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This paper considers the oscillators coupled in the form of a ring through resistors. It is described for the case of five oscillators that, for the symmetrical solutions, the analysis based on the conjugate class in group theory is useful. Major problems in tie coupled oscillators are the existence of the synchronized periodic solution and its stability. In the case of the weakly nonlinear system, the problems of the synchronized periodic solution and the stability are almost answered by techniques that include mode analysis. In the case of the strongly nonlinear system, on the other hand, the periodic solutions with symmetry have been classified mostly based on the subgroup in the group theory when the number of coupled oscillators is less. When the number of oscillators is increased, however, the number of subgroups becomes tremendous, which makes systematic analysis difficult. This paper uses the conjugate class and shows that tie symmetrical solutions can be classified systematically. It is also shown that the approximate periodic solution with symmetry can be approximately determined by the averaging based on the symmetry. [ABSTRACT FROM AUTHOR]

A preconditioning methodology for an iterative solution of discrete stress analysis problems based on a space decomposition and subspace correction framework is analysed in this paper. The principle idea of our approach is a decomposition of a global discrete system into the series of subproblems each of which correspond to the different Cartesian co-ordinates of the solution (displacement) vector. This enables us to treat the matrix subproblems in a segregated way. A host of well-established scalar solvers can be employed for the solution of subproblems. In this paper we constrain ourselves to an approximate solution using the scalar algebraic multigrid (AMG) solver, while the subspace correction is performed either in block diagonal (Jacobi) or block lower triangular (Gauss–Seidel) fashion. The preconditioning methodology is justified theoretically for the case of the block-diagonal preconditioner using Korn's inequality for estimating the ratio between the extremal eigenvalues of a preconditioned matrix. The effectiveness of the AMG-based preconditioner is tested on stress analysis 3D model problems that arise in microfabrication technology. The numerical results, which are in accordance with theoretical predictions, clearly demonstrate the superiority of a component decomposition AMG preconditioner over the standard ILU preconditioner, even for the problems with a relatively small number of degrees of freedom. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field. [ABSTRACT FROM AUTHOR]

To make possible the application of the three-dimensional finite element method to electrically large problems, it is combined with the analytical solutions of arbitrary large cavity with aperture. In this paper, detailed analysis for this hybrid method is presented. The element matrices necessary for coupling the finite element method to analytical solutions are given. The proposed hybrid method significantly reduces the number of unknown since only the inhomogeneity needs to be discretized. Thus computer memory and storage demands are reduced. In addition, this hybrid method employs the edge element which is not expected to produce spurious solutions. Also, the formulation presented in this paper preserves the sparsity of the finite element matrix, and does not require any matrix inversion. This new hybrid method is used to compute the field distributions in various partially filled rectangular enclosures. The results match well with the pure edge- based finite element method and analytical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider three-dimensional incompressible magnetohydrodynamics equations. By using interpolation inequalities in anisotropic Lebesgue space, we provide regularity criteria involving the velocity or alternatively involving the fractional derivative of velocity in one direction, which generalize some known results. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In representation theory of algebras the notion of ‘mutation’ often plays important roles, and two cases are well known, that is, ‘cluster tilting mutation’ and ‘exceptional mutation’. In this paper we focus on ‘tilting mutation’, which has a disadvantage that it is often impossible, that is, some of summands of a tilting object cannot be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing ‘silting mutation’ for silting objects as a generalization of ‘tilting mutation’. We shall develop a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with ‘silting mutation’ by generalizing the theory of Riedtmann–Schofield and Happel–Unger. We show that iterated silting mutations act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally, we give a bijection between silting subcategories and certain t-structures. [ABSTRACT FROM PUBLISHER]

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students `use of graphing calculators to solve the problems is discussed. [ABSTRACT FROM AUTHOR]

