extended convergence regions for the aor method abstract in this paper we give sufficient conditions for the convergence of the aor method when the matrix a for ax b is a strictly diagonally dominant matrix these results improve the conclusions obtained in the theorem 4 10 with the notion of generalized diagonal dominant matrix we enlarge the convergence regions given in theorem 9 10 when a is a nonsingular h matrix in the last section we generalize theorem 6 of robert 11 and we present some results which extend the convergence regions for the aor method

iterative solution of fuzzy linear systems abstract linear systems have important applications to many branches of science and engineering in many applications at least some of the parameters of the system are represented by fuzzy rather than crisp numbers so it is immensely important to develop numerical procedures that would appropriately treat general fuzzy linear systems will be denoted by fls and solve them in this paper firstly a general fuzzy linear system using the embedding approach has been investigated and then several well known numerical algorithms for solving system of linear equations such as richardson extrapolated richardson jacobi jor gaus seidel egs sor aor esor ssor ussor ema and msor are extended for solving fls the iterative methods are followed by convergence theorems and the presented algorithms are tested by solving some numerical examples hackbusch noticed that gauss spelling is less correct than gaus w hackbusch iterative solution of large sparse systems of equations springer verlag inc new york 1994 9

optimal 2 cyclic msor for bowtie spectra and the continuous manteuffel algorithm abstract for the numerical solution of a linear system whose matrix coefficient is block 2 cyclic consistently ordered with the eigenvalues of the associated block jacobi matrix lying in a bowtie region several efficient stationary iterative methods have been proposed among others by chin and manteuffel 1988 elman and golub 1990 de pillis 1991 and eiermann niethammer and varga 1992 we propose as an alternative the stationary modified successive overrelaxation msor method or an equivalent two step method applied to the cyclically reduced linear system it is shown both theoretically and experimentally that the application of a continuous version of manteuffel s algorithm to derive the optimal parameters of the two step method produces an iterative method that is asymptotically faster than all the aforementioned methods

optimum accelerated overrelaxation method in a special case absbact in this paper we give the optimum parameters for the accelerated overrelaxation aor method in the special case where the matrix coefficient of the linear system which is solved is consistently ordered with nonvanishing diagonal elements under certain assumptions concerning the eigenvalues of the corresponding jacobi matrix it is shown that the optimum aor method gives better convergence rates than the optimum sor does while in the remaining cases the optimum aor method coincides with the optimum sor one

on solutions of generalized complementarity and eigenvector problems abstract from a new fan browder type fixed point theorem due to the second author we deduce an existence theorem for a solution of an equilibrium problem in section 3 this theorem is applied to generalized complementarity problems in section 4 and to eigenvector problems in section 5

on the modified successive overrelaxation method new forms of the modified successive overrelaxation msor mksor mksor1 and mksor2 are introduced these forms depend on the updated successive overrelaxation method ksor and the combination of the sor and the ksor methods we present analysis of the convergence of the mksor methods with fixed parameters for solving linear system of equations ax b where a belongs to a subclass of consistently ordered matrices these matrices appear in the numerical treatment of elliptic partial differential equations corresponding to the red black ordering also functional relations for the eigenvalues of the iteration matrices of the jacobi the mksor methods and the relaxation parameters are established a general representative numerical example illustrating this treatment is considered different values of the relaxation parameters are used on each set of equations we have implemented our treatment using computer algebra software mathematica 8 0 new results on this subject are presented and compared with those of young this approach is promising and will help in the numerical treatment of boundary value problems other extensions and applications for further work are mentioned

the convergence rate of the chebyshev sim under a perturbation of a complex line segment spectrum abstract the chebyshev semiiterative method chsim is probably the best known and most often used method for the iterative solution of linear system x t x c where the spectrum of t is located in a complex line segment excluding 1 the asymptotic convergence factor acf of the chsim under a perturbation of is considered several formulae for the approximation to the acfs up to the second order of a perturbation are derived this generalizes the results about the sensitivity of the asymptotic rate of convergence to the estimated eigenvalues by hageman and young in the case that both and are real two numerical examples are given to illustrate the theoretical results

