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Newton's Method (newton + method)

Distribution by Scientific Domains

Mathematics and Statistics	48%
Engineering	48%

Selected Abstracts

A dual mortar approach for 3D finite deformation contact with consistent linearization

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
Alexander Popp
Abstract In this paper, an approach for three-dimensional frictionless contact based on a dual mortar formulation and using a primal,dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two-dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi-smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis. Copyright © 2010 John Wiley & Sons, Ltd. [source]

Numerical derivation of contact mechanics interface laws using a finite element approach for large 3D deformation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2004
Alex Alves Bandeira
Abstract In this work a homogenization method is presented to obtain by numerical simulation interface laws for normal contact pressure based on statistical surface models. For this purpose and assuming elastic behaviour of the asperities, the interface law of Kragelsky et al. (Friction and Wear,Calculation Methods, Pergamon, 1982) is chosen for comparison. The non-penetration condition and interface models for contact that take into account the surface micro-structure are investigated in detail. A theoretical basis for the three-dimensional contact problem with finite deformations is shortly presented. The augmented Lagrangian method is then used to solve the contact problem with friction. The algorithms for frictional contact are derived based on a slip rule using backward Euler integration like in plasticity. Special attention was dedicated to the consistent derivation of the contact equations between finite element surfaces. A matrix formulation for a node-to-surface contact element is derived consisting of a master surface segment with four nodes and a contacting slave node. It was also necessary to consider the special cases of node-to-edge contact and node-to-node contact in order to achieve the desired asymptotic quadratic convergence in the Newton method. A numerical example is selected to show the ability of the contact formulation and the algorithm to represent interface law for rough surfaces. Copyright © 2003 John Wiley & Sons, Ltd. [source]

The modeling and numerical analysis of wrinkled membranes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2003
Hongli Ding
Abstract In this paper three fundamental issues regarding modeling and analysis of wrinkled membranes are addressed. First, a new membrane model with viable Young's modulus and Poisson's ratio is proposed, which physically characterizes stress relaxation phenomena in membrane wrinkling, and expresses taut, wrinkled and slack states of a membrane in a systematic manner. Second, a parametric variational principle is developed for the new membrane model. Third, by the variational principle, the original membrane problem is converted to a non-linear complementarity problem in mathematical programming. A parametric finite element discretization and a smoothing Newton method are then used for numerical solution. The proposed membrane model and numerical method are capable of delivering convergent results for membranes with a mixture of wrinkled and slack regions, without iteration of membrane stresses. Three numerical examples are provided. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Improved multi-level Newton solvers for fully coupled multi-physics problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2003
J. Y. Kim
Abstract An algorithm is suggested to improve the efficiency of the multi-level Newton method that is used to solve multi-physics problems. It accounts for full coupling between the subsystems by using the direct differentiation method rather than error prone finite difference calculations and retains the advantage of greater flexibility over the tightly coupled approaches. Performance of the algorithm is demonstrated by solving a fluid,structure interaction problem. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Shape reconstruction of an inverse boundary value problem of two-dimensional Navier,Stokes equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010
Wenjing Yan
Abstract This paper is concerned with the problem of the shape reconstruction of two-dimensional flows governed by the Navier,Stokes equations. Our objective is to derive a regularized Gauss,Newton method using the corresponding operator equation in which the unknown is the geometric domain. The theoretical foundation for the Gauss,Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the boundary curve in the sense of a domain derivative. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Non-linear additive Schwarz preconditioners and application in computational fluid dynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002
Xiao-Chuan Cai
Abstract The focus of this paper is on the numerical solution of large sparse non-linear systems of algebraic equations on parallel computers. Such non-linear systems often arise from the discretization of non-linear partial differential equations, such as the Navier,Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non-linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel non-linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier,Stokes equations are reported. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Convergence analysis of the Newton,Shamanskii method for a nonsymmetric algebraic Riccati equation

