implementation Line Search with Negative Curvature Detection with direction for CR

[farhadrclass]

The algorithm adapts its behavior upon detecting nonpositive curvature:

At iteration k = 0: Returns the right-hand side b with solver.npc_dir set to the preconditioned initial residual.

At iteration k > 0: Returns the solution from iteration k-1 with solver.npc_dir updated to the most recent residual.

This modification ensures that the CR algorithm can be effectively utilized within inexact Newton methods that employ a linesearch strategy.

[codecov]

Codecov Report

Attention: Patch coverage is 83.18584% with 19 lines in your changes missing coverage. Please review.

Project coverage is 97.48%. Comparing base (9536ef7) to head (711eab0).
Report is 80 commits behind head on main.

Files with missing lines	Patch %	Lines
src/cr.jl	63.46%	19 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #985      +/-   ##
==========================================
+ Coverage   94.68%   97.48%   +2.80%     
==========================================
  Files          45       49       +4     
  Lines        8027     9347    +1320     
==========================================
+ Hits         7600     9112    +1512     
+ Misses        427      235     -192     

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[farhadrclass]

@dpo I’ve updated the PR.

I introduced num_dir in stats to indicate how many negative-curvature directions we’re using. However, I’m not entirely confident in the approach or whether I placed everything correctly (some lines are marked with #TODO). Would you be able to take a look now so I can finalize it and then write the test cases?

The idea is that we can detect 0, 1, or 2 negative directions. npc_dir is set to either r or p, and if both directions are needed, we use solver.p as well.

Let me know what you think!

[farhadrclass]

added npc_dir logic for when radius > 0 and nonpositive curvature is detected, the behavior depends on the iteration and follow similar logic as linesearch;