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Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems [article]

Pierre Lairez
2019 arXiv   pre-print
The new algorithm relies on numerical continuation along rigid continuation paths.  ...  How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system?  ...  Beyond that, the continuation paths in the rigid setting preserve the geometry of the input equations. This opens a way for studying the average complexity of solving certain structured systems.  ... 
arXiv:1711.03420v2 fatcat:ldlka37r7zfopihfnqpghlq2ly

Rigid continuation paths II. Structured polynomial systems [article]

Peter Bürgisser, Felipe Cucker, Pierre Lairez
2021 arXiv   pre-print
This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used.  ...  Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L.  ...  Rigid continuation paths. In Part I, we introduced rigid continuation paths to obtain, in the case of random Gaussian systems, the bound (2.10) ErKs polypn, δq.  ... 
arXiv:2010.10997v2 fatcat:k2oucawmqfdrlkp2kduwlgofdu

Master index volumes 11–20

1987 Journal of Computational and Applied Mathematics  
the convergence rates of stationary iterative schemes for the solution of complex linear systems 15 (1) (1986) l-11 Axelsson, O., Incomplete block matrix factorization preconditioning methods.  ...  G. and Rach, R., Evaluation of integrals by decomposition 16 (2) (1986) 233-235 Agarwai, R.P. and Usmani, R.A., Iterative methods for solving right focal point boundary value problems 14 (3) (1986) 371  ...  Magnus, A.P., Toeplitz matrix techniques and convergence of complex weight Pad6 approximants Makinson, G.J. and Shah, A.A., An iterative solution method for solving sparse nonsymmetric linear systems Manoranjan  ... 
doi:10.1016/0377-0427(87)90187-7 fatcat:ijnwtaltozdqfe5kt72oly7jja

Subject index volumes 101–150

1995 Theoretical Computer Science  
Gallier and -monadic rewrite system 417 sample complexity 6 I?  ...  Lyndon words, average cost of - 518, 756 average cost of - 518 Dyck paths 2.54 semi--language 5 IX set 420 Dymond and Tompa 562 dynamic adequacy 594 algebras 544 algorithms 234 atomicity  ... 
doi:10.1016/0304-3975(95)80002-6 fatcat:azbu2orp7vc75pwux4jxnvxl7m

Two-Layer Shallow Water System: A Relaxation Approach

Rémi Abgrall, Smadar Karni
2009 SIAM Journal on Scientific Computing  
The two-layer shallow water system is an averaged flow model. It forms a nonconservative system which is only conditionally hyperbolic.  ...  Numerical results are shown for unsteady flows as well as for smooth and nonsmooth steady flows.  ...  The shallow water system, being a layer-averaged flow model, does not offer a clear definition of a path.  ... 
doi:10.1137/06067167x fatcat:elnvke5rz5f3xcm3su3pubha5m

Quantized global parametrization

Marcel Campen, David Bommes, Leif Kobbelt
2015 ACM Transactions on Graphics  
Figure 1 : Left: Global parametrization with continuous rigid transitions; notice the mismatch of the depicted isolines across the red cuts.  ...  We present a novel method that allows us to solve it, i.e. to find valid as well as good quality quantizations.  ...  Acknowledgments We thank Max Lyon and Moritz Ibing for their help with the implementation and evaluation.  ... 
doi:10.1145/2816795.2818140 fatcat:4pesvbxsxff2xncszksnpvuptm

Stability of shear shallow water flows with free surface [article]

Alexander Chesnokov, Gennady El, Sergey Gavrilyuk, Maxim Pavlov
2016 arXiv   pre-print
Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated.  ...  This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations.  ...  The authors thank IMéRA foundation of Aix-Marseille Université for hospitality. They also grateful to P. V. Kovtunenko for performing symbolic computations in Appendix.  ... 
arXiv:1610.04331v1 fatcat:xoirae5ewvbjfh26sbq7euanua

Functional norms, condition numbers and numerical algorithms in algebraic geometry [article]

Felipe Cucker and Alperen A. Ergür and Josué Tonelli-Cueto
2021 arXiv   pre-print
algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry, and the computation of zeros of systems of complex quadratic polynomials  ...  In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of L_p norms for numerical algebraic geometry, with an emphasis on L_∞.  ...  The second author is grateful to Hakan and Bahadır Ergür for their cheerful response to his sudden all-day availability throughout the pandemic times.  ... 
arXiv:2102.11727v1 fatcat:3jdrw5pwazbd5dvgwjkoohskme