In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We generalize this and minimize over derivatives of polynomials of arbitrary odd degree. We analyze the cubic case, which assumes first derivative boundary data and show that the minimization has a unique, infinitely smooth solution with cubic precision. We have not been able to prove that the solution satisfies the Hermite interpolation conditions but numerical examples strongly indicate that it does for a wide variety of planar domains and that it behaves nicely. [ABSTRACT FROM AUTHOR]

The electrical activity in the heart is governed by the bidomain equations. In this paper, we analyse an order optimal method for the algebraic equations arising from the discretization of this model. Our scheme is defined in terms of block Jacobi or block symmetric Gauss–Seidel preconditioners. Furthermore, each block in these methods is based on standard preconditioners for scalar elliptic or parabolic partial differential equations (PDEs). Such preconditioners can be realized in terms of multigrid or domain decomposition schemes, and are thus readily available by applying ‘off-the-shelves’ software. Finally, our theoretical findings are illuminated by a series of numerical experiments. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper analyses a novel method for constructing preconditioners for diagonally dominant symmetric positive-definite matrices. The method discussed here is based on a simple idea: we construct M by simply dropping offdiagonal non-zeros from A and modifying the diagonal elements to maintain a certain row-sum property. The preconditioners are extensions of Vaidya's augmented maximum-spanning-tree preconditioners. The preconditioners presented here were also mentioned by Vaidya in an unpublished manuscript, but without a complete analysis. The preconditioners that we present have only O(n+t2) nonzeros, where n is the dimension of the matrix and 1⩽t⩽n is a parameter that one can choose. Their construction is efficient and guarantees that the condition number of the preconditioned system is O(n2/t2) if the number of nonzeros per row in the matrix is bounded by a constant. We have developed an efficient algorithm to construct these preconditioners and we have implemented it. We used our implementation to solve a simple model problem; we show the combinatorial structure of the preconditioners and we present encouraging convergence results. [ABSTRACT FROM AUTHOR]

This paper studies the possibility of combining interior point strategy with a steepest descent method when solving convex programming problems, in such a way that the convergence property of the interior point method remains valid but many iterations do not request the solution of a system of equations. Motivated by this general idea, we propose a hybrid algorithm which combines a primal–dual potential reduction algorithm with the use of the steepest descent direction of the potential function. The $O(\sqrt{n}\left|{\ln {\epsilon}} \right|)$ complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. Our numerical experiments are also reported. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In FDTD analysis, the numerical dispersion caused by the discretization of time and space becomes a problem. Previously, the numerical dispersion was corrected by changing the speed of light. However, such an approach is limited to isotropic meshes. In this paper, a generalized formulation is introduced such that this correction method can also be used in nonisotropic meshes. As an application example, the method is used for the analysis of a large cavity, demonstrating the effectiveness of the method. © 2002 Scripta Technica, Electron Comm Jpn Pt 2, 85(2): 61–69, 2002 [ABSTRACT FROM AUTHOR]

The second part of this paper deals with the systolic implementation of the computational kernel for factorial data analysis, defined in Part I, on special-purpose hardware. The framework of the study is that a sequence of different algorithms has to be performed on a unique hardware array. This fact has led us to the design of the programmable systolic array SARDA: this is a triangular array which consists of programmable nodes with local memory and programmable orthogonal connections. [ABSTRACT FROM AUTHOR]

One of the solution methods for algebraic equations is the continuous iterative method. This is an iterative method in which a differential equation is constructed with the solution of a given algebraic equation as the equilibrium solution, and the solution of the difference equation obtained by discretizing that equation is sought until the solution converges. The method is an extension of various basic iterative methods, and coincides with the original iterative method when the discretization step width is 1. This paper shows first that the continuous iterative method converges globally if the function of the given algebraic equation is a product of a positive-definite matrix, and the gradient of a scalar function and the step width is somewhat small. Especially, for the case where the algebraic equation is linear, it is shown that the continuous iterative method may converge even if the original iterative method diverges, or that it converges faster than the original iterative method if the parameters are appropriately set. Then a case is considered where the function of the given algebraic equation is the gradient of a scalar function. A method is proposed which adjusts the coefficients of the continuous iterative method adaptively so that the convergence is as fast as possible. The effectiveness of the method is verified by simple examples.

