block gauss elimination followed by a classical iterative method for the solution of linear systems in the last two decades many papers have appeared in which the application of an iterative method for the solution of a linear system is preceded by a step of the gauss elimination process in the hope that this will increase the rates of convergence of the iterative method this combination of methods has been proven successful especially when the matrix a of the system is an m matrix the purpose of this paper is to extend the idea of one to more gauss elimination steps consider other classes of matrices a e g p cyclic consistently ordered and generalize and improve the asymptotic convergence rates of some of the methods known so far

the numerical solution of equality constrained quadratic programming problems this paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem the constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints among this class of convergent iterative schemes are generalizations of the relaxed jacobi gauss seidel and symmetric gauss seidel schemes

extrapolated gauss seidel i and sor methods for least squares problems recently special attention has been given in the literature to the problems of accurately computing the least squares solution of very largescale over determined systems of linear equations which occur in geodetic applications in particular it has been suggested that one can solve such problems iteratively by applying the block sor successive overrelaxation and egs1 extrapolated gauss seidel 1 plus semi iterative methods to a linear system with coefficient matrix 2 cyclic or 3 cyclic the comparison of 2 block sor and 3 block sor was made in 1 and showed that the 2 block sor is better in 6 the authors also proved that 3 block egs1 si is better than 3 block sor here we first show that the 2 block dj double jacobi si gs si and egs1 si methods are equivalent and all of them are equivalent to the 3 block egs1 si method then we prove that the composite methods and 2 block sor have the same asymptotic rate of convergence but the former has a better average rate of convergence finally numerical experiments are reported and confirm that the 3 block egs1 si is better than the 2 block sor

transient acoustic radiation from impulsively accelerated bodies by the finite element method bodies under impulsive motion immersed in an infinite acoustic fluid severely put to test any numerical method for the transient exterior acoustic problem such problems in the context of the finite element method fem are not well studied fe modeling of such problems requires truncation of the infinite fluid domain at a certain distance from the structure the volume of computation depends upon the extent of this domain as well as the mesh density the modeling of the fluid truncation boundary is crucial to the economy and accuracy of solution and various boundary dampers have been proposed in the literature for this purpose the second order damper leads to unsymmetric boundary matrices and this necessitates the use of an unsymmetric equation solver for large problems the present paper demonstrates the use of fem with zeroth first and second order boundary dampers in conjunction with an unsymmetric out of core banded equation solver for impulsive motion problems of rigid bodies in an acoustic fluid the results compare well with those obtained from analytical methods

transient spheroidal elements for unbounded wave problems a wave envelope element numerical scheme is applied to the solution of unbounded wave problems the scheme is based on a fourier transformation of a discrete model formulated in the frequency domain this yields a discrete system of ordinary differential equations in time which are local in space oblate and prolate spheroidal element geometries are used the accuracy of the scheme is demonstrated by a comparison of computed and analytic solutions for axisymmetric test cases time harmonic and transient solution are presented an indirect solution procedure is also presented which permits the sparsity of the transient equations to be utilised more effectively

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

nonreflecting boundary conditions in elastodynamics for finite element methods based upon off surface boundary integral equations in this work an off surface boundary integral osbi method is presented as a mesh termination scheme for solving large or infinite domain problems of elastodynamics the boundary integral equation is discretized using finite element shape functions and the neumann boundary condition term is solved for in terms of the dirichlet boundary condition term this expression is then substituted into associated finite element formulation for the interior problem comparisons are made using the new osbi technique the dtn method of givoli and keller and several other local nonreflecting boundary conditions the proposed boundary condition is shown to be accurate is well suited for use with finite element methods and is competitive with the dtn method

some properties of solutions to the weakly dissipative degasperis procesi equation abstract in this paper we consider the weakly dissipative degasperis procesi equation the present paper is concerned with some aspects of existence of global solutions persistence properties and propagation speed first we try to discuss the local well posedness and blow up scenario then establish the sufficient conditions on global existence of the solution finally persistence properties on strong solutions and the propagation speed for the weakly dissipative degasperis procesi equation are also investigated

