Generalising the Fast Reciprocal Square Root Algorithm

Classés selon la proximité de leur titre et de leur résumé avec ceux de cet article. Sélectionnez-en pour les lire ensemble.

Les articles les plus proches

1. Apparenté Pas pertinent

   Fractional Newton-Raphson Method

   A. Torres-Hernandez, F. Brambila-Paz · 2017 · Applied Mathematics and Sciences: An International Journal (MathSJ), 8:1-13, 2021 · Pas encore résumé

   Résumé scientifique

   The Newton-Raphson (N-R) method is useful to find the roots of a polynomial of degree n. However, this method is limited since it diverges for the case in which polynomials only have complex roots if a real initial condition is taken. In the present work, we explain an iterative method that is created using the fractional calculus, which we will call the Fractional Newton-Raphson (F N-R) Method, which has the ability to enter the space of complex numbers given a real initial condition, which allows us to find both the real and complex roots of a polynomial unlike the classical Newton-Raphson method.

2. Apparenté Pas pertinent

   Isomeric trees and the order of Runge--Kutta methods

   John C. Butcher, Helmut Podhaisky · 2021 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   The conditions for a Runge--Kutta method to be of order $p$ with $p\ge 5$ for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge--Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge--Kutta trees into "atomic stumps" and enumerate those conditions up to $p=20$. Using a special search procedure over unsatisfied order conditions, new Runge--Kutta methods of "ambiguous orders" five and six are constructed. These are used to verify the validity of the results.

3. Apparenté Pas pertinent

   Convergence Analysis for Nonlinear GMRES

   Yunhui He · 2025 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   In this work, we revisit nonlinear generalized minimal residual method (NGMRES) applied to nonlinear problems. NGMRES is used to accelerate the convergence of fixed-point iterations, which can substantially improve the performance of the underlying fixed-point iterations. We consider NGMRES with a finite window size $m$, denoted as NGMRES($m$). However, there is no convergence analysis for NGMRES($m$) applied to nonlinear systems. We prove that for general $m>0$, the residuals of NGMRES($m$) converge r-linearly under some conditions. For $m=0$, we prove that the residuals of NGMRES(0) converge q-linearly.

4. Apparenté Pas pertinent

   An embedded corrector problem for homogenization. Part II: Algorithms and discretization

   Eric Cancès, Virginie Ehrlacher, Frederic Legoll et al. · 2018 · Analyse numérique · Pas encore résumé

   Résumé scientifique

   This contribution is the numerically oriented companion article of the work [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Canc\`es, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv p…

5. Apparenté Pas pertinent

   Tensor-Product Split-Simplex Summation-By-Parts Operators

   Zelalem Arega Worku, Jason E. Hicken, David W. Zingg · 2024 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   We present an approach to construct efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The operators are constructed by splitting the simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto the subdomains, and assembling back using a continuous-Galerkin-type procedure. These tensor-product split-simplex operators do not have repeated degrees of freedom at the interior interfaces between the split subdomains. Furthermore, they satisfy the SBP property by construction, leading to stable discretizations. The accuracy and sparsity of the operators substantially enhance the efficiency of SBP dis…

6. Apparenté Pas pertinent

   DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation Solvers

   David Widmann, Chris Rackauckas · 2022 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation, leads to a design that is effective for DDEs while using all of the ODE internals. We highlight som…

7. Apparenté Pas pertinent

   A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation

   Ritukesh Bharali, Somdatta Goswami, Cosmin Anitescu et al. · 2021 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adop…

8. Apparenté Pas pertinent

   Thirty years of the finite volume method for solid mechanics

   Philip Cardiff, Ismet Demirdžić · 2018 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs staggered vs vertex-centred), solution algorithm (implicit vs explicit), and stabilisation strategy (Rhie-Chow vs Jameson-Schmidt-Turkel vs Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finish…