The Yasuura method (mode matching method) was applied to the problem of electromagnetic wave scattering by an obstacle in a rectangular waveguide to prove the usefulness of this method as a general-purpose analysis method. In this paper, the scattering problem of an inductive obstacle with edges placed in a rectangular waveguide is analyzed numerically, taking into account the singularity of the solution at the edges. The inductive obstacle is considered to be a perfect conductor with an asymmetrical triangular cross section with an arbitrary apex angle. Each comer of the triangle is treated as a singularity. The algorithm is presented which takes into consideration the singularity of the solution at the edge, and then the accuracy of the analysis is discussed. As a numerical example, the frequency characteristics of reflection and transmission coefficients versus various parameters are shown including the behaviors of higher-order modes. By taking advantage of the fact that the magnitude of the apex angle can be changed arbitrarily, the effect of the slant angle of the triangle on the incident side exerted on the scattering is discussed. Since the structure can be computed for an apex angle starting from 0°, the solution for an apex angle of 0° or the inductive window is compared with the exact solution by Wiener-Hopf method so that the analysis error is carried out. [ABSTRACT FROM AUTHOR]

The paper presents a simple algorithm for solving a system of inhomogeneous high order differential equations with variable coefficients. The method also provides a numerical solution to non-linear ordinary differential equations. The technique is based on reducing the high order equations into a system of first order rate equations. Through a simple translation process, the variables in the reduced set of equations are solved simultaneously by an iterative scheme using the TLM multicompartment model. The numerical technique is demonstrated by solving well-known second order differential equations. The numerical solutions are compared with the analytical solutions to the differential equations. [ABSTRACT FROM AUTHOR]

Given a solution u to a linear, homogeneous, second-order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set 풷( u)u { x :|δ u|( x) = 0}, as well as the first estimates on the effective critical set 풷 r( u), which roughly consists of points x such that the gradient of u is small somewhere on B r( x) compared to the nonconstancy of u. The results are new even for harmonic functions on ℝ n. Given such a u, the standard first-order stratification { l k} of u separates points x based on the degrees of symmetry of the leading-order polynomial of u- u( x). In this paper we give a quantitative stratification [ABSTRACT FROM AUTHOR]

