Eigenvalue Dan Eigenvector Dari Matriks Polinomial Dalam Aljabar Max-Plus

Pada penelitian ini, akan dibahas mengenai eigenvalue dan eigenvector dari polinomial dalam bentuk matriks di dalam Aljabar Max-Plus. Teorema Perron-Frobenius diterapkan seperti halnya pada aljabar biasa dengan membentuk korespondensi satu-satu antara eigenvalue dan eigenvector dari matriks polinomial dengan eigenvalue dan eigenvector dari matriks Companion. Proses perhitungan akan digunakan Program Scilab. Kata Kunci : eigenvalue, eigenvector, matriks polinomial, Aljabar Max-Plu

Tropical bounds for eigenvalues of matrices

We show that for all k = 1,...,n the absolute value of the product of the k largest eigenvalues of an n-by-n matrix A is bounded from above by the product of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute value), up to a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1.Comment: 17 pages, 1 figur

Economic Networks: Theory and Computation

This textbook is an introduction to economic networks, intended for students and researchers in the fields of economics and applied mathematics. The textbook emphasizes quantitative modeling, with the main underlying tools being graph theory, linear algebra, fixed point theory and programming. The text is suitable for a one-semester course, taught either to advanced undergraduate students who are comfortable with linear algebra or to beginning graduate students.Comment: Textbook homepage is https://quantecon.github.io/book-networks/intro.htm

Jet_fitting_3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting

Surfaces of R^3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in Computer Graphics, Computer Aided Design, Computational Geometry, Computer Vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpolation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, extraction of the differential properties. This paper presents a generic C++ software package solving these problems. On the theoretical side and as established in a companion paper, the interpolation and approximation methods provided achieve the best asymptotic error bounds known to date. On the implementation side and following state-of-the-art coding rules in Computational Geometry, genericity of the package is achieved thanks to three template classes accounting for (a) the type of the input points (b) the internal geometric computations and (c) the linear algebra operations. An instantiation within the Computational Geometry Algorithms Library (CGAL, version 3.3) and using CLAPACK is also provided

Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and P\'olya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated.Comment: 36 pages, 19 figure