ENH: Add derivative point evaluation via coefficient differentiation

[oyamad]

Summary

Extend the point-evaluation kernel (#93) to partial derivatives of the interpolant, as preparation for derivative-based inner optimization (#7, #73) and as part of the functionality proposed for upstreaming (#94, QuantEcon/BasisMatrices.jl#82).

Rather than adding per-point derivative recurrences, derivatives are obtained by coefficient differentiation: for each basis family, the derivative of an interpolant is itself an interpolant on the differentiated parameters (Chebyshev of degree n-1, spline of degree k-1, piecewise linear on midpoints), with coefficients given by a sparse linear operator — both provided by BasisMatrices.derivative_op. This is the same construction evalbase uses internally, so agreement with BasisMatrices is by construction, and the already-tested order-0 kernels are reused verbatim.

• DerivFunEvalCache(basis, order::NTuple{N,Int}): workspace for the partial derivative of orders order (one per dimension; mixed and higher orders supported, e.g. (1, 1) or (2, 0)). For SplineParams the order must be less than the spline degree, mirroring evalbase's restriction.
• set_coefs!(dfec, C): transforms the coefficients with a single precomputed sparse matvec (kron of the per-dimension operators); call once whenever C changes (e.g. once per sweep in a solver).
• funeval_point!(dfec, x): non-allocating single-point evaluation of the derivative interpolant.
• Project.toml: add the SparseArrays stdlib dependency.

No solver code is touched; this PR is purely additive to src/point_eval.jl.

A BasisMatrices limitation found along the way

funeval with derivative orders routes through the Direct/SplineSparse path, whose LinParams method errors with "derivatives un-supported right now" — so BasisMatrices' own funeval cannot evaluate LinParams derivatives, although the dense evalbase(p, x, order) supports them. The agreement tests therefore use the expanded derivative basis row built from per-dimension evalbase calls as the reference. Worth mentioning in the QuantEcon/BasisMatrices.jl#82 discussion.

Verification

• 124 new agreement tests: Cheb (orders 1, 2), splines of degree 2-3 with even and uneven breaks (orders 1, 2), Lin (order 1), and mixed 2-D/3-D tensor bases with gradient, mixed-partial (1,1), and second-order (2,0) combinations, at points inside and outside the domain; plus coefficient-update and zero-allocation checks and error cases (spline order >= degree, negative order).
• Full test suite passes (384 point-eval tests total).

🤖 Generated with Claude Code (Claude Fable 5)

[Copilot]

Pull request overview

This PR extends the existing non-allocating point-evaluation kernel to support evaluating partial derivatives of tensor-product interpolants by differentiating coefficients (via BasisMatrices.derivative_op) and then reusing the existing order-0 point-evaluation machinery.

Changes:

• Add DerivFunEvalCache plus set_coefs! for precomputing derivative-interpolant coefficients using a sparse (Kronecker-composed) linear operator.
• Add a funeval_point!(dfec, x) entry point for allocation-free single-point derivative evaluation.
• Expand the point-eval test suite with derivative agreement, coefficient-update, and allocation checks; add SparseArrays as a stdlib dependency.

Reviewed changes

Copilot reviewed 2 out of 3 changed files in this pull request and generated no comments.

File	Description
src/point_eval.jl	Adds derivative point-evaluation support via coefficient differentiation and sparse precomputed operators.
test/test_point_eval.jl	Adds derivative agreement/reference tests and allocation checks for derivative point evaluation.
Project.toml	Adds SparseArrays stdlib dependency and compat entry to support sparse operators used by derivative caches.