Confusing terminology in singular-poisson demo

[Nikolaus Rath]

(repost from https://bugs.launchpad.net/dolfin/+bug/1267578 - didn't realize that the issue tracker has moved)

http://fenicsproject.org/documentation/dolfin/1.3.0/python/demo/documented/singular-poisson/python/documentation.html says:

[Let ] AU=b, where U gives the coefficient for the basis functions
expressing u.

Since we have pure Neumann boundary conditions, the matrix A
is singular. There exists a vector e such that Ae=0.

span {e} is the null space of A, and by removing the components
of b that lie in the null space we make the system solvable.

The last part does not make sense to me. Let A be a linear operator from V to W. Then the null space is a subspace of V, but b is defined in W. Therefore, it doesn't make sense to talk about components of b that lie in the null space.

I think a better formulation might be

[Let] AU=b, where U gives the coefficient for the basis functions
expressing u.

Since we have pure Neumann boundary conditions, the matrix A
is singular. There exists a vector e such that Ae=0. We make the
system solvable by removing the components of b that do not lie
in the column space of A, which are also the components that lie
in the null space of transpose(A).

[Anders Logg (Chalmers)]

Fix issue #186: Confusing terminology in singular-poisson demo

→ <<cset 25c9fa4c96f0>>

[Anders Logg (Chalmers)]

Changed to this:

Since we have pure Neumann boundary conditions, the matrix :math:`A`
is singular. There exists a non-trival vector :math:`e` such that

.. math::
    Ae=0.

span :math:`\{ e \}` is the null space of A. Consequently, the matrix
:math:`A` is rank deficient and the right-hand side vector :math:`b`
may fail to be in the column space of :math:`A`. We therefore need
to remove the components of :math:`b` that do not lie in the column
space to make the system solvable.

[Prof Garth Wells]

Removing milestone: 1.4 (automated comment)