Topic
Algebraic equation
About: Algebraic equation is a research topic. Over the lifetime, 9252 publications have been published within this topic receiving 156854 citations.
Papers published on a yearly basis
1 / 3
Papers
TL;DR: The current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness, are described.
Abstract: SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.
2,696 citations
TL;DR: In this paper, a Rayleigh-Ritz procedure is introduced which replaces arbitrary variations with parametric variations, and previously unsolved nonlinear equations become solvable algebraic equations in the Rayleigh Ritz approximation.
Abstract: An effective action and potential for composite operators is obtained. The formalism is used to analyze, by variational techniques, dynamical symmetry breaking and coherent solutions to field theory. A Rayleigh-Ritz procedure is introduced which replaces arbitrary variations with parametric variations. Previously unsolved nonlinear equations become, in the Rayleigh-Ritz approximation, solvable algebraic equations.
1,587 citations
TL;DR: Taylor-series estimation as mentioned in this paper gives a least-sum-squared-error solution to a set of simultaneous linearized algebraic equations and provides the statistical spread of the solution errors.
Abstract: Taylor-series estimation gives a least-sum-squared-error solution to a set of simultaneous linearized algebraic equations. This method is useful in solving multimeasurement mixed-mode position-location problems typical of many navigational applications. While convergence is not proved, examples show that most problems do converge to the correct solution from reasonable initial guesses. The method also provides the statistical spread of the solution errors.
1,422 citations
1 Sep 1982
TL;DR: The algorithms and strategies used in DASSL, for the numerical solution of implicit systems of differential/algebraic equations, are outlined, and some of the features of the code are explained.
Abstract: This paper describes a new code DASSL, for the numerical solution of implicit systems of differential/algebraic equations. These equations are written in the form F(t,y,y') = 0, and they can include systems which are substantially more complex than standard form ODE systems y' = f(t,y). Differential/algebraic equations occur in several diverse applications in the physical world. We outline the algorithms and strategies used in DASSL, and explain some of the features of the code. In addition, we outline briefly what needs to be done to solve a problem using DASSL.
1,161 citations
TL;DR: A linear hyperbolic system is constructed with a stiff lower order term that approximates the original system with a small dissipative correction and can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally.
Abstract: We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.
1,011 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2026 | 1 |
| 2025 | 29 |
| 2024 | 109 |
| 2023 | 256 |
| 2022 | 415 |
| 2021 | 461 |