MatrixEquationsAD

Automatic differentiation rules for selected MatrixEquations.jl solvers, plus a Klein/Sims first-order policy-map primitive. AD rules (ForwardDiff Dual dispatches and Enzyme forward / reverse) load through package extensions when the relevant AD package is in scope.

See the documentation for derivations, worked examples, and the per-solver API:

• Klein Policy Map — first-order DSGE policy (g_x, h_x)
• Discrete Lyapunov (Schur) — lyapd(A, C) solving A X A^\top - X + C = 0
• Kronecker Discrete Lyapunov — lyapdkr(A, C) Kronecker LU variant
• Generalised Sylvester — gsylv / gsylvkr for A X B + C X D = E
• Algebraic Riccati (DARE) — ared(A, B, R, Q, S) with stabilising gain F

Install

using Pkg
Pkg.add(url = "https://github.com/QuantEcon/MatrixEquationsAD.jl")

Example

Differentiate a small discrete Lyapunov solve with ForwardDiff:

using ForwardDiff
using MatrixEquations
using MatrixEquationsAD

A = [0.55 0.08; -0.04 0.42]
C = [1.0  0.2;  0.2 0.7]

lyapd_sum(x) =
    sum(lyapd(reshape(x[1:4], 2, 2), reshape(x[5:8], 2, 2)))

ForwardDiff.gradient(lyapd_sum, [vec(A); vec(C)])

The same loss with Enzyme reverse mode:

using Enzyme: Active, Const, Duplicated, Reverse, autodiff, make_zero
using LinearAlgebra: dot
using MatrixEquations
using MatrixEquationsAD

A = [0.55 0.08; -0.04 0.42]
C = [1.0  0.2;  0.2 0.7]
W = [0.4 -0.1; -0.1 0.7]

weighted(A, C, W) = dot(W, lyapd(A, C))

A_bar = make_zero(A); C_bar = make_zero(C)
autodiff(
    Reverse, weighted, Active,
    Duplicated(A, A_bar), Duplicated(C, C_bar), Const(W),
)