an iterative method for nonsymmetric systems with multiple right hand sides we propose a method for the solution of linear systems ax b where a is a large possibly sparse nonsymmetric matrix of order n and b is an arbitrary rectangular matrix of order n times s with s of moderate size the method uses a single krylov subspace per step as a generator of approximations a projection process and a richardson acceleration technique it thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple right hand sides numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple right hand sides

rapid solution of parameter dependent linear systems for electromagnetic problems in the frequency domain the minimal residual interpolation method reduces the number of iterations in an iterative method for multiple right hand sides it uses computed solutions to initialize an iterative solver with an accurate guess this work demonstrates the efficiency of the method for frequency sweeps and solving scattering problems by plane waves incident from multiple angles a bound on the number of solutions required for plane wave scattering before the remaining solutions obtained by minimal residual interpolation only is given we discuss the performance of the method compared to iterative seed techniques in a numerical example a reduction factor of 60 is obtained on the number of matrix vector multiplications

fourth order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems in this paper we study numerical methods for a class of two dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains a compact finite difference method with an anisotropic mesh is proposed for the problems the existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions and two sufficient conditions for the uniqueness of the solution are also given the optimal error estimate in the discrete l norm is obtained under certain conditions the error estimate shows the fourth order accuracy of the proposed method when two spatial mesh sizes are proportional by using an upper solution or a lower solution as the initial iteration an almost optimal picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently numerical results are presented to confirm our theoretical analysis

a block qmr method for computing multiple simultaneous solutions to complex symmetric systems the solution of complex symmetric indefinite systems of equations where multiple solutions are required is considered the quasi minimum residual qmr method ideally suited for these matrices is generalized using the block lanczos algorithm to solve multiple solutions simultaneously this modification alone is shown through numerical examples involving large sparse matrices from finite element discretization of maxwell s equations to accelerate the convergence by a factor almost as great as the number of simultaneous solutions a natural convergence criterion for this method is presented that is shown to be as effective as and easier to compute than the usual equation residual finally a numerical comparison of the classical incomplete cholesky and a variant of the ilu t preconditioners is given showing superior performance by the latter

a conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields an iterative method based on the conjugate gradient cg algorithm is developed for the efficient treatment of equations involving multiple excitations examples show that significant time savings can be obtained as compared to treating each excitation individually with the conjugate gradient algorithm however these savings are not obtained without the drawback of increased memory requirements to store the additional excitations residuals and solutions the efficiency of this algorithm tends to increase as additional excitations are added

monotone schwarz iterates for a semilinear parabolic convection diffusion problem this paper deals with a discrete monotone iterative algorithm for solving a nonlinear singularly perturbed convection diffusion problem of parabolic type on each time level the monotone method known as the method of lower and upper solutions is applied to computing a nonlinear upwind difference scheme obtained after discretisation of the continuous problem a monotone domain decomposition algorithm based on a modification of the schwarz alternating method is constructed the rate of convergence of the monotone schwarz method is estimated uniform convergence properties of the monotone domain decomposition algorithm are studied numerical experiments are presented

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems this paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems monotone sequences based on the accelerated monotone iterative method are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem this monotone convergence leads to the existence uniqueness theorem an analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given numerical experiments are presented

monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem this paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic reaction diffusion problem firstly the monotone method known as the method of lower and upper solutions is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem secondly a monotone domain decomposition algorithm based on a modification of the schwarz alternating method is constructed this monotone algorithm solves only linear discrete systems at each iterative step of the iterative process the rate of convergence of the monotone domain decomposition algorithm is estimated numerical experiments are presented

a uniform monotone alternating direction scheme for nonlinear singularly perturbed parabolic problems this paper deals with a monotone alternating direction adi scheme for solving nonlinear singularly perturbed parabolic problems monotone sequences based on the method of upper and lower solutions are constructed for a nonlinear difference scheme which approximates the nonlinear parabolic problem the monotone sequences possess quadratic convergence rate an analysis of uniform convergence of the monotone adi scheme to the solutions of the nonlinear difference scheme and to the continuous problem is given numerical experiments are presented

analysis of projection methods for solving linear systems with multiple right hand sides we analyze a class of krylov projection methods but mainly concentrate on a specific conjugate gradient cg implementation by smith peterson and mittra ieee transactions on antennas and propogation 37 1989 pp 1490 1493 to solve the linear system ax b where a is symmetric positive definite and b is a multiple of right hand sides this method generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems orthogonally onto the generated krylov subspace to get the approximate solutions the whole process is repeated with another unsolved system as a seed until all the systems are solved we observe in practice a superconvergence behavior of the cg process of the seed system when compared with the usual cg process we also observe that only a small number of restarts is required to solve all the systems if the right hand sides are close to each other these two features together make the method particularly effective in this paper we give theoretical proof to justify these observations furthermore we combine the advantages of this method and the block cg method and propose a block extension of this single seed method

a hybrid approach for rapid computation of two dimensional monostatic radar cross section problems with the multilevel fast multipole algorithm this communication proposes a hybrid acceleration technique for the efficient computation of large two dimensional monostatic radar cross section rcs problems with the multilevel fast multipole algorithm mlfma for three dimensional scattering objects the described approach combines a spectral low rank preconditioner with an efficient angular response computation based on a low rank compression of the right hand side matrix in this communication the low rank decomposition is done using the adaptive cross approximation aca since conventional approaches are hardly applicable to large excitation matrices the proposed technique turns out to be highly efficient and accurate for the solution of large 2d monostatic rcs problems involving 3d geometries

