ENH: Accept a function for v_init, evaluated at the solver's basis nodes

Motivation

A good v_init is naturally a function of the state, but the keyword takes a vector of values at the solver's collocation nodes, which couples the caller to two pieces of solver-internal information: the node count (length(basis)) and the node ordering (the gridmake convention, first dimension fastest). The canonical construction is boilerplate that every user has to rediscover:

S = nodes(basis)[1]
v_init = [v_guess(S[i, :]) for i in 1:size(S, 1)]

The need is not hypothetical: for the monetary policy model (MF §9.7.5), PFI from the default v_init = zeros(...) fails — the reward is independent of x, so with a constant value guess the inner objective is flat and the first greedy policy is indeterminate, after which Newton-style policy iteration cycles between corner policies (unstable uncontrolled dynamics + out-of-domain extrapolation in the policy evaluation). Any reasonable non-constant shape (e.g. a rough quadratic $-|s - s^*|^2$, or the LQA solution) fixes it. Users should be able to express such a guess directly as a function, robustly to later changes of the basis size.

Proposal

Let v_init also accept a callable, evaluated internally at the solver's nodes:

res = solve(cdp, solver; v_init = s -> -norm(s - s_target)^2)

• The callable receives one state point per node, following the same convention as f and g (a scalar for N == 1, a length-N vector for N >= 2), and must return a real number.
• The vector form stays as is (values at nodes(solver.basis)[1], in order), including its length validation and error message.
• Applies to both CollocationSolver and LQASolver entrances (they share the same v_init handling).

Implementation sketch

A new _check_v_init(v_init, interp::Interp) method for the callable case: evaluate at _row(interp.S, i) for i in 1:interp.length into a Vector{Float64}; dispatch keeps nothing (zeros) and AbstractVector behavior untouched. Non-breaking 0.3.x addition.

Tests

• Callable vs. manually evaluated vector give identical results (N = 1 and N = 2).
• The flat-degeneracy case: a non-constant callable v_init as the documented remedy (cf. the monetary policy example).
• Wrong-length vector error unchanged; a callable returning a non-real value errors informatively.

🤖 Generated with Claude Code (Claude Fable 5)

Start at the shared v_init handling and the proposed _check_v_init(v_init, interp::Interp) entry point, then trace how CollocationSolver and LQASolver pass through it. Run the existing solver tests first, add coverage for N = 1 and N = 2 callable inputs, and verify vector behavior and invalid callable results remain clear and unchanged.