In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A*X-αA = Q with α ∈ (0, ∞) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Several representations for the interpolating polynomial exist: Lagrange, Newton, orthogonal polynomials, etc. Each representation is characterized by some basis functions. In this paper we investigate the transformations between the basis functions which map a specific representation to another. We show that for this purpose the LU- and the QR-decomposition of the Vandermonde matrix play a crucial role. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kernel polynomials are considered. Theoretical results as well as numerical experiments illustrate the effectiveness of our approach. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we present some new absolute and relative perturbation bounds of eigenvalues of normal matrices. The bounds depend upon the closeness of perturbed matrices to normal matrices and improve those previous results (Duke Math. J. 1953; 20:37–39, Linear Algebra Appl. 1996; 246:215–223). Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we discuss some linear algebra issues concerning the implementation of blended implicit methods (J. Comput. Appl. Math. 2000; 116:41–62, Appl. Numer. Math. 2002; 42:29–45, J. Comput. Appl. Math. 2004; 164–165:145–158, In Recent Trends in Numerical Analysis, Trigiante D (ed.), Nova Science Publication Inc.: New York, 2001; 81–105) for the numerical solution of ODEs. In particular, we describe the strategies, used in the numerical code BiM (J. Comput. Appl. Math. 2004; 164–165:145–158), for deciding whether re-evaluating the Jacobian and/or the factorization involved in the non-linear splitting for solving the discrete problem. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The symmetric Sinc–Galerkin method developed by Lund (Math. Comput. 1986; 47:571–588), when applied to second-order self-adjoint boundary value problems on d dimensional rectangular domains, gives rise to an N × N positive definite coefficient matrix which can be viewed as the sum of d Kronecker products among d - 1 real diagonal matrices and one symmetric Toeplitz-plus-diagonal matrix. Thus, the resulting coefficient matrix has a strong structure so that it can be advantageously used in solving the discrete system. The main contribution of this paper is to present and analyse a multigrid method for these Sinc–Galerkin systems. In particular, we show by numerical examples that the solution of a discrete symmetric Sinc–Galerkin system can be obtained in an optimal way only using O(N log N) arithmetic operations. Numerical examples concerning one- and two-dimensional problems show that the multigrid method is practical and efficient for solving the above symmetric Sinc–Galerkin linear system. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper defines topology relations of elements treated as overlapping lists of nodes. In particular, the element topology makes use of element faces, element vertices and boundary faces which coincide with the actual (geometrical) faces, vertices and boundary faces in the case of true finite elements. The element topology is used in an agglomeration algorithm to produce agglomerated elements (a non-overlapping partition of the original elements) and their topology is then constructed, thus allowing for recursion. The main part of the algorithms is based on operations on Boolean sparse matrices and the implementation of the algorithms can utilize any available (parallel) sparse matrix format. Applications of the sparse matrix element topology to AMGe (algebraic multigrid for finite element problems), including elementwise constructions of coarse non-linear finite element operators are outlined. An algorithm to generate a block nested dissection ordering of the nodes for generally unstructured finite element meshes is given as well. The coarsening of the element topology is illustrated on a number of fine-grid unstructured triangular meshes. Published in 2002 by John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block-diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg-method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M-matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block-diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)-factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher-order finite elements. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H0(curl,Ω). The finite element spaces are generated by Nédélec's edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl-operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper deals with two-step W-methods for stiff initial value problems which allow the parallel solution of the stage equations. Special methods of order 3 and 4 are tested for both sequential and parallel implementations and compared with efficient sequential codes. The results show that the methods are already competitive in a sequential implementation and have a high level of parallelization especially in combination with Krylov solvers for linear equations of large dimensions. [ABSTRACT FROM AUTHOR]

In this paper we define the polyadic Pavelka algebras as algebraic structures for Rational Pavelka predicate calculus (RPL∀). We prove two representation theorems which are the algebraic counterpart of the completness theorem for RPL∀. [ABSTRACT FROM AUTHOR]

We consider meshless collocation methods for the numerical solution of transport processes described by hyperbolic conservation laws. The future goal is the construction of a robust and reliable meshfree discretization method for the equations of gas dynamics in complex geometries. In this paper we start with the simplest scalar model problems and analyze basic problems occurring in the grid-free approach. A topological condition on clouds of points is derived and several possible versions of a generalized Lax-Friedrichs scheme are discussed with respect to their numerical dissipation. A moving least-squares approach is followed to construct a positive discretization for solutions with shocks which is thoroughly analyzed and applied to test problems. [ABSTRACT FROM AUTHOR]