principal minor assignment isometries of hilbert spaces volumes of parallelepipeds and rescaling of sesqui holomorphic functions abstract in this article we consider the following equivalence relation on the class of all functions of two variables on a set x we will say that l m x x c are rescalings if there are non vanishing functions f g on x such that m x y f x g y l x y for any x y x we give criteria for being rescalings when x is a topological space and l and m are separately continuous or when x is a domain in c n and l and m are sesqui holomorphic a special case of interest is when l and m are symmetric and f g only has values 1 this relation between m and l in the case when x is finite and so l and m are square matrices is known to be characterized by the equality of the principal minors of these matrices we extend this result to the case when x is infinite as an application we characterize restrictions of isometries of hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors

over and underrelaxation for linear systems with weakly cyclic jacobi matrices of index p abstract d young s results from 1954 concerning the application of the successive overrelaxation sor method to linear systems ax b with matrices possessing property a were generalized by r s varga in 1959 to systems with jacobi matrices ja which are weakly cyclic of index p if the eigenvalues of jpa are nonnegative then for the optimal parameter opt there holds 1 opt 1 1 p 1 here a different proof of varga s results is given and it is shown that 1 1 p 1 opt 1 holds if the eigenvalues of jpa are nonpositive further exact intervals for those parameters which yield convergence of the sor method are given in terms of chebyshev polynomials

the application of leja points to richardson iteration and polynomial preconditioning abstract let a x b be a linear system of algebraic equations with a large nonhermitian matrix a and let a denote the spectrum of a assume that there is an explicitly known compact set k in the complex plane such that a k and 0 k we introduce sequences of leja points zj j 0 for k and discuss convergence and stability properties of the richardson iteration method with relaxation parameters j 1 z j by replacing k with a finite set km and using reciprocal values of the leja points for km as relaxation parameters we obtain a practical scheme for determining relaxation parameters for richardson iteration with a suitable choice of km this scheme can be used to order any given sequence of relaxation parameters so as to avoid large amplification of roundoff errors we also show how leja points can be used to determine polynomial preconditioners

a survey of the iterative methods for the solution of linear systems by extrapolation relaxation and other techniques abstract for the iterative solution of linear systems of algebraic equations ax b 1 with a c n n x b c n and det a 0 numerous methods exist although a classification of them seems not to be possible one may note that the first step for the construction of an iterative method usually begins with a splitting of a in 1 thus a is written as a m n where det m 0 and m is easily inverted so that 1 is equivalent to x tx c 2 t m 1n c m 1 the discussion will be restricted to the so called linear stationary iterative methods although some nonstationary ones will be mentioned and emphasis will be given to those which fall into the categories of extrapolation relaxation and similar ones for this it will be assumed that the spectrum t of t in 2 is contained in a well defined compact region r whose complement with respect to the complex plane is simply connected and 1 r under these assumptions and for each specific class of methods described each time an attempt will be made to present the optimum one the optimum is the one out of the class of methods for which the sequence of vectors yielded converges asymptotically to the unique solution of 1 as fast as possible

a young eidson s type algorithm for complex p cyclic sor spectra abstract in a recent work of ours we have solved the problem of the minimization of the spectral radius of the iteration matrix of a p cyclic successive overrelaxation sor method for the solution of the linear system ax b when the matrix a is block p cyclic consistently ordered for what is known as the one point problem for any p 3 particular cases of the one point problem were solved by young varga kjellberg kredell russell and others in the present work we develop a theory using the results of our previous one and solve first the two point problem special cases of which were solved by wrigley eiermann niethammer ruttan noutsos and others secondly we generalize and extend our theory to cover the many point problem and develop a young eidson s type algorithm for its solution as possible application areas we mention among others the best block p cyclic repartitioning for the sor method are the solution of large scale systems arising in queueing network problems in markov analysis

on the class of chebyshev approximation problems which arise in connection with a conjugate gradient type method we investigate approximation problems of the form mathop min limits p p 0 1 mathop max p lambda limits lambda in s