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From: Matthias Troyer (troyer_at_[hidden])
Date: 2004-07-06 04:54:11

On Jul 6, 2004, at 9:51 AM, Peter Schmitteckert (boost) wrote:
> On Tuesday 06 July 2004 03:06, Jeremy Graham Siek wrote:
>> A *linear algebra* algorithm will never need to be generalized
>> in the fashion you advocate below. That's because matrices are
>> about systems of equations. A system of equations has a certain
>> number of unknowns and a certain number of equations.
>> Once the system is represented as a matrix, the number of rows
>> corresponds to the number of equations, and the number of
>> columns corresponds to the number of unknowns. That's it.
>> There's no more numbers to talk about.
> Here I have to disagree. But the problem lies in the fact, that I'm a
> theoretical
> physicist, who learned numerics by doing. For me, matrices are far
> more, than just
> objects used to solve linear set of equations.
> Matrices are representations of algebras, they can be fused (e.g.
> tensor products),
> and one can calculate general functions on matrices, like the matrix
> exponential.
> In my work, vectors have double indices, i.e. a matrix has four
> indices,
> but it is still a matrix, the double index is just an implementation
> feature,
> since vectors are representated by a dyadic product of other vectors.
> You can now argues, wether this is still a vector, but in physics it
> is called
> a vector.

As another theoretical physicist I want to disagree with this
definition. I would call the object with four indices a linear
operator, but not a matrix. Matrices for me are representation of
linear operators with two indices.

You however point out an important requirement for generic algorithms
on vector spaces: they should not require that a vector can be accessed
with operator[] and a single subscript, or that once can construct a
vector by passing just the size to the constructor. These too narrow
requirements of the Iterative Template Library ITL, caused us to
introduce the "vector space" concept in the our Iterative Eigenvalue
Template Library IETL.


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