Re: [Axiom-developer] symbolic matrix multiply

[Mike Dewar]

Tim,

James Davenport had a post-doc, Christele Faure, who did a lot of work
on what you are calling indefinite integers in Axiom about 10 years ago.
The only reference I have is:
  James H. Davenport and Christ`ele Faure.
  The "unknown" in computer algebra.
  Programmirovanie, 1(1), 1994.
I'm not sure what happened to the code - it was quite complicated if I
recall correctly.  Last I heard Christele was at INRIA in
Sophia-Antipolis working on automatic differentiation, James may have more
up-to-date information.

Cheers, Mike.



On Tue, Jun 22, 2004 at 09:58:42AM -0400, Tim Daly wrote:
> Richard,
> 
> Your paper
> http://www.cs.berkeley.edu/~fateman/papers/symmat2.pdf
> is very interesting.
> 
> We've been looking at the issue in terms of indefinites.
> In the current version of Axiom if you type:
> 
>   x+1
> 
> 'x' is known to be a symbol
> '1' is known to be an integer
> there is no plus which takes +(symbol,integer)
> so 'x' gets promoted to polynomial over integers
> '1' gets promoted to polynomial over integers
> '+(POLY(INT),POLY(INT)) exists
> and the result is
>   x + 1
>          Type: Polynomial Integer
> 
> Now it is often convenient, and especially important for further
> research work we want to do, to be able to specify that 'x' is
> an "indefinite integer". Thus there can be a signature 
>   +(Indefinite(Integer),Integer) -> Indefinite(Integer)
> so that
>   x + 1
>          Type: Indefinite Integer
> 
> This is a slightly more primitive notion than matrices of 
> indefinite size but the ideas are essentially the same.
> 
> Indeed, the idea of Indefinite(R) where R is a domain is
> the generalization. Thus, for your example, in Axiom the 
> appropriate type would be 
>   Matrix(Indefinite(Integer),Indefinite(Integer))
> 
> We can clearly construct such types in Axiom. What the 
> mathematically correct reasoning would be and what algorithms
> apply is an interesting question that we need to explore.
> 
> The key issue is that symbolic computation systems do very
> little "symbolic" computation (hasty generalization to make
> the point). We'd like to be able to do computation "along
> the theorem line" (that is, reasoning with known theorems)
> rather than basic algebra.
> 
> Comments?
> 
> Tim
> 
> 
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