The modified Gram–Schmidt (MGS) orthogonalization process—used for example in the Arnoldi algorithm—often constitutes the bottleneck that limits parallel efficiencies. Indeed, a number of communications, proportional to the square of the problem size, are required to compute the dot-products. A block formulation is attractive but it suffers from potential numerical instability. In this paper, we address this issue and propose a simple procedure that allows the use of a block Gram—Schmidt algorithm while guaranteeing a numerical accuracy close to that of MGS. The main idea is to determine the size of the blocks dynamically. The main advantages of this dynamic procedure are two-fold: first, high performance matrix–vector multiplications can be used to decrease the execution time. Next, in a parallel environment, the number of communications is reduced. Performance comparisons with the alternative Iterated CGS also show an improvement for a moderate number of processors. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Given convex scattered data in R3 we consider the constrained interpolation problem of finding a smooth, minimal Lp-norm (1 < p < ∞) interpolation network that is convex along the edges of an associated triangulation. In previous work the problem has been reduced to the solution of a nonlinear system of equations. In this paper we formulate and analyse a Newton-type algorithm for solving the corresponding type of systems. The correctness of the application of the proposed method is proved and its superlinear (in some cases quadratic) convergence is shown. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Research on map processing using computers has been active recently. Data of road networks in a map often are required for processing an urban map. If these data are fed manually into a computer, not only is labor considerable but many errors would occur. A solution to this problem is to extract road networks automatically from the map using a computer. This paper proposes a skip-scan method which automatically extracts parallel, lines from 1/25,000-scaled maps. A given map is scanned at constant intervals so that each cross point of the scanning line and the central line of a road are extracted as feature points. Many short vectors are formed by connecting the feature points. Referring to the original map, connection of the vectors, elimination of noise, and interpolation of parts of roads interrupted by characters on the map are carried out automatically. An algorithm used for this operation is simple, since almost all the elements are vectors. This paper describes the algorithm, and examples of result applications to actual 1/25,000-scaled maps. These confirm the usefulness of the method. [ABSTRACT FROM AUTHOR]

In this paper we study block diagonal preconditioners for mixed systems derived from the Dirichlet problems for second order elliptic equations. The main purpose is to discuss how an embedding of the original computational domain into a simpler extended can be utilized in this case. We show that a family of uniform preconditioners for the corresponding problem on the extended, or fictitious, domain leads directly to uniform preconditioners for the original problem. This is in contrast to the situation for the standard finite element method, where the domain embedding approach for the Dirichlet problem is less obvious. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In recent years multilevel preconditioners like BPX have become more and more popular for solving second-order elliptic finite element discretizations by iterative methods. P. Oswald has adapted these methods for discretizations of the fourth order biharmonic problem by rectangular conforming Bogner-Fox-Schmit elements and non-conforming Adini elements and has derived optimal estimates for the condition numbers of the preconditioned linear systems. In this paper we generalize the results from Oswald to the construction of parallel BPX and multilevel diagonal scaling (MDS-BPX) preconditioners for the elasticity problem of thin smooth shells in connection with Koiter's shell theory. We use the two discretizations mentioned above and the preconditioned conjugate gradient method as iterative method. The parallelization concept is based on a non-overlapping domain decomposition data structure. We describe the implementations of the parallel multilevel preconditioners. Finally, we show numerical results for some shells representing elliptic, parabolic, hyperbolic and more complicated classes. In addition, the influence of the thickness parameter and the loading on the preconditioner are investigated experimentally. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The incomplete Cholesky decomposition is known as an excellent smoother in a multigrid iteration and as a preconditioner for the conjugate gradient method. However, the existence of the decomposition is only ensured if the system matrix is an M-matrix. It is well-known that finite element methods usually do not lead to M-matrices. In contrast to this restricting fact, numerical experiments show that, even in cases where the system matrix is not an M-matrix the behaviour of the incomplete Cholesky decomposition apparently does not depend on the structure of the grid. In this paper the behaviour of the method is investigated theoretically for a model problem, where the M-matrix condition is violated systematically by a suitable perturbation. It is shown that in this example the stability of the incomplete Cholesky decomposition is independent of the perturbation and that the analysis of the smoothing property can be carried through. This can be considered as a generalization of the results for the so called square-grid triangulation, as has been established by Wittum in [12] and [11]. [ABSTRACT FROM AUTHOR]

The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s > 0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

