Speed up lattice precision p-adics

[saraedum]

• Drop labels keyword in parents & make parents more unique
• Switch to templates for elements
• Switch to cython for precision lattice
• Specialized data structure for matrix
• Make pickling and conversion more robust (pickling should dump the full lattice matrix and unpickling restores the matrix but puts it next to the existing matrix)
• Share precision lattice between all parents with the same p.

Depends on #23505

CC: @roed314 @xcaruso @sagetrac-TristanVaccon

Component: padics

Keywords: lattice precision

Issue created by migration from https://trac.sagemath.org/ticket/24809

[xcaruso]

comment:1

About pickling/unpickling, I propose the following interface:

• (usual syntax) if x is a p-adic, dumps(x) saves x as approx + O(p^prec) but diffused digits implying x are not saved
• if R is a parent (and/or a precision lattice maybe), R.dumps_elements(x,y,z,...) saves the approximation of all listed elements together with a lattice representing the precision on this set of elements
• loads(obj) loads the stored variables in obj (they can be many of them if obj was constructed by dumps_elements) as new independant variables.

How does it sound?

[saraedum]

comment:2

I am not sure how dumps_elements would integrate with the usual pickling workflow, say when pickling something like a matrix.

I think a simpler interface is not harder implement and makes things less surprising:

• Whenever we pickle an element we pickle its entire precision lattice. So, if we happen to pickle many elements at the same time, say a polynomial, we pickle their lattice (once.)
• Restoring elements just appends the pickled lattice to the existing lattice and does a "drop unused columns".

While this is not a very space-efficient format (but I guess we don't care) it gives us both behaviours you describe above automatically, doesn't it?