Re: [boost] numeric: linear algebra concepts

From: Jeremy Siek (jsiek_at_[hidden])
Date: 2001-03-29 11:02:29

I've updated the linear algebra concepts with semantics. There
still a couple missing, but it is 90% there.

Cheers,
Jeremy

On Thu, 29 Mar 2001, Ullrich Koethe wrote:

koethe> Jeremy Siek wrote:
koethe> > 
koethe> > I've briefly sketched out the concepts for a "linear algebra" that we were
koethe> > planning on using for MTL v3.0.
koethe> > 
koethe> > http://www.lsc.nd.edu/~jsiek/tlc/multi_array/libs/numeric/doc/LinearAlgebraConcepts.html
koethe> > 
koethe> > Cheers,
koethe> > Jeremy
koethe> 
koethe> Very good list! However, the necessary constraints are missing, for
koethe> example:
koethe> 
koethe> Additive Abelian Group:
koethe> 
koethe> 1.1) a + b == b + a
koethe> 1.2) a - a == zero(a)
koethe> 1.3) a + zero(a) == a
koethe> 1.4) zero(a) - a == -a
koethe> 
koethe> Multiplicative Ring with Identity:
koethe> 
koethe> 2.1) (a * b) * c == a * (b * c)
koethe> 2.2) a * identity(a) == a
koethe> 
koethe> Field
koethe> 
koethe> 3.1) (a + b) + c == a + (b + c)
koethe> 3.2) a * b == b * a
koethe> 3.3) a * (b + c) == a * b + a * c
koethe> 3.4) a / b is undefined if and only if b == zero(a)
koethe> 3.5) a * b == zero(a) if and only if a == zero(a) or b == zero(a)
koethe> 
koethe> Unfortunately, these constraints lead to serious practical problems,
koethe> because none of the build-in types fulfills them: 
koethe> 
koethe> float and double fail because of limited accuracy (especially on
koethe> conditions 2.1 and 2.2 - what does the IEEE floating point standard say
koethe> here ?)
koethe> 
koethe> the integer types fail on the condition (3.5), e.g.
koethe> 
koethe> unsigned char a = 16;
koethe> a * a == zero(a)
koethe> 
koethe> How can these problems be solved?
koethe> 
koethe> Regards
koethe> Ulli
koethe> -- 
koethe>  ________________________________________________________________
koethe> |                                                                |
koethe> | Ullrich Koethe  Universität Hamburg / University of Hamburg    |
koethe> |                 FB Informatik / Dept. of Computer Science      |
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koethe> |                                                                |
koethe> | Phone: [phone]                Vogt-Koelln-Str. 30 |
koethe> | Fax:   [phone]                D - 22527 Hamburg   |
koethe> | Email: u.koethe_at_[hidden]               Germany             |
koethe> |        koethe_at_[hidden]                        |
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----------------------------------------------------------------------
 Jeremy Siek                        www: http://www.lsc.nd.edu/~jsiek/
 Ph.D. Candidate                  email: jsiek_at_[hidden]
 Univ. of Notre Dame         work phone: [phone]
----------------------------------------------------------------------