Zur numerischen Bestimmung des Abbildungsgrades im \(R^ n\). II (Q1819545)

This paper uses techniques and results from part I [ibid. 3, 337-355 (1984; Zbl 0594.65035] to derive various formulas for computing the topological degree d(F,D,\(\theta)\) of a function F relative to a domain D at the zero vector \(\theta\). These formulas include those of \textit{F. Stenger} [Numer. Math. 25, 23-38 (1975; Zbl 0316.55007)], the reviewer [Numer. Math. 32, 109-127 (1979; Zbl 0386.65016)], and \textit{M. Stynes} [Numer. Math. 33, 147-156 (1979; Zbl 0387.65035)]. These formulas are of ''n-cube'' type. This paper also derives formulas of ''n-octahedron'' type, based on a new idea in part I. In addition to presenting formulas, the paper studies properties of triangulations which may be used to compute the topological degree. It also gives a point in a given simplex for approximating a solution of \(F(x)=\theta\) and a bound on the distance of this point from the true solution. Finally, the paper presents numerical results using Stenger's formula. The test functions are Brown's function, a system of two quadratics introduced by \textit{C. Harvey} and \textit{F. Stenger} [Quart. Appl. Math. 33, 351-368 (1976; Zbl 0365.65032)]), and the Gheri-Manciano function. Correct computation of the degree depends on using a triangulation with a sufficiently fine mesh; the results illustrate computed approximations to the degree as the mesh changes. The paper does not present ideas on how to determine the appropriate mesh a priori. The results are possibly relevant in a numerical setting where the function cannot be evaluated accurately, when the function is not smooth, etc.