optimal p cyclic sor for complex spectra abstract we consider the successive overrelaxation sor method for the solution of a linear system ax b when the matrix a has a block p p partitioned p cyclic form and its associated block jacobi matrix jp is weakly cyclic of index p following the pioneering work by young and varga in the 1950s many researchers have considered various cases for the spectrum jp and have determined optimal values for the relaxation factor 0 2 so that the sor method converges as fast as possible after the most recent work on the best block p cyclic repartitioning and that on the solution of large scale systems arising in queueing network problems in markov analysis the optimization of the convergence of the p cyclic sor for more complex spectra jp has become more demanding here we state the one point problem for the general p cyclic complex sor case the existence and the uniqueness of its solution are established by analyzing and developing further the theory of the associated hypocycloidal curves for the determination of the optimal parameter s an algorithm is presented and a number of illustrative numerical examples are given

a space time finite element method for structural acoustics in infinite domains part 1 formulation stability and convergence abstract a space time finite element method for solution of the exterior structural acoustics problem involving the interaction of vibrating elastic structures submerged in an infinite acoustic fluid is formulated in particular time discontinuous galerkin and galerkin least squares gls variational formulations for coupled structural acoustics in unbounded domains are developed and analyzed for stability and convergence the formulation employs a finite computational fluid domain surrounding the structure and incorporates time dependent non reflecting boundary conditions on the fluid truncation boundary energy estimates are obtained which allow us to prove the unconditional stability of the method for the coupled fluid structure problem with absorbing boundaries the methods developed are especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space time meshes are used to track waves propagating along space time characteristics an important feature of the space time formulation is the incorporation of temporal jump operators which allow for finite element interpolations that are discontinaous in time for additional stability least squares operators based on local residuals of the structural acoustics equations including the non reflecting boundary conditions are incorporated the energy decay estimates and high order accuracy predicted by our a priori error estimates are demonstrated numerically in a simple canonical example

the analysis ofk step iterative methods for linear systems from summability theory using the theory of euler methods from summability theory we investigate general iterative methods for solving linear systems of equations in particular for a given euler method a regions of the complex plane is determined such that ak step iterative method converges if the eigenvalues of an iteration operatort are contained ins for a givens optimal methods are described and upper and lower bounds are derived for the associated asymptotic rate of convergence special attention is given to two step methods with complex parameters

three dimensional wave envelope elements of variable order for acoustic radiation and scattering part i formulation in the frequency domain a variable order infinite wave envelope element scheme is formulated for transient unbounded acoustical problems the transient formulation which is local in space and time is obtained by applying an inverse fourier transformation to a time harmonic wave envelope model whose formulation is described in a companion article this procedure yields a coupled system of second order differential equations which can be integrated in time to yield transient pressure histories at discrete nodal points far field transient pressures can also be obtained at adjusted times the method can be applied quite generally to two dimensional and three dimensional problems and is compatible with a conventional finite element model in the near field the utility of the method is confirmed by the presentation of transient solutions for axisymmetric and fully three dimensional test problems an implicit time integration scheme is used and computed results are compared to analytic solutions and to solutions obtained from alternative numerical schemes close correspondence is demonstrated and the scheme is shown to be stable for the problems which are presented cpu times for a large three dimensional problem are shown to compare favorably with those required for an equivalent transient boundary element computation

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

a parallelizable sor like method systems with plus shaped and linear spectra abstract the classical sor method replaces an invertible cyclic linear system in b x f with a family of systems i n b x f where for some optimal scalar b the spectrum of b b is circular the eigenvalues of b are linearly placed falling on the real line moreover the spectral radius of b b is less than that of b similarly this paper explores a tensor mechanism which converts more general systems in b x f to equivalent systems i 2n b x f where the spectrum of b may be plus shaped the spectrum of b is circular and x col 0 x as in the sor theory the spectral radius of b is less than that of b among other things we recapture the results of the sor theory improve on some known results and extend our theory directly to matrices in b which need not be symmetric yet have real plus and minus valued eigenvalues

