feat(linalg): Add weighted and generalized least squares solvers

[aamrindersingh]

Resolves #1047

This PR adds two interfaces to stdlib_linalg for weighted least-squares problems. The weighted_lstsq interface handles diagonal weight matrices where each observation has a different importance, common in heteroscedastic regression and outlier downweighting. Usage: x = weighted_lstsq(w, A, b) where w is a vector of positive weights. It transforms to ordinary least-squares via row scaling and uses LAPACK's GELSD. The generalized_lstsq interface handles correlated errors described by a symmetric positive definite covariance matrix: x = generalized_lstsq(W, A, b). This minimizes the Mahalanobis distance (Ax-b)^T W^{-1} (Ax-b) using LAPACK's GGGLM.

The key design decisions are:-

(1) weighted_lstsq validates that all weights are positive since negative weights have no statistical meaning and would cause numerical issues with sqrt(w).
(2) generalized_lstsq always copies W internally to protect user data from GGGLM's destructive operations. This applies regardless of the prefactored_w flag , the input W matrix is never modified.
(3) Both interfaces follow the overwrite_a pattern from solve_lu where copy_a defaults to .true. to preserve A unless the user explicitly opts into destruction for performance.
(4) A handle_ggglm_info error handler was added to parse LAPACK return codes, consistent with existing handle_gelsd_info and handle_gglse_info patterns.

Testing includes:-

• Basic WLS solves with uniform weights
• Verification that non-uniform weights change solutions
• Error handling for negative weights
• Basic GLS solves with correlated errors
• Verification that GLS with identity covariance equals OLS
• All real types (sp/dp/qp/xdp) covered following existing lstsq patterns

[jalvesz]

Hi @aamrindersingh I just skimmed through your implementation. For the moment I won't comment on the design, just a minor very important detail:
I saw this kind of loops a few times in your code

do i = 1, m
      a_scaled(i, :) = sqrt_w(i) * a(i, :)
end do

In Fortran, indexes go from left to right in terms of memory access (as opposed to C based languages, where indexes go from right to left). This implies that this loop is forcing cache misses at every j for all i rows. Please rewrite your loops either using do concurrent with a j column indexe in front or a plain nested do loop where j is the outmost level.

[aamrindersingh]

Thanks for the review @jalvesz!

Fixed all row-wise loops to use column-major access . It was a good learning moment on how Fortran's left to right indexing means do j=1,n; a(:,j)=...; end do stays contiguous while row-wise forces cache misses.

Did a full dry run after fixing using critical edge cases, found a few more things:- workspace using int() instead of ceiling(), missing cleanup on error paths, and GGGLM needing upper triangle zeroed since potrf only sets the lower part.

Eager to discuss design further

[jvdp1]

Thank you @aamrindersingh . Could you split this PR in two smaller one (weighted_lstsq and generalized_lstsq), please? Or are these two implementations dependent of each other?

[aamrindersingh]

No, they are not dependent on each other , Will start on splitting them into two PRs.
Thanks

[aamrindersingh]

@jvdp1, Below are the Two PR's

• feat(linalg): add weighted least squares solver #1104 :- weighted_lstsq
• feat(linalg): add generalized least squares solver #1105 :- generalized_lstsq

[codecov]

Codecov Report

❌ Patch coverage is 0% with 20 lines in your changes missing coverage. Please review.
✅ Project coverage is 68.44%. Comparing base (b32b42a) to head (42b28a4).
⚠️ Report is 31 commits behind head on master.

Files with missing lines	Patch %	Lines
example/linalg/example_generalized_lstsq.f90	0.00%	11 Missing ⚠️
example/linalg/example_weighted_lstsq.f90	0.00%	9 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1096      +/-   ##
==========================================
- Coverage   68.55%   68.44%   -0.11%     
==========================================
  Files         396      398       +2     
  Lines       12746    12766      +20     
  Branches     1376     1376              
==========================================
  Hits         8738     8738              
- Misses       4008     4028      +20     

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[loiseaujc]

Closing this PR as it has been split between #1104 and #1105.