Improvements for Orthonormalize Symmetric and Canonical

[EricaCMitchell]

Description

Modifies orthonormalization routines for symmetric and canonical.

Details

• Fixes transpose to conjugate transpose in orthonormalize_symmetric
• orthonormalize_symmetric keeps track of linear dependencies based on a tolerance (default = 1e-12)
• orthonormalize_symmetric warns user if linear dependencies are detected
• orthonormalize_canonical performs more in-place modifications of U and s rather than storing them in new tensors
• orthonormalize_canonical does not require lindep tolerance (default = 1e-12)
• Added tests to testvmra for orthonormalize_symmetric and orthonormalize_canonical
• Convenience routines for orthonormalize_* perform symmetric/Hermitian matrix_inner
• Typos

[Copilot]

Pull request overview

This PR updates the symmetric and canonical orthonormalization routines to better support complex-valued overlaps (via conjugate transpose), add/standardize linear-dependence handling, and adds new VMRA tests covering both routines.

Changes:

• Update orthonormalize_symmetric to use conjugate transpose and introduce a linear-dependence tolerance parameter with user warnings.
• Update orthonormalize_canonical to do more in-place work and optionally remove linearly dependent modes.
• Add new tests in testvmra.cc to validate orthonormality for both routines (real and complex, small and non-small test modes).

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 4 comments.

File	Description
src/madness/mra/vmra.h	Modifies symmetric/canonical orthonormalization logic, tolerance handling, and overlap computation flags.
src/madness/mra/testvmra.cc	Adds tests for both orthonormalization routines and wires them into main().

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[Copilot]

Eigenvalues at/below the linear-dependence threshold are still inverted via 1.0 / sqrt(s_i). For near-zero (or slightly negative) eigenvalues this will produce inf/NaN and corrupt X. Only apply the inverse-sqrt when s_i > lindep (and > 0), and set the scaling to 0 (or otherwise handle via pseudoinverse / truncation) for dependent modes.