The analysis of variance (ANOVA) is widely used in biological studies, yet there remains considerable confusion among researchers about the interpretation of hypotheses being tested. Ambiguities arise when statistical designs are unbalanced, and in particular when not all combinations of design factors are represented in the data. This paper clarifies the relationship among hypothesis testing, statistical modelling and computing procedures in ANOVA for unbalanced data. A simple two-factor fixed effects design is used to illustrate three common parametrizations for ANOVA models, and some associations among these parametrizations are developed. Biologically meaningful hypotheses for main effects and interactions are given in terms of each parametrization, and procedures for testing the hypotheses are described. The standard statistical computing procedures in ANOVA are given along with their corresponding hypotheses. Throughout the development unbalanced designs are assumed and attention is given to problems that arise with missing cells. [ABSTRACT FROM AUTHOR]

A large collection of factorial data analysis methods can be characterized by the following matrices: X, the k × n matrix of data, and A, B the symmetric positive definite matrices of size n, k which represent the chosen norms of Rn, Rk, respectively. All methods amount to computing the largest eigenvalues of U = XAXTB or the largest singular values of E = B½XA½. In Part I of this paper we begin by a geometrical and probabilistic interpretation of the various methods, showing how U and E are defined in each case. We then define the computational kernel for factorial data analysis. We conclude by devising the numerical aspects of software implementation for this kernel on microcomputers and presenting the package INDA. [ABSTRACT FROM AUTHOR]

The finite element method (FEM) with local absorbing boundary conditions has been recently applied to compute electromagnetic scattering from large 3-D geometries. In this paper, we present details pertaining to code implementation and optimization. Various types of sparse matrix storage schemes are discussed and their performance is examined in terms of vectorization and net storage requirements. The system of linear equations is solved using a preconditioned biconjugate gradient (BCG) algorithm and a fairly detailed study of existing point and block preconditioners (diagonal and incomplete LU) is carried out. A modified ILU preconditioning scheme is also introduced which works better than the traditional version for our matrix systems. The parallelization of the iterative sparse solver and the matrix generation/assembly as implemented on the KSR1 multiprocessor is described and the interprocessor communication patterns are analysed in detail. Near-linear speed-up is obtained for both the iterative solver and the matrix generation/assembly phases. Results are presented for a problem having 224,476 unknowns and validated by comparison with measured data. [ABSTRACT FROM AUTHOR]

The paper presents a combined fine-coarse mesh TLM method. This technique reduces the memory storage and the computational time when modelling complex structures. In the combined fine-coarse mesh model, the region of interest is covered with a set of transmission-line fine meshes. The remaining part is covered with regular (or irregular) coarse meshes. The fine-mesh nodes are connected through busbars to the adjacent coarse-mesh nodes. The combined fine-coarse mesh TLM technique is applied to some diffusion problems. Significant savings in computational time and memory storage are obtained without loss of accuracy. [ABSTRACT FROM AUTHOR]

The boundary residual method is a powerful technique for solving EM boundary-value problems. This technique produces matrices difficult to solve using many of the standard numerical techniques. This paper discusses lesser-known numerical aspects of this method, and it shows efficient and well-conditioned methods to solve both the homogeneous' and non-homogeneous boundary residual problem. [ABSTRACT FROM AUTHOR]

