287 research outputs found
Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices
The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 Ă— 3 and 4 Ă— 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants
Solving the Selesnick-Burrus Filter Design Equations Using Computational Algebra and Algebraic Geometry
In a recent paper, I. Selesnick and C.S. Burrus developed a design method for
maximally flat FIR low-pass digital filters with reduced group delay. Their
approach leads to a system of polynomial equations depending on three integer
design parameters . In certain cases (their ``Region I''), Selesnick and
Burrus were able to derive solutions using only linear algebra; for the
remaining cases ("Region II''), they proposed using Gr\"obner bases. This paper
introduces a different method, based on multipolynomial resultants, for
analyzing and solving the Selesnick-Burrus design equations. The results of
calculations are presented, and some patterns concerning the number of
solutions as a function of the design parameters are proved.Comment: 34 pages, 2 .eps figure
The Multivariate Resultant is NP-hard in any Characteristic
The multivariate resultant is a fundamental tool of computational algebraic
geometry. It can in particular be used to decide whether a system of n
homogeneous equations in n variables is satisfiable (the resultant is a
polynomial in the system's coefficients which vanishes if and only if the
system is satisfiable). In this paper we present several NP-hardness results
for testing whether a multivariate resultant vanishes, or equivalently for
deciding whether a square system of homogeneous equations is satisfiable. Our
main result is that testing the resultant for zero is NP-hard under
deterministic reductions in any characteristic, for systems of low-degree
polynomials with coefficients in the ground field (rather than in an
extension). We also observe that in characteristic zero, this problem is in the
Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In
positive characteristic, the best upper bound remains PSPACE.Comment: 13 page
Solving geoinformatics parametric polynomial systems using the improved Dixon resultant
Improvements in computational and observational technologies in geoinformatics, e.g., the use of laser scanners that produce huge point cloud data sets, or the proliferation of global navigation satellite systems (GNSS) and unmanned aircraft vehicles (UAVs), have brought with them the challenges of handling and processing this “big data”. These call for improvement or development of better processing algorithms. One way to do that is integration of symbolically presolved sub-algorithms to speed up computations. Using examples of interest from real geoinformatic problems, we will discuss the Dixon-EDF resultant as an improved resultant method for the symbolic solution of parametric polynomial systems. We will briefly describe the method itself, then discuss geoinformatics problems arising in minimum distance mapping (MDM), parameter transformations, and pose estimation essential for resection. Dixon-EDF is then compared to older notions of “Dixon resultant”, and to several respected implementations of Gröbner bases algorithms on several systems. The improved algorithm, Dixon-EDF, is found to be greatly superior, usually by orders of magnitude, in both CPU usage and RAM usage. It can solve geoinformatics problems on which the other methods fail, making symbolic solution of parametric systems feasible for many problems
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