This thesis is a collection of the main work I did during my graduate study on copositive matrix and homology computation. For copositive matrix defined on polyhedron, I define its extension and give a sufficient condition for the existence for its copositive extension. For standard higher dimensional copositive matrix, I generalize the algorithm given by Anderson et al., for determining the copositivity of a square matrix by classifying the structure of a polyhedron. For homology computation, I establish the algorithm for computing the homology group of a manifold from the sampling point data. I provide the general theory and give a probabilistic asymptotic analysis on the correctness of the computation.
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