منابع مشابه
Solving quadratic equations using reduced unimodular quadratic forms
Let Q be an n × n symmetric matrix with integral entries and with detQ = 0, but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to run in polynomial time. We also describe an algorithm for the minimization of a ternary quadratic form: when a...
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Now, the formula does not provide us with a solution that is written in decimal form; to get such a solution we need to evaluate the above expression. To this end, since most computers perform additions, subtractions, multiplications, and even divisions very fast, we should focus on the need to take a squareroot. In short, the “explicit” solution for the quadratic equation actually reduces our ...
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Let p(z) be a monic cubic complex polynomial with distinct roots and distinct critical points. We say a critical point has the Voronoi property if it lies in the Voronoi cell of a root θ, V (θ), i.e. the set of points that are closer to θ than to the other roots. We prove at least one critical point has the Voronoi property and characterize the cases when both satisfy this property. It is known...
متن کاملSolving Underdefined Systems of Multivariate Quadratic Equations
The security of several recent digital signature schemes is based on the difficulty of solving large systems of quadratic multivariate polynomial equations over a finite field F. This problem, sometimes called MQ, is known to be NP-hard. When the number m of equations is equal to the number n of variables, and if n < 15, Gröbner base algorithms have been applied to solve MQ. In the overdefined ...
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This paper deals with finding an n-dimensional solution x to a system of quadratic equations yi = |〈ai,x〉|, 1 ≤ i ≤ m, which in general is known to be NP-hard. We put forth a novel procedure, that starts with a weighted maximal correlation initialization obtainable with a few power iterations, followed by successive refinements based on iteratively reweighted gradient-type iterations. The novel...
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ژورنال
عنوان ژورنال: Journal of Chemical Education
سال: 1987
ISSN: 0021-9584,1938-1328
DOI: 10.1021/ed064p472.3