Approximate GCD in a Bernstein Basis

Corless, Robert M.; Sevyeri, Leili Rafiee

doi:10.1007/978-3-030-41258-6_6

Cited by 3 publications

(3 citation statements)

References 24 publications

(24 reference statements)

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“…In particular, we choose α = 2, empirically chosen to represent a reasonable spread between simple and smooth interfacial geometry, to complex geometry involving high degrees of curvature. 14 We further characterise this class of randomly generated as follows:…”

Section: Randomly Generated Geometrymentioning

confidence: 99%

“…A variety of approaches have been developed to compute the roots of univariate Bernstein polynomials, including through computation of companion matrices and their eigenvalues [76,77]; using Bernstein subdivision algorithms and convex hull properties to first isolate (real) roots and then polish with an iterative method such as Newton's [21,63]; and through generalized eigenvalue problems [33,14], among a number of other possibilities. For a discussion of some of these methods, see the dissertation of Spencer [63].…”

Section: Appendix C Computing Roots Of Univariate Bernstein Polynomialsmentioning

confidence: 99%

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High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

Saye

2021

Preprint

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A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of accompanying numerical experiments demonstrates the quadrature algorithm on two-and three-dimensional problems, including randomly generated geometry involving multiple high curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyper-rectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

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Section: Randomly Generated Geometrymentioning

confidence: 99%

Section: Appendix C Computing Roots Of Univariate Bernstein Polynomialsmentioning

confidence: 99%

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

Saye

2021

Preprint

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“…the subresultant matrix). Moreover, we note that there are also algorithms [31,32,4,21,28,7] using other bases (e.g. the Bernstein basis) instead of the monomial basis.…”

Section: Introductionmentioning

confidence: 99%

Relaxed NewtonSLRA for Approximate GCD

Nagasaka

2021

Computer Algebra in Scientific Computing

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We propose a better algorithm for approximate GCD in terms of robustness and distance, based on the NewtonSLRA algorithm that is a solver for the structured low rank approximation (SLRA) problem. Our algorithm mainly enlarges the tangent space in the Newton-SLRA algorithm and adapts it to a certain weighted Frobenius norm. By this improvement, we prevent a convergence to a local optimum that is possibly far from the global optimum. We also propose some modification using a sparsity on the NewtonSLRA algorithm for the subresultant matrix in terms of computing time.

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Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation

Nagasaka

2025

Journal of Symbolic Computation

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Approximate GCD in a Bernstein Basis

Cited by 3 publications

References 24 publications

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

Relaxed NewtonSLRA for Approximate GCD

Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation

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