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
Papers
More filters
Book•
01 Jan 1984
TL;DR: The history of numerical device modeling can be traced back to the early 1970s as mentioned in this paper, when the basic Semiconductor Equations were defined and the goal of modeling was to identify the most fundamental properties of numerical devices.
Abstract: 1. Introduction.- 1.1 The Goal of Modeling.- 1.2 The History of Numerical Device Modeling.- 1.3 References.- 2. Some Fundamental Properties.- 2.1 Poisson's Equation.- 2.2 Continuity Equations.- 2.3 Carrier Transport Equations.- 2.4 Carrier Concentrations.- 2.5 Heat Flow Equation.- 2.6 The Basic Semiconductor Equations.- 2.7 References.- 3. Proeess Modeling.- 3.1 Ion Implantation.- 3.2 Diffusion.- 3.3 Oxidation.- 3.4 References.- 4. The Physical Parameters.- 4.1 Carrier Mobility Modeling.- 4.2 Carrier Generation-Recombination Modeling.- 4.3 Thermal Conductivity Modeling.- 4.4 Thermal Generation Modeling.- 4.5 References.- 5. Analytical Investigations About the Basic Semiconductor Equations.- 5.1 Domain and Boundary Conditions.- 5.2 Dependent Variables.- 5.3 The Existence of Solutions.- 5.4 Uniqueness or Non-Uniqueness of Solutions.- 5.5 Sealing.- 5.6 The Singular Perturbation Approach.- 5.7 Referenees.- 6. The Diseretization of the Basic Semiconductor Equations.- 6.1 Finite Differences.- 6.2 Finite Boxes.- 6.3 Finite Elements.- 6.4 The Transient Problem.- 6.5 Designing a Mesh.- 6.6 Referenees.- 7. The Solution of Systems of Nonlinear Algebraic Equations.- 7.1 Newton's Method and Extensions.- 7.2 Iterative Methods.- 7.3 Referenees.- 8. The Solution of Sparse Systems of Linear Equations.- 8.1 Direct Methods.- 8.2 Ordering Methods.- 8.3 Relaxation Methods.- 8.4 Alternating Direction Methods.- 8.5 Strongly Implicit Methods.- 8.6 Convergence Acceleration of Iterative Methods.- 8.7 Referenees.- 9. A Glimpse on Results.- 9.1 Breakdown Phenomena in MOSFET's.- 9.2 The Rate Effect in Thyristors.- 9.3 Referenees.- Author Index.- Table Index.
2,550 citations
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.
2,124 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.
1,273 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.
1,060 citations
01 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.
1,043 citations