local high order radiation boundary conditions for the two dimensional time dependent structural acoustics problem the time dependent structural acoustics problem involving solution of the coupled wave equation over an infinite fluid domain is posed as a coupled problem over a finite fluid domain with local time dependent radiation boundary conditions applied to the fluid truncation boundary the proposed radiation boundary conditions are based on an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a nondimensional wave number a sequence of differential operators that match the leading terms of the asymptotic expansion provide boundary conditions that are of progressively higher order and increasing accuracy time dependent boundary conditions are obtained through an inverse fourier transform the relationship of these approximate local operators to the exact nonlocal dirichlet to neumann map is examined to illustrate their effectiveness the boundary conditions are employed in a finite element formulation for the time dependent structural acoustics problem in contrast to nonlocal boundary conditions based on the dirichlet to neumann map or retarded potential integral formulations the proposed local boundary conditions preserve the data structure of the standard finite element method and do not require storage of a large pool of historical data during the solution process numerical results illustrate the accuracy of the proposed boundary conditions as effected by the operator order acoustic wave number radiation directionality and distance from the acoustic source

convergence of block iterative methods applied to sparse least squares problems abstract recently special attention has been given in the mathematical literature to the problems of accurately computing the least squares solutions of very large scale overdetermined systems of linear equations such as those arising in geodetical network problems in particular it has been suggested that one solve such problems iteratively by applying the block sor successive overrelaxation iterative method to a consistently ordered block jacobi matrix that is weakly cyclic of index 3 here we obtain new results theorem 1 giving the exact convergence and divergence domains for such iterative applications it is then shown how these results extend and correct the literature on such applications in addition analogous results theorem 2 are given for the case when the eigenvalues of the associated block jacobi matrix are nonnegative

a simple impedance infinite element for the finite element solution of the three dimensional wave equation in unbounded domains this paper is concerned with the development of an efficient and accurate impedance infinite element that can be used either in the frequency or directly in the time domain for the modeling and solution of problems described by the scalar three dimensional wave equation in infinite or semi infinite domains the infinite domain is truncated and the effect of the truncated infinite region is simulated by the introduction of an absorbing boundary condition prescribed on the truncation boundary a systematic procedure for the construction of a family of such conditions of increasing accuracy and complexity is developed with explicit formulas given for approximations up to second order a central feature of this high order approximation is that it can be expressed within the context of a finite element formulation as a set of local infinite elements located at the boundary of the computational domain with each element defined by a pair of symmetric time invariant stiffness and damping matrices this makes it possible to incorporate readily the new local boundary element into finite element software developed for purely interior regions for applications involving steady state harmonic or transient excitations whereas the theory has been developed formally for arbitrary smooth boundary surfaces here details are provided for ellipsoidal and spherical boundaries thus far only the latter has been implemented and tested in problems involving cavities and rigid scatterers of spherical and cubic shape numerical experiments in both the frequency and time domain attest to the efficacy and accuracy of the proposed new element

mapped spheroidal wave envelope elements for unbounded wave problems this paper describes a family of axisymmetric spheroidal wave envelope elements for modelling exterior wave problems they are of variable radial order and can be used to represent steady and transient wave fields the formulation is presented for the axisymmetric case using elements which are based on oblate and prolate spheroidal geometries these offer the prospect of reduced dimensionality in comparison to conventional spherically formulated elements when used to represent wave fields in the vicinity of slender or flat objects conjugated weighting functions are used to give frequency independent acoustic mass stiffness and damping matrices this facilitates a simple extension of the method to transient problems the effectiveness and accuracy of the method is demonstrated by a comparison of computed and analytic solutions for sound fields generated by a rigid sphere in steady harmonic oscillation by a rigid sphere excited from rest and by a circular plate vibrating in a plane baffle

finite element computation of absorbing boundary conditions for time harmonic wave problems this paper proposes a new method in the frequency domain to define absorbing boundary conditions for general two dimensional problems the main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general it is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software boundary conditions at various orders of accuracy can be obtained in a simple way this is then applied to study some examples for which analytical or numerical results are available good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method

optimality relationships forp cyclic sor the optimality question for blockp cyclic sor iterations discussed in young and varga is answered under natural conditions on the spectrum of the block jacobi matrix in particular it is shown that repartitioning a blockp cyclic matrix into a blockq cyclic form q p results in asymptotically faster sor convergence for the same amount of work per iteration as a consequence block 2 cyclic sor is optimal under these conditions