Glas List: Re: [glas] norms and inner products for vectors of matrices

From: Toon Knapen (toon.knapen_at_[hidden])
Date: 2006-01-10 03:48:51

What exactly a vector of matrices is is clearly interpreted differently
by different users. We had a similar situation with the interpretation
of the product of 2 vectors. However the vector of matrices has no
direct algebraic meaning so I suggest that Glas would not directly
support it. However Glas should provide the necessary flexibility such
that the user is able to extend Glas to make this possible and to make
it do exactly what the he wants.

Karl Meerbergen wrote:
> I suggest we first agree what we mean by vector< matrix<T> > ? Why do we 
> need it? What do we want to use it for?
> 
> I do not interpret vector<matrix<T> > as a matrix, but a vector with 
> matrix<T> as value_type.
> Strictly algebraically speaking, vector<matrix<T> > is an unusual 
> concept, so I would avoid its use if possible.
> 
> One possibility is to consider vector<matrix> as a matrix, but which is 
> its size? Suppose v[0] is a 2x3 matrix and v[1] a 4x1 matrix, which is 
> the number of columns of v?
> 
> An argument in favour of vector<matrix> is the possibility for blocking 
> in algorithms. But I think there is a better way than vector<matrix> for 
> doing this.
> 
> Perhaps we could introduce the notion of grid, where a matrix (or a 
> vector) can be mapped onto this grid for performing blocking operations. 
> For example, a matrix mapped onto a (1x10) grid still is a matrix and 
> not a vector<matrix>. The algorithms use the grid for computational 
> purposes or for accessing the data. To the outside world, the matrix 
> behaves like a matrix, while the algorithmic internals use the blocking 
> information.
> 
> The grid_type could be a template argument of vector and matrix.
> 
> 
> Karl
> 
> 
> 
> 
> Andrew Lumsdaine wrote:
> 
> 
>>I think the definition of the norm for a vector of matrices depends  
>>on what is meant mathematically by a vector of matrices.  If the  
>>vector of matrices is a block row (or block column) sliced out of a  
>>block matrix, then the definitions shown would not be the expected  
>>norms for that portion of the matrix without the blocking.  And I  
>>think, at least in some cases, one would want those norms to be the  
>>same.
>>
>> 
>>
> 
> 
> 
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-- 
Toon Knapen
Skype Name: toonknapen
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