A discontinuous Galerkin finite-element model for fast channelized lava flows v1.0

Colton J. Conroy, Einat Lev
2021 Geoscientific Model Development  
We verify the model against an analytic test case and demonstrate convergence rates of P+1/2 for polynomials of degree 𝒫.  ...  Direct observations recorded by unoccupied aerial systems (UASs) during the Kīlauea eruption provide inlet conditions, constrain input parameters, and serve as a benchmark for model evaluation.  ...  The unknown polynomial basis coefficients that define the solution over a given element, j , are advanced in time from t m to t m+1 via 1. SetŨ (i) 0 ←Ũ (i) m , for i = 1, 2, 3, 4. 2.  ... 
doi:10.5194/gmd-14-3553-2021 fatcat:auxhsnxxx5ce7biwpozoan5cee

Constraint Control of Nonholonomic Mechanical Systems [article]

Vakhtang Putkaradze, Stuart Rogers
2018 arXiv   pre-print
We show that it is possible to control the system using the constraint ξ(t) and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.  ...  We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control.  ...  followpath, available in the MATLAB toolbox Chebfun, is able to utilize Chebfun's quasilinearization ODE BVP solver bvpsolve to solve ODE BVPs via continuation, where the continuation parameter may have  ... 
arXiv:1610.02595v17 fatcat:dxzbkpmur5d2fgg4sla56wswam

Global Dynamics, Blow-Up, and Bianchi Cosmology [article]

Nitsan Ben-Gal, Bernhard Brehm, Johannes Buchner, Juliette Hell, Anna Karnauhova, Stefan Liebscher, Alan Rendall, Brian Smith, Hannes Stuke, Martin Väth, Bernold Fiedler
2016 arXiv   pre-print
Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type.  ...  This means that the embedded i(v)-cells W u (v) possess embedded sphere boundaries ∂W u (v) ∼ = S i(v)−1 , so that the attaching maps onto the boundary complexes are homeomorphisms, rather than just continuous  ...  Stuke proves that solutions on the complex unstable manifold of u * stay bounded on time paths Γ ± p parallel to the real axis.  ... 
arXiv:1607.04600v1 fatcat:mn5bqodp25aehmpt4n7vsnlhym

Asymptotic invariants, complexity of groups and related problems

Mark Sapir
2011 Bulletin of Mathematical Sciences  
We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.  ...  For this, the key idea is to solve systems of equations.  ...  [155] . 2.2.I Complexity classes and algebraic systems.  ... 
doi:10.1007/s13373-011-0008-1 fatcat:mqv7bb7rz5e4lpjvkz2cvndho4

A Recent Development of Computer Methods for Solving Singularly Perturbed Boundary Value Problems

Manoj Kumar, Parul
2011 International Journal of Differential Equations  
The authors also extend their appreciation to anonymous reviewers for their valuable suggestions to revise this paper.  ...  To derive the method, the authors used a polynomial function S Δ x of class C 2 0, 1 in the following form: S Δ x a i b i x − x i c i x − x i 2 d i x − x i 3 , i 0, 1, 2, . . . , N − 1, Δ : 0 x 0 < x 1  ...  or not rigid.  ... 
doi:10.1155/2011/404276 fatcat:boyo2ksfifgjperfeonk2yyyne

Gonihedric string equation I

G.K Savvidy, K.G Savvidy
1998 Journal of High Energy Physics  
Explicit formulas for the mass spectrum allow to compute the string tension and thus demonstrate the string character of the theory.  ...  The spectrum of the theory consists of particles and antiparticles of increasing half-integer spin lying on quasilinear trajectories of different slope.  ...  We are thankful to Pavlos Savvidis, who has found the recurrent relation for the computation of the characteristic polynomials of the Jacoby matrices which finally leads to the basic solution for Γ 0 .  ... 
doi:10.1088/1126-6708/1998/04/008 fatcat:d4eztrkuavh35mmyumsm3u3fbm

Asymptotic invariants, complexity of groups and related problems [article]

Mark Sapir
2011 arXiv   pre-print
We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.  ...  For this, the key idea is to solve systems of equations.  ...  For example we do not talk about the average case complexity, generic solutions of algorithmic problems, search problems and other complexity related issues used in group based cryptography [168] .  ... 
arXiv:1012.1325v4 fatcat:aoac6a3vcraghisr7qwb4vc5v4
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