Standardize solver refactoring and shared factorizations

[jcapriot]

Introduces a public factor(A=None) method across all solvers for consistent matrix updates. Some solvers already supported this. This just makes it a guaranteed interface across all solvers. This enables efficient re-factorization when the matrix values change but the sparsity pattern and analysis can be reused.

For direct solvers like Mumps and Pardiso, a new _SharedFactorBase ensures that transposed and conjugated views share the same underlying factorization. This was to make it so that when the underlying solver object is refactored, any other transposed or conjugate views stay in sync with the updated solver.

[codecov]

Codecov Report

❌ Patch coverage is 99.45946% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 99.66%. Comparing base (200d44b) to head (ea89bb2).

Files with missing lines	Patch %	Lines
tests/test_Wrappers.py	94.11%	1 Missing ⚠️

Additional details and impacted files

@@                  Coverage Diff                   @@
##           maint/deprecations      #55      +/-   ##
======================================================
- Coverage              100.00%   99.66%   -0.34%     
======================================================
  Files                      16       17       +1     
  Lines                    1059     1197     +138     
======================================================
+ Hits                     1059     1193     +134     
- Misses                      0        4       +4     

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[santisoler]

It looks good. My only comment would be why instead of "stacking" the inheritance like:

Base --(parent of)--> _SharedFactorBase --(parent of)--> Mumps

we make multiple inheritance like:

class Mumps(_SharedFactorBase, Base):
    ...

We could even make _SharedFactorBase an ABC class and force child classes to implement the factor method.

Just an idea, a bit of nit picking too. What do you think?

[jcapriot]

It looks good. My only comment would be why instead of "stacking" the inheritance like:

Base --(parent of)--> _SharedFactorBase --(parent of)--> Mumps

we make multiple inheritance like:

class Mumps(_SharedFactorBase, Base):
    ...

We could even make _SharedFactorBase an ABC class and force child classes to implement the factor method.

Just an idea, a bit of nit picking too. What do you think?

I like the idea of making it a mix in class. I don't think the factor method is what distinguishes it though. These are meant for classes that wrap some other solver that owns the factorization, not necessarily if they can be factored.