The numerical solution of systems of polynomials - arising in engineering and science

By Andrew J. Sommese, Charles W. Wampler II. 2005

962 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fff443a3982dd16781df3aca4d54d0913f1bda10

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In Computational Theory and Mathematics

Citations in the corpus

Mechanism mobility and a local dimension test (2011, background): “Numerical algebraic geometry [22, 23], which is based on continuation methods, is able to solve larger systems than symbolic methods.”
Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases (2015, background): “Generally, random choices for coefficients ci are sufficient to ensure these properties almost surely [28].”
Resultants and Loop Closure (2006, background): “Topological considerations become important and powerful continuation and other topological methods can be brought to bear to identify feasible solution sets and speed up the solution process [33].”
Challenges and opportunities in quantum optimization (2023, neutral): “2,3,44) is another exact optimization algorithm related to homotopy methods in numerical mathematics 229 .”
Homotopy techniques for tensor decomposition and perfect identifiability (2014, background): “, see [10,59].”
Vevacious: a tool for finding the global minima of one-loop effective potentials with many scalars (2013, methodology): “method [22] to solve the tree-level tadpole equations.”
Reconfiguration analysis of a 3-DOF parallel mechanism using Euler parameter quaternions and algebraic geometry method (2014, methodology): “The methods based on algebraic geometry [10,11] or numerical algebraic geometry [12] as well as computer algebra systems [13]”
Energy landscapes for machine learning. (2017, positive): “119 Recently, a complementary study of the minima and saddle points of the p -spin model has been initiated using the numerical polynomial homotopy continuation method, 120,121 which guarantees to find all the minima, transition states and higher index saddles for moderate sized systems.”
Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis (2016, methodology): “We refer the reader to [50] for a detailed explanation of PHC.”
Algebraic boundaries of Hilbert’s SOS cones (2011, background): “An introduction to numerical algebraic geometry can be found in [29].”
Finding All Real Points of a Complex Curve (2006, methodology): “This can be done using a diagonal homotopy [19, 20, 23 ] to solve the system ”
HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia (2017, methodology): “. An example for a custom-deﬁned homotopy for a family of overdetermined systems is given in Section 3.1. In order to deal with singular solutions, an endgame strategy which combines the power series [18,8] and Cauchy endgame [18] is implemented. The solution can be computed in serial-processing as ”