Refactor DLE (_dle.py) to use native scalars and 1-D arrays instead of (1, 1)/2-D matrix conventions

Background

This came out of the review of #842 (which fixed #839). @oyamad noted that quantecon/_dle.py is written in a very "matrix-oriented" style that differs from the rest of the library and is the root cause of the NumPy 2.4 breakage that #842 patched.

In _dle.py, conceptually scalar quantities are stored as (1, 1) arrays and conceptually 1-D vectors are stored as 2-D row arrays. Concretely: self.beta is a (1, 1) array (a scalar discount factor), e1 is built as a (1, nc) 2-D row selector, and the bond-price expressions therefore reduce to size-1 (ndim > 0) arrays rather than plain scalars.

Why it matters

Proposed change

The idea is to make the representation match the mathematics:

• Store scalar parameters (e.g. beta) as native Python/NumPy scalars rather than (1, 1) arrays.
• Store conceptual vectors as 1-D arrays, and build selectors such as e1 as 1-D.
• Assemble the price/return series from genuine scalars, which removes the need for the .item() coercions added in #842 and makes the price loop read directly as its closed form.

Scope and risks

This is a global rewrite of _dle.py, not a local edit — which is why it was deliberately kept out of #842.

It is potentially breaking: public attributes such as DLE.beta (and possibly others) would change shape from (1, 1) to scalar, which could affect downstream code and the DLE lectures in lecture-python-advanced.myst (cattle_cycles, growth_in_dles, irfs_in_hall_model, lucas_asset_pricing_dles, permanent_income_dles). This warrants an audit of which attributes are part of the public API and whether a deprecation path is needed.

#842 added the first regression tests for compute_sequence (test_compute_sequence, test_compute_sequence_with_pay). Before attempting the refactor, test coverage should be expanded further (e.g. irf and the other public attributes) so the current numerical behaviour can be pinned and compared before/after.

References

• Regression that triggered this: #839
• Workaround / minimal fix: #842
• Style point raised by @oyamad in the #842 review

🤖 Generated with Claude Code

Start with quantecon/_dle.py and the existing test_compute_sequence and test_compute_sequence_with_pay regression tests added for #842. Audit the public attributes and the DLE lecture references named in the issue before changing representations. Done means scalar and 1-D conventions are consistent, numerical behavior is pinned by expanded tests including irf, and API compatibility or deprecation needs are documented.