ENH: Improve polynomial equations and analytical eigendecompositions

Summary

Analytical eigendecomposition routines show irregular numerical instabilities particularly for cases wherein off-diagonal elements of a tensor involve noise. The issue revealed to be caused by the polynomialEqn containers, particularly cubicEqn.

In addition, the analytical eigendecomposition routines seem to be mathematically inconsistent since tensors are not allowed to return complex types, which is nevertheless the norm for asymmetric matrices.

Accordingly, this set of commits aims to improve the numerical stability of analytical eigendecompositions, and to provide a group of verification tests to prevent future breaks in the workflow.

unit-test-notebooks-17-Feb-20.zip

Resolved bugs

#1311 (closed) #1312 (closed) #1527 #1575 #1596

Risks

Low.

polynomialEqns and analytical eigendecomposition routines were considerably isolated from the rest of the code.

@mark

changed milestone to %v2006

added enhancement label

looks good

• tempted to remove support for label (remove enable_if etc)
• test clang build
• think about how to set return value on test scripts

The closest I have to a unit-test would be applications/primitives/Test-primitives.

added 15 commits

• cf887553...4fea7b3b - 11 commits from branch develop
• d12e2f49 - ENH: improve stability in polynomialEqns
• f98e4ada - ENH: improve funcs and opers in Tensor types
• 3e56f513 - STYLE: reorder global funcs and opers in Tensor types
• d2b0be83 - ENH: improve analytical eigendecompositions

Compare with previous version

changed the description

added 3 commits

• 6e2ea423 - ENH: improve funcs and opers in Tensor types
• 44b427e4 - STYLE: reorder global funcs and opers in Tensor types
• 3c8c9b06 - ENH: improve analytical eigendecompositions

Compare with previous version

unmarked as a Work In Progress

changed the description

added bug label

• removed label support by removing enable_ifs
• can compile with Clang 5.0.2 & gcc 7.4.1
• designed as unit tests by using hard-coded NumPy-SciPy benchmarks (see the attached notebooks)

added 7 commits

• 3c8c9b06...97bdd5bc - 3 commits from branch develop
• 8ca724ff - ENH: improve stability in polynomialEqns
• 66b02ca5 - ENH: improve funcs and opers in Tensor types
• 6a53794e - STYLE: reorder global funcs and opers in Tensor types
• 55e7da67 - ENH: improve analytical eigendecompositions

Compare with previous version

All looks good

mentioned in commit 0e1c9b0a

merged