Differentiate a finite fixed-point iteration as a finite composed map

Input a finite-dimensional $C^1$ map $G(z,\theta )$, a $C^1$ initial-state map $z_0(\theta )$, a parameter $\theta$ in an open subset of ${\mathbb{R}}^p$, and an integer budget $K\ge 0$; the states lie in ${\mathbb{R}}^m$, with $p,m\ge 0$. Use the specified maps $z_0$ and $G$, with their exact derivative evaluations, as primitive maps on their declared open domains. Evaluate $z_0(\theta )$ and then $z_{k+1}(\theta )=G(z_k(\theta ),\theta )$ for $k=0,\dots ,K-1$. A primitive-domain failure returns DOMAIN_ERROR; a branch boundary or ambiguous active branch returns NONDIFFERENTIABLE and no derivative. The admissible case requires every evaluation to remain in its declared domain and, for piecewise primitives, on a unique strict active branch.

Allocate an $m\times p$ Jacobian array $J$. For each state coordinate $a=1,\dots ,m$, form the finite scalar-output graph for $f_a(\theta )=(z_K(\theta ){)}_a$ by appending the coordinate projection to the finite recursion. For each parameter coordinate $\mathrm{\ell }=1,\dots ,p$, call Forward-mode automatic differentiation on a finite computational graph on this graph at $\theta$ with direction $e_{\mathrm{\ell }}$, the $\mathrm{\ell }$th standard basis vector of ${\mathbb{R}}^p$, and put its returned tangent in $J_{a\mathrm{\ell }}$. Propagate any DOMAIN_ERROR or NONDIFFERENTIABLE status from a call. Return $(z_K(\theta ),J)$ with status ITERATES_DIFFERENTIATED. For $K=0$ the recursion contains only the initial-state map. If $m=0$ or $p=0$, the corresponding loops are empty and $J$ has the specified empty shape; the primal recursion is still evaluated. This procedure differentiates the $K$-step map actually specified by $G,z_0,K$; it does not substitute the derivative of an exact fixed point.