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2008
Yiqin Lin
Abstract In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We apply the Newton,Shamanskii method to solve the vector equation. Convergence analysis shows that the sequence of vectors generated by the Newton,Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the Newton,Shamanskii method is feasible and effective, and outperforms the Newton method. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Parallel Newton two-stage methods based on ILU factorizations for nonlinear systems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2006
J. Arnal
Abstract Parallel iterative algorithms based on the Newton method and on two of its variants, the Shamanskii method and the Chord method, for solving nonlinear systems are proposed. These algorithms are based on two-stage multisplitting methods where incomplete LU factorizations are considered as a mean of constructing the inner splittings. Convergence properties of these parallel methods are studied for H -matrices. Computational results of these methods on two parallel computing systems are discussed. The reported experiments show the effectiveness of these methods. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Mesh-independent convergence of the modified inexact Newton method for a second order non-linear problem,

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2006
T. Kim
Abstract In this paper, we consider an inexact Newton method applied to a second order non-linear problem with higher order non-linearities. We provide conditions under which the method has a mesh-independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non-linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd. [source]

State waypoint approach to continuous-time nonlinear optimal control problems

ASIAN JOURNAL OF CONTROL, Issue 6 2009
Mohamadhadi Honarvarmahjoobin
Abstract In this paper, we propose an optimal control technique for a class of continuous-time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit-form suboptimal solution, by means of linear approximation. Thus the original continuous-time nonlinear control problem reduces to a finite-dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior-reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source]

A computer method based on simulated annealing to identify aquifer parameters using pumping-test data

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2008
Yen-Chen Huang
Abstract Conventional graphical or computer methods for identifying aquifer parameters have their own inevitable limitations. This paper proposes a computer method based on a drawdown model and a heuristic approach of simulated annealing (SA) to determine the best-fit aquifer parameters of the confined and unconfined aquifer systems. The drawdown model for the confined aquifer is the Theis solution and the unconfined aquifer is the Neuman solution. The estimated results of proposed method have better accuracy than those of the graphical methods and agree well with those of the computer methods based on the extended Kalman filter and Newton's method. Finally, the sensitivity analyses for the control parameters of SA indicate that the proposed method is very robust and stable in parameter identification procedures. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Steady-state 3D rolling-contact using boundary elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2007
R. Abascal
Abstract This work presents a new approach to the steady-state rolling contact problem for 3D elastic bodies. The problem solution is achieved by minimizing a general function representing the equilibrium equation and the rolling-contact restrictions. The boundary element method is used to compute the elastic influence coefficients of the surface points involved in the contact (equilibrium equations); while the contact conditions are represented with the help of projection functions. Finally, the minimization problem is solved by the generalized Newton's method with line search. Classic rolling problems are also solved and commented. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Fully stressed frame structures unobtainable by conventional design methodology

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2001
Keith M. Mueller
Abstract A structure is said to be fully stressed if every member of the structure is stressed to its maximum allowable limit for at least one of the loading conditions. Fully stressed design is most commonly used for small and medium size frames where drift is not a primary concern. There are several potential methods available to the engineer to proportion a fully stressed frame structure. The most commonly used methods are those taught to all structural engineering students and are very easy to understand and to implement. These conventional methods are based on the intuitive idea that if a member is overstressed, it should be made larger. If a member is understressed, it can be made smaller, saving valuable material. It has been found that a large number of distinct fully stressed designs can exist for a single frame structure subjected to multiple loading conditions. This study will demonstrate that conventional methods are unable to converge to many, if not most, of these designs. These unobtainable designs are referred to as ,repellers' under the action of conventional methods. Other, more complicated methods can be used to locate these repelling fully stressed designs. For example, Newton's method can be used to solve a non-linear system of equations that defines the fully stressed state. However, Newton's method can be plagued by divergence and also by convergence to physically meaningless solutions. This study will propose a new fully stressed design technique that does not have these problems. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Implicit symmetrized streamfunction formulations of magnetohydrodynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008
K. S. Kang
Abstract We apply the finite element method to the classic tilt instability problem of two-dimensional, incompressible magnetohydrodynamics, using a streamfunction approach to enforce the divergence-free conditions on the magnetic and velocity fields. We compare two formulations of the governing equations, the standard one based on streamfunctions and a hybrid formulation with velocities and magnetic field components. We use a finite element discretization on unstructured meshes and an implicit time discretization scheme. We use the PETSc library with index sets for parallelization. To solve the nonlinear problems on each time step, we compare two nonlinear Gauss-Seidel-type methods and Newton's method with several time-step sizes. We use GMRES in PETSc with multigrid preconditioning to solve the linear subproblems within the nonlinear solvers. We also study the scalability of this simulation on a cluster. Copyright © 2008 John Wiley & Sons, Ltd. [source]

