By Luke Y.L.

ISBN-10: 0124599400

ISBN-13: 9780124599406

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**Extra info for Algorithms for computations of mathematical functions**

**Example text**

12. For a generic matrix A, it holds that generic-rank A = term-rank A. 10) Proof. Consider a k × k term-nonsingular submatrix of A, where k = term-rank A. The deﬁning expansion of its determinant contains a term, say, Ai1 j1 Ai2 j2 · · · Aik jk , which cannot be cancelled out. Hence generic-rank A ≥ term-rank A. 8), in which rank A = generic-rank A. 13. 3 for terminology). Consider a bipartite graph G = (V + , V − ; A) has the vertex bipartition (V + , V − ) = (C, R) corresponding to the column set C and the row set R, and the arc set A˜ deﬁned by A˜ = {(j, i) | i ∈ R, j ∈ C, Aij = 0}.

T -part: The matrix T (s) is a structured matrix in that the nonzero coeﬃcients are algebraically independent. 7) in particular). Combination: Each of fQ (I, J) and fT (I, J) enjoys a combinatorially nice property, being a variant of a valuated matroid. Therefore, the maximum can be computed by a straightforward application of an algorithmic scheme for the valuated matroid intersection problem, where the functions fQ (I, J) and fT (I, J) are evaluated polynomially many times (polynomial in n). If the stronger condition (MP-Q2) on Q(s) is satisﬁed, the maximization can be reduced to an easier problem (the ordinary weighted matroid intersection problem).

Notes. 3 is due to Murota [200]. This chapter is an improved version of a presentation (Murota [223]) at ICIAM 95. 2. Matrix, Graph, and Matroid This chapter lays the mathematical foundation for combinatorial methods of systems analysis. Combinatorial properties of numerical matrices can be stated and analyzed with the aid of matroid theory, whereas those of polynomial matrices are formulated in the language of valuated matroids in Chap. 5. Emphasis is laid also on the general decomposition principle based on submodularity, and accordingly the Dulmage–Mendelsohn decomposition, which serves as a fundamental tool for the generic-case analysis of matrices, is presented in a systematic manner.

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