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

**From:** Peter Gottschling (*pgottsch_at_[hidden]*)

**Date:** 2006-01-11 20:13:09

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I wonder what vectors of matrices are intended to be used for.

Matrices of matrices are as far as I see used for blocking, e.g.

matrix< matrix<T, fixed_size<2, 3> >. The corresponding vector types

to this matrix would also be blocked but this are vectors of vectors.

In the example it would be vector< vector<T, fixed_size<3> > x and

vector< vector<T, fixed_size<2> > y to compute y= A*x in the canonical

way.

If there is no other reason for having matrix< matrix> I don't see a

need to have vector< matrix > and vector< vector > is what we need to

consider.

Concerning the norms, I would like to limit the result type to real

values as I have seen it in all mathematical definitions. Does

somebody sees a reason to vectors or matrices as results?

The definitions of vector<vector> norms are straight forward:

- norm_1 := sum (norm_1(x[i])

- norm_2 := sqrt(sum(norm_2(x[i])^2)) // there might be more

efficient ways to compute this

- norm_inf := max(norm_inf(x[i])

Best,

Peter

On 09.01.2006, at 11:22, 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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>

------------

Peter Gottschling

Research Associate

Open Systems Laboratory

Indiana University

301i Lindley Hall

Bloomington, IN 47405

Tel.: +1 812 855-8898 Fax: +1 812 856 0853

http://www.osl.iu.edu/~pgottsch

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