A 3-D non-hydrostatic pressure model for small amplitude free surface flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2006
J. W. Lee
Abstract A three-dimensional, non-hydrostatic pressure, numerical model with k,, equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non-hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non-hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure,Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Elastohydrodynamics of tensioned web roll coating process

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003
M. S. Carvalho
Abstract Coating process is an important step in the manufacturing of different products, such as paper, adhesive and magnetic tapes, photographic films, and many other. The tensioned web roll coating is one the several methods used by different industries. It relies on the elastohydrodynamic action between the fluid and the tensioned substrate for transferring and applying the liquid. The main advantage of this method is its ability to apply very thin liquid layers with less sensitivity to mechanical tolerance at relative small cost. Despite its industrial application, theoretical analysis and fundamental understanding of the process are limited. This work analyses this elastohydrodynamic action by solving the differential equations that govern the liquid flow, described by the Navier,Stokes equation, and the web deformation, modelled by the cylindrical shell approximation. The goal is to determine the operating conditions at which the process is two dimensional and defect free. The equations are discretized by the Galerkin/finite-element method. The resulting non-linear system of equations is solved by Newton's method coupled with pseudo-arc-length continuation in order to obtain solutions around turning points. The theoretical results are used to construct an operating window of the process that is in agreement with limited experimental data. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Numerical computation of cross-coupled algebraic Riccati equations related to H2/H, control problem for singularly perturbed systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 8 2004
Hiroaki Mukaidani
Abstract In this paper, we present a numerical algorithm to the cross-coupled algebraic Riccati equations(CARE) related to H2/H, control problems for singularly perturbed systems (SPS) by means of Newton's method. The resulting algorithm can be widely used to solve Nash game problems and robust control problems because the CARE is solvable even if the quadratic term has an indefinite sign. We prove that the resulting iterative algorithm has the property of the quadratic convergence. Using the solution of the CARE, we construct the high-order approximate H2/H, controller. Copyright © 2004 John Wiley & Sons, Ltd. [source]

A bi-order kinetic model for poly(methyl methacrylate) decomposition in HNO3 using microwave irradiation

AICHE JOURNAL, Issue 8 2009
Chun-Chih Lin
Abstract In this study, a novel bi-order model combined with zero- and first-order kinetics was proposed for the decomposition of PMMA (MW = 120,000 g/mol) in concentrated HNO3 by microwave irradiation. To develop and validate this model, Fourier Transform Infrared spectroscopy, scanning electron microscopy, fractional-life method, the gravimetric analysis and Newton's method were utilized. Rate constants, activation energies, the pre-exponential factors and the weight fractions (,) via main-chain scission for the decomposition at 423,453 K were derived from this model. The zero-order reaction was observed dominant at 423,443 K, while the first-order reaction dominated at 453 K and 473 K. The digestion efficiency increased as HNO3 was increased to >3 mL at 423 K,443 K. At 473 K, the digestion was almost 100% when HNO3 volume was >3 mL. The estimated , values increased with HNO3 volume at 423 and 443 K, but varied insignificantly at 453 and 473 K. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source]

The multiroute maximum flow problem revisited

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2006
Donglei Du
Abstract We are given a directed network G = (V,A,u) with vertex set V, arc set A, a source vertex s , V, a destination vertex t , V, a finite capacity vector u = {uij}(i,j),A, and a positive integer m , Z+. The multiroute maximum flow problem (m -MFP) generalizes the ordinary maximum flow problem by seeking a maximum flow from s to t subject to not only the regular flow conservation constraints at the vertices (except s and t) and the flow capacity constraints at the arcs, but also the extra constraints that any flow must be routed along m arc-disjoint s - t paths. In this article, we devise two new combinatorial algorithms for m -MFP. One is based on Newton's method and another is based on an augmenting-path technique. We also show how the Newton-based algorithm unifies two existing algorithms, and how the augmenting-path algorithm is strongly polynomial for case m = 2. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(2), 81,92 2006 [source]

