Zero-compatible modified Patankar Euler in rate-factor form

Let $K$ be a compact nonnegative state set with a source-factored transfer representation Source-factored transfer rates for zero compartments. Given $y^n\in {\mathbb{R}}^m$ and a step size $h$, if $y^n\mathrm{\notin }K$ or $h\le 0$, return an inadmissible-input status before evaluating any factor. Otherwise form $B(y^n)\in {\mathbb{R}}^{m\times m}$ by

$B_{ij}(y^n)=a_{ij}(y^n)(i\mathrm{\ne }j),B_{jj}(y^n)=-{\sum }_{i\mathrm{\ne }j}{a}_{ij}(y^n)\mathrm{.}$

Return the unique solution $y^{n+1}$ of

$(I-hB(y^n))y^{n+1}=y^n\mathrm{.}$

Equivalently, component $i$ satisfies

$y_i^{n+1}=y_i^n+h{\sum }_{j\mathrm{\ne }i}(a_{ij}(y^n)y_j^{n+1}-a_{ji}(y^n)y_i^{n+1})\mathrm{.}$

This procedure uses no denominator $y_j^n$ and is defined when some donor coordinates vanish. It is the factorized extension of the first-order Patankar Euler formula to the zero-state boundary. In exact arithmetic its linear system is uniquely solvable by Strict column dominance gives an entrywise nonnegative inverse.