On Piecewise Linear Approximations to Smooth Mappings

Given a continuously differentiable mapping f from Rⁿ into Rⁿ, in this work we study piecewise linear approximations to it on certain subdivisions of Rⁿ. It is shown that several properties of the subdivision are critical when the Jacobians of the pieces of linearity of the approximation are required to be close to the Jacobians of f. In addition, it is shown that even under arbitrary scaling of the triangulations used in fixed point algorithms, good approximations of the derivatives result.