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2009
Zheng-Jian Bai
Abstract In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive-definite matrices with the mass matrix being diagonal and the damping and stiffness matrices being tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newton-type algorithm for the optimization problem, which improves a pre-existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2008
Peter Benner
Abstract We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linear-quadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2006
Alexei A. Mailybaev
Abstract The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Mesh-independent convergence of the modified inexact Newton method for a second order non-linear problem,

The effect of overall discretization scheme on Jacobian structure, convergence rate, and solution accuracy within the local rectangular refinement method

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2001
Beth Anne V. Bennett
Abstract The local rectangular refinement (LRR) solution-adaptive gridding method automatically produces orthogonal unstructured adaptive grids and incorporates multiple-scale finite differences to discretize systems of elliptic governing partial differential equations (PDEs). The coupled non-linear discretized equations are solved simultaneously via Newton's method with a Bi-CGSTAB linear system solver. The grids' unstructured nature produces a nonstandard sparsity pattern within the Jacobian. The effects of two discretization schemes (LRR multiple-scale stencils and traditional single-scale stencils) on Jacobian bandwidth, convergence speed, and solution accuracy are studied. With various point orderings, for two simple problems with analytical solutions, the LRR multiple-scale stencils are seen to: (1) produce Jacobians of smaller bandwidths than those resulting from the traditional single-scale stencils; (2) lead to significantly faster Newton's method convergence than the single-scale stencils; and (3) produce more accurate solutions than the single-scale stencils. The LRR method, including the LRR multiple-scale stencils, is finally applied to an engineering problem governed by strongly coupled, highly non-linear PDEs: a steady-state lean Bunsen flame with complex chemistry, multicomponent transport, and radiation modeling. Very good agreement is observed between the computed flame height and previously published experimental data. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Numerical solution of the free-surface viscous flow on a horizontal rotating elliptical cylinder

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2008
Roland Hunt
Abstract The numerical solution of the free-surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time-dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free-surface is determined as part of the solution. For the time-dependent case a simple time-marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break-down occurs. Time-dependent solutions do not produce the periodic solution after a long time-scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub-maximum weight solutions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 [source]

Error estimate and regularity for the compressible Navier-Stokes equations by Newton's method

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2003
Sang Dong Kim
Abstract The finite element discretization error estimate and H1 regularity are shown for the solution generated by Newton's method to the stationary compressible Navier-Stokes equations by interpreting Newton's method as an equivalent iterative method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 511,524, 2003 [source]

AN EVALUATION OF NON-ITERATIVE METHODS FOR ESTIMATING THE LINEAR-BY-LINEAR PARAMETER OF ORDINAL LOG-LINEAR MODELS

AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2009
Eric J. Beh
Summary Parameter estimation for association and log-linear models is an important aspect of the analysis of cross-classified categorical data. Classically, iterative procedures, including Newton's method and iterative scaling, have typically been used to calculate the maximum likelihood estimates of these parameters. An important special case occurs when the categorical variables are ordinal and this has received a considerable amount of attention for more than 20 years. This is because models for such cases involve the estimation of a parameter that quantifies the linear-by-linear association and is directly linked with the natural logarithm of the common odds ratio. The past five years has seen the development of non-iterative procedures for estimating the linear-by-linear parameter for ordinal log-linear models. Such procedures have been shown to lead to numerically equivalent estimates when compared with iterative, maximum likelihood estimates. Such procedures also enable the researcher to avoid some of the computational difficulties that commonly arise with iterative algorithms. This paper investigates and evaluates the performance of three non-iterative procedures for estimating this parameter by considering 14 contingency tables that have appeared in the statistical and allied literature. The estimation of the standard error of the association parameter is also considered. [source]