Linear Algebra API - Part I : Vector and Matrix

[ratnania]

Up to now, we were relying on sympy.Matrix and its variants to handle our Linear Algebra operations.
It is now time to have our own Linear Algebra building blocks, and it starts with the Vector and Matrix objects.
This issue aims to:

• have a new API for Vector and Matrix
• remove the use of Tuple from all our examples (sympde and psydac)
• remove the use of sympy.Matrix-like objects appearing inside sympde, when possible

The new objects should be used in two different ways:

• As containers. example:

b = Vector([1.,2.,3.], name='b')
M = Matrix([[1.,2.],[3.,4.]], name='M')

• as variables. example:

b = Vector(name='b', is_real=True)
M = Matrix(name='M', is_complex=True)

in the which case, these variables will appear as free-variables in our expressions (linear, bilinear forms, etc)

[yguclu]

I feel that we need to clarify what kind of entries are allowed in a Vector or Matrix.

If any expression of the spatial coordinates is allowed, then it behaves just like a Function, but without specifying the arguments. In fact, it would seem natural to subclass Function to get the classes ScalarFunction, VectorFunction, and MatrixFunction. Of course this would create a conflict with the pre-exististing classes ScalarFunction and VectorFunction, which represent elements of a ScalarFunctionSpace and VectorFunctionSpace, respectively.

[ratnania]

Yes, we should differentiate between a (Scalar/)Vector/Matrix valued functions and other expressions.
This means that we need to check, at the construction, that Vector/Matrix expressions do not contain the coordinates.
For the (Scalar/)Vector/Matrix valued functions, we'll move the discussion to the associated issue.

[ratnania]

This issue is closely related to #149