Backward Euler step equation and algebraic residual

Let $f:D\to {\mathbb{R}}^d$ define a first-order initial-value problem on an open time-state domain $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$. Given a time $t_n$, step $h_n>0$, and state $y_n$ with $(t_n,y_n)\in D$, an admissible endpoint $y_{n+1}$ must satisfy $(t_n+h_n,y_{n+1})\in D$. The backward Euler endpoint is any such $y_{n+1}$ satisfying

$y_{n+1}=y_n+h_{n}f(t_n+h_n,y_{n+1})\mathrm{.}$

Its update residual at an approximate endpoint $z$ with $(t_n+h_n,z)\in D$ is

$G_n^{\mathrm{B}\mathrm{E}}(z)=z-y_n-h_{n}f(t_n+h_n,z)\mathrm{.}$

When $f$ is defined at the endpoint, the derivative residual is $G_n^{\mathrm{B}\mathrm{E}}(z)\mathrm{/}{h}_n$. The equation can have no, one, or several roots on a restricted admissible branch set; this definition by itself asserts neither existence nor uniqueness.