A variable rate refining triangulation

A new variable rate refining subdivision B4 for the solution of equations with piecewise linear homotopies and restart homotopies is described. The essential virtue of the subdivision B4 is that it offers vast latitude to the user. In particular, the parameters of B4 can be set so one restriction of B4 is that J3 triangulation and another restriction is the octahedral subdivision. More generally, full flexibility is available in the placement of local focal points, and perhaps most importantly, B4 permits the new capability of refining coordinates at variable and independent rates.