Introduction to Combinatorial Analysis

This is a textbook that teaches the bridging topics between numerical analysis, parallel computing, code performance, large scale applications.
The field of high performance scientific computing lies at the crossroads of a number of disciplines and skill sets, and correspondingly, for someone to be successful at using high performance computing in science requires at least elementary knowledge of and skills in all these areas.

This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the nonsmooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop inverse and implicit function theorems for piecewise differentiable equations.