get that all straight proceed to isomorphism, and avoid
reproving things about other things. 

-- 
Chuck F (cbfalconer@yahoo.com) (cbfalconer@worldnet.att.net)
   Available for consulting/temporary embedded and systems.
   <http://cbfalconer.home.att.net>  USE worldnet address!

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