 # Glas :

## Re: [glas] concepts, algorithms etc

From: Matthias Troyer (troyer_at_[hidden])
Date: 2005-03-11 10:18:29

On Mar 11, 2005, at 6:16 AM, Karl Meerbergen wrote:
> My question is: why do we need a VectorSpace concept?

When writing the iterative eigensolver template library (IETL), a
generalization of the ITL to eigenvalue problems, we encountered the
fact that the concepts used in the ITL were not general enough and had
to introduce a vector space comcepts.(see
http://www.comp-phys.org/software/ietl/ to obtain the IETL library)

The problems that made us introduce a vector space concept were:

1.) creation of new vectors
2.) obtain the dimension of a vector space
3.) projecting vectors from a "representation" vector space to the real
"physical" vector space.

Before explaining the above points let me quickly show what functions
and types we introduced:

For a model VS of the concept vector space we defined the following
types

vectorspace_traits<VS>::vector_type
vectorspace_traits<VS>::scalar_type
vectorspace_traits<VS>::size_type

and the following functions:

typename ietl::vectorspace_traits<VS>::vector_type new_vector(const
VS& vs)
typename ietl::vectorspace_traits<VS>::size_type vec_dimension(const
VS& vs)
void project(typename ietl::vectorspace_traits<VS>::vector_type& v,
const VS& vs)

Now let me explain why we needed these functions and concepts by
comparing to what the ITL did:

1.) new_vector:

Here the ITL assumed that a new vector of type V could be created by
calling a constructor taking the size of the vector, e.g. given the
dimension d of the vector space it assumed that a new vector can be
constructed by

V x(s);

In our applications this caused several problems, where I will just
mention two typical problems:
a) to solve the eigenvalue problem for a 3D partial differential
equation on a regular mesh we most naturally use a three-dimensional
Blitz array (or boost::multi_array) as our data type. These 3D arrays
form a vector space, but have no constructor of the above type.
b) for parallel vectors we sometimes want to pass information on how
the vector should be distributed over the nodes in the constructor.

The new_vector(vs) function solves these problems.

2.) vec_dimension:

allows us to obtain the dimension of the vector space, without creating
a vector.

3.) project:

came up when we realized that in some applications the actual vector
space we are interested is actually a subspace of the vector space
spanned by the data type, defined by some constraints. One example is
again solving a partial differential equation, with fixed boundary
conditions. The variables might again be stored in a 3D array, but
where the values in the boundary layer are fixed. Solving such an
eigenvalue problem by an iterative eigensolver requires starting from a
random starting vector. Obviously we cannot just fill the entire vector
with random numbers, since this would give us wrong values on the
boundaries. The project() function can then be used to make sure that
the boundary values are correct, thus "projecting" the vector from the
vector space spanned by the data type (with arbitrary boundary values)
to the desired vector space (with fixed boundary values).

A student of mine will soon start a term project to add these vector
space features to the ITL library, so that we can also use the ITL on
our data types. We'll keep you updated on our progress.

Matthias