General reductions and a second scalar-shooting subclass

For rational $A$ and $b$ such that every $A-\operatorname{diag}(d)$, $d\in[-1,1]^n$, is nonsingular, the note gives a deterministic algorithm of polynomial bit complexity that computes the exact solution of $Ax-|x|=b$ when the leading $(n-1)\times(n-1)$ block of $A$ is lower triangular with diagonal entries of modulus greater than one; the last row and column are arbitrary. Taking the last coordinate as a parameter, scalar inversions of $z\mapsto az-|z|$ solve the first $n-1$ equations, regularity makes the last residual strictly monotone, and bisection, sign recovery and one exact linear solve give the solution. The class contains dense arrow matrices that are not permutation-similar to Hessenberg matrices. The note also shows that a residual of polynomially bounded bit length suffices for exact recovery, and gives polynomial reductions in both directions between such equations and P-matrix linear complementarity problems, a known equivalence.