affine curves on which every point is a set theoretic complete intersection we consider here those reduced and irreducible affine algebraic curves on which every point is the set theoretic complete intersection of the given curve and a hypersurface these curves for historical reasons explained in section 1 are called prefactorial our joint research in this area stems from our common interest in a certain result due to mm thy and pedrini 1121 over algebraically closed ground fields of characteristic 0 prefactorial curves are necessarily rational and nollsingular presented here are the results of our efforts to understand prefactoriaiity in the absence of algebraically closed and or characteristic 0 our methods are mainly algebraic depending largely on an analysis of the grayer vietoris sequence of the conductors as in 112 and on certain well known arithmetic geometric facts which we learned from a paper of rosen 16j and which seem not to be so readily avaitable elsewhere in the literature it is also of interest to examine pr fa toriality from the viewpoint of the classical theory of aigebraic curves section 4 contains some of the product of our tllinking along those lines

a new family of global methods for linear systems with multiple right hand sides the global bi conjugate gradient gl bcg method is an attractive matrix krylov subspace method for solving nonsymmetric linear systems with multiple right hand sides but it often show irregular convergence behavior in many applications in this paper we present a new family of global a biorthogonal methods by using short two term recurrences and formal orthogonal polynomials which contain the global bi conjugate residual gl bcr algorithm and its improved version finally numerical experiments illustrate that the proposed methods are highly competitive and often superior to originals

numerical methods for semilinear parabolic equations the method of upper lower solutions for continuous parabolic equations is extended to some finite difference system for numerical solutions the idea of this method is that by using the upper or lower solution as the initial iteration one can obtain a monotone sequence that converges to a unique solution of the problem the aim of this paper is to present two iterative schemes for the construction of the monotone sequence and to show that both schemes are numerically stable these two schemes are modified jacobi method and gauss seidel method for nonlinear algebraic equations an advantage of this approach is that each of the two methods yields an error estimate between the true solution and the computed mth iteration on the other hand the standard picard type of iterative scheme is used to show that the finite difference system converges to the continuous parabolic equations

a note on principal ideals and smooth curves abstract let a be the coordinate ring of an affine piece of a smooth curve v defined over a field k this paper investigates for each positive integer n the principal ideals of a of degree n the main tool is the picard bundle of v of degree n for k of characteristic zero v a general curve of genus g v and n v the lang swinnerton dyer version of modell s conjecture implies there are but finitely many linear equivalence classes of principal ideals of a of degree n conjectured conditions for v to be general are explicit and effectively computable

accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems abstract an accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented the mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite difference or finite element method it is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions this result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions an application is given to a tabular reactor model from chemical engineering for numerical solutions and the number of iterations are compared with that by the regular monotone iterative scheme

on the lanczos method for solving symmetric linear systems with several right hand sides this paper analyzes a few methods based on the lanczos algorithm for solving large sparse symmetric linear systems with several right hand sides the methods examined are suitable for the situation when the right sides are not too different from one another as is often the case in time dependent or parameter dependent problems we propose a theoretical error bound for the approximation obtained from a projection process onto a krylov subspace generated from processing a previous right hand side

qmr based projection techniques for the solution of non hermitian systems with multiple right hand sides in this work we consider the simultaneous solution of large linear systems of the form ax j b j j 1 k where a is sparse and non hermitian we describe single seed and block seed projection approaches to these multiple right hand side problems that are based on the qmr and block qmr algorithms respectively we use block qmr to solve the block seed system and generate the relevant biorthogonal subspaces approximate solutions to the nonseed systems are simultaneously generated by minimizing their appropriately projected block residuals after the initial block seed has converged the process is repeated by choosing a new block seed from among the remaining nonconverged systems and using the previously generated approximate solutions as initial guesses for the new seed and nonseed systems we give theory for the single seed case that helps explain the convergence behavior under certain conditions implementation details for both the single seed and block seed algorithms are discussed and advantages of the block seed algorithm in cache based serial and parallel environments are noted the computational savings of our methods over using qmr to solve each system individually are illustrated in two examples

a block qmr algorithm for non hermitian linear systems with multiple right hand sides abstract many applications require the solution of multiple linear systems that have the same coefficient matrix but differ in their right hand sides instead of applying an iterative method to each of these systems individually it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously in this paper we propose a block version of freund and nachtigal s quasi minimal residual qmr method for the iterative solution of non hermitian linear systems the block qmr method uses a novel lanczos type process for multiple starting vectors which was recently developed by aliaga boley freund and hernandez to compute suitable basis vectors for the underlying block krylov subspaces we describe the basic block qmr method and also give important implementation details in particular we show how to incorporate deflation to drop converged linear systems and to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences numerical results are reported that illustrate typical features of the block qmr method

convergence properties of block gmres and matrix polynomials abstract this paper studies convergence properties of the block gmres algorithm when applied to nonsymmetric systems with multiple right hand sides a convergence theory is developed based on a representation of the method using matrix valued polynomials relations between the roots of the residual polynomial for block gmres and the matrix e pseudospectrum are derived and illustrated with numerical experiments the role of invariant subspaces in the effectiveness of block methods is also discussed

uniform convergence of a weighted average scheme for a nonlinear reaction diffusion problem this paper deals with a weighted average scheme or q scheme for solving a nonlinear singularly perturbed parabolic reaction diffusion problem the uniform convergence of the weighted average scheme on piecewise uniform and log meshes is established numerical experiments are presented

generalized global conjugate gradient squared algorithm in this paper a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right hand sides is presented the method can be derived by using products of two nearby global bicg polynomials and formal orthogonal polynomials of which global cgs and global bicgstab are just particular cases we also show to apply the method for solving the sylvester matrix equation finally numerical examples are given to illustrate the effectiveness of the proposed method