Feature/add analytic test rpcholesky

[gw265981]

PR Type

• Bugfix
• Documentation content changes
• Tests

Description

• Added an analytic test for RPCholesky on an independently derived example.
• In the process, fixed a bug in the algorithm.
• Added a documentation page with the derivation and example code.

How Has This Been Tested?

Test A: pytest -k "test_rpcholesky_analytic_unique"

Does this PR introduce a breaking change?

The RPCholesky has been modified but this shouldn't introduce a breaking change.

Checklist before requesting a review

• [x ] I have made sure that my PR is not a duplicate.
• [x ] My code follows the style guidelines of this project.
• [x ] I have commented my code, particularly in hard-to-understand areas.
• [x ] I have performed a self-review of my code.
• [x ] I have made corresponding changes to the documentation.
• [x ] My changes generate no new warnings.
• [x ] I have added tests that prove my fix is effective or that my feature works.
• [x ] New and existing unit tests pass locally with my changes.
• [x ] Any dependent changes have been merged and published in downstream modules.

[CLAassistant]

All committers have signed the CLA.

[github-actions]

Performance review

Commit 46eb6dc - Merge 3bf7cce into bc5acf0

Statistically significant changes

• basic_rpc:
  • OLD: compilation 0.6218 units ± 0.02447 units; execution 0.1587 units ± 0.0008577 units
  • NEW: compilation 0.6074 units ± 0.02588 units; execution 0.1006 units ± 0.001554 units
  • Significant decrease in execution time (-36.60%, p=1.304e-21)

---

Normalisation values for new data:
Compilation: 1 unit = 372.35 ms
Execution: 1 unit = 803.06 ms

[github-actions]

Performance review

Commit a93cdd1 - Merge 35ac1b6 into bc5acf0

Statistically significant changes

• basic_rpc:
  • OLD: compilation 0.6218 units ± 0.02447 units; execution 0.1587 units ± 0.0008577 units
  • NEW: compilation 0.606 units ± 0.03039 units; execution 0.1028 units ± 0.0007012 units
  • Significant decrease in execution time (-35.19%, p=7.238e-29)

---

Normalisation values for new data:
Compilation: 1 unit = 372.04 ms
Execution: 1 unit = 789.33 ms