Weakly compressible and advection approximations of incompressible isothermal flows were developed and tested in (Commun. Numer. Methods Eng. 2006; 22:831–847). In this paper, we extend the method to solve equations governing incompressible thermal flows. The emphasis is again on the reconstruction of unconditionally stable numerical scheme such that, restriction on time steps, projection procedures, solution of linear system of algebraic equations and staggered grids are completely avoided in its implementation. These features are achieved by combining a low-Mach asymptotic in compressible flow equations with a semi-Lagrangian method for the weakly compressible approach. The time integration is carried out using an explicit Runge–Kutta with variable stages. The method is applied to the natural convection flows in a squared cavity for both steady and transient computations. The numerical results demonstrate high resolution of the proposed method and confirm its capability to provide accurate and efficient simulations for thermal flow problems. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This is an investigative paper which reports the results of comparisons of two numerical techniques for the solution of the Burton Cabrera and Frank (BCF) equation for the growth on crystal surfaces under steady state conditions. A successive over-relaxation (SOR) scheme for the equivalent finite difference equation gives rapid convergence to the static solution. It is known that a suitable choice of scattering parameters in a transmission line matrix (TLM) network analogue of the Laplace equation yields ultra-fast convergence. The results of numerical experiments which are reported here suggests that a similar situation also applies to the solution of the Poisson equation with shunt losses (the BCF equation), although the choice of optimum conditions appears to be different for different spatial positions within the solution space. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite. We prove that, for such A, the growth factor in Gaussian elimination is less than 3. Moreover, a slightly larger bound $${3\sqrt{2}}$$ holds true for a broader class of complex matrices A=B+iC, where B and C are Hermitian and positive definite. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The integrals required in the computation of influence coefficient matrices of the boundary element method (BEM) depend on the distance r(x,x′) from the collocation point or field point x to the source or load point x′. As a consequence, a distinction must be made between the case where the collocation point does not belong to the integration domain (proper integrals) and the case where the collocation point does belong to the integration domain (improper integrals). Moreover, situations arise where x comes close to x′ and the integrals, albeit of a regular character, behave almost as improper, this case being referred to as nearly singular integration. Analytical integration captures best the singular or nearly singular kernel behavior, but this technique can only be carried out in very simple situations as, for instance, boundary integrals over straight elements. In the present paper a set of useful analytical integration formulas for the 2D BEM with curved elements is deduced, employing a symbolic computational algebra system. © 1997 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of two- or three-dimensional contact problems with Coulomb friction and large deformation. The classical approach is based on two minimum principles or two variational inequalities: the first for unilateral contact and the second for friction. A coupled approach using only one principle or one inequality is presented. This new approach allows us to extend the notion of normality law to dissipative behaviors with a non-associated flow rule, such as surface friction. Non-differentiable contact potentials are regularized by means of the augmented Lagrangian method. Using the C++ language, an object-oriented finite element database is created, which allows us to implement the contact and friction in an existing code in a very simple and neat way. Numerical examples are carried out in many difficult cases such as shock absorber and three-dimensional contact. The numerical results prove that this approach is robust and efficient concerning numerical stability. [ABSTRACT FROM AUTHOR]

In this paper I re-examine the mechanics of the residual bending flexibility correction and show that it is an extravariational trick. [ABSTRACT FROM AUTHOR]

In this paper, I discuss the convergence and divergence rates for a class of generalized matrix multisplitting relaxation methods in a detailed manner. In particular, when the coefficient matrix is an L-matrix, I obtain several sufficient and necessary conditions ensuring the convergence of these methods. [ABSTRACT FROM AUTHOR]

In the paper we present a superconvergent patch recovery technique for obtaining higher-order-accurate finite-element solutions and thus a postprocessed type of L2 norm error estimate. Two modifications make our procedure different from the one proposed by Zienkiewicz and Zhu (1992), in which higher- order-accurate derivatives of the finite-element solution at nodes are determined. Firstly, the recovery process is made for elements, not for nodes. An 'element patch', which represents the union of an element under consideration and the surrounding elements, is introduced. Secondly, the local error estimate is calculated directly from the improved solution for this element. Numerical tests on both 1D and 2D model problems show that this method can provide an asymptotically exact a posteriori L2 norm error estimate if the used element possesses superconvergent points for the solutions. [ABSTRACT FROM AUTHOR]

The paper deals with the solution of the quasi-harmonic equation by the finite element method in a domain with corners on its boundary. A mapping technique, which can be used for locating the nodes of standard isoparametric elements in a subgrid (patch) of finite elements in the vicinity of a corner singularity point, is presented. The equations of the mapping are based on the analytical singular solution of the problem. Some analytical and numerical results to show the efficiency of the technique are presented. [ABSTRACT FROM AUTHOR]