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Raphson Algorithm (raphson + algorithm)

Distribution by Scientific Domains
Mathematics and Statistics60%
Engineering40%

Selected Abstracts

Simulation of special loading conditions by means of non-linear constraints imposed through Lagrange multipliers

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2002
M. A. Gutiérrez
Abstract This paper discusses the necessity and handling of non-linear constraint equations to describe the behaviour of properties of the loading system such as, e.g. smooth free-rotating loading platens. An exact, non-linear formulation for a smooth loading platen is derived and its incorporation into the equilibrium equations is presented. For this purpose, the Lagrange multiplier method is used. The solution of the equilibrium equations by means of a Newton,Raphson algorithm is also outlined. The proposed approach is validated on a patch of two finite elements and applied to a compression-bending test on a pre-notched specimen. It is observed that use of a linearized approximation of the boundary constraint can lead to errors in the description of the motion of the constrained nodes. Thus, the non-linear formulation is preferable. Copyright © 2002 John Wiley & Sons, Ltd. [source]

A comparative study of GLS finite elements with velocity and pressure equally interpolated for solving incompressible viscous flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2009
Yongtao Wei
Abstract A comparative study of the bi-linear and bi-quadratic quadrilateral elements and the quadratic triangular element for solving incompressible viscous flows is presented. These elements make use of the stabilized finite element formulation of the Galerkin/least-squares method to simulate the flows, with the pressure and velocity fields interpolated with equal orders. The tangent matrices are explicitly derived and the Newton,Raphson algorithm is employed to solve the resulting nonlinear equations. The numerical solutions of the classical lid-driven cavity flow problem are obtained for Reynolds numbers between 1000 and 20 000 and the accuracy and converging rate of the different elements are compared. The influence on the numerical solution of the least square of incompressible condition is also studied. The numerical example shows that the quadratic triangular element exhibits a better compromise between accuracy and converging rate than the other two elements. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Maximum likelihood estimation of a latent variable time-series model

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 1 2001
Francesco Bartolucci
Abstract Recently, Fridman and Harris proposed a method which allows one to approximate the likelihood of the basic stochastic volatility model. They also propose to estimate the parameters of such a model maximising the approximate likelihood by an algorithm which makes use of numerical derivatives. In this paper we propose an extension of their method which enables the computation of the first and second analytical derivatives of the approximate likelihood. As will be shown, these derivatives may be used to maximize the approximate likelihood through the Newton,Raphson algorithm, with a saving in the computational time. Moreover, these derivatives approximate the corresponding derivatives of the exact likelihood. In particular, the second derivative may be used to compute the standard error of the estimator and confidence intervals for the parameters. The paper presents also the results of a simulation study which allows one to compare our approach with other existing approaches. Copyright © 2001 John Wiley & Sons, Ltd. [source]

L1 Penalized Estimation in the Cox Proportional Hazards Model

BIOMETRICAL JOURNAL, Issue 1 2010
Jelle J. Goeman
Abstract This article presents a novel algorithm that efficiently computes L1 penalized (lasso) estimates of parameters in high-dimensional models. The lasso has the property that it simultaneously performs variable selection and shrinkage, which makes it very useful for finding interpretable prediction rules in high-dimensional data. The new algorithm is based on a combination of gradient ascent optimization with the Newton,Raphson algorithm. It is described for a general likelihood function and can be applied in generalized linear models and other models with an L1 penalty. The algorithm is demonstrated in the Cox proportional hazards model, predicting survival of breast cancer patients using gene expression data, and its performance is compared with competing approaches. An R package, penalized, that implements the method, is available on CRAN. [source]

Semiparametric Estimation Exploiting Covariate Independence in Two-Phase Randomized Trials

BIOMETRICS, Issue 1 2009
James Y. Dai
Summary Recent results for case,control sampling suggest when the covariate distribution is constrained by gene-environment independence, semiparametric estimation exploiting such independence yields a great deal of efficiency gain. We consider the efficient estimation of the treatment,biomarker interaction in two-phase sampling nested within randomized clinical trials, incorporating the independence between a randomized treatment and the baseline markers. We develop a Newton,Raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate (SPMLE). Our algorithm accommodates both continuous phase-one outcomes and continuous phase-two biomarkers. The profile information matrix is computed explicitly via numerical differentiation. In certain situations where computing the SPMLE is slow, we propose a maximum estimated likelihood estimator (MELE), which is also capable of incorporating the covariate independence. This estimated likelihood approach uses a one-step empirical covariate distribution, thus is straightforward to maximize. It offers a closed-form variance estimate with limited increase in variance relative to the fully efficient SPMLE. Our results suggest exploiting the covariate independence in two-phase sampling increases the efficiency substantially, particularly for estimating treatment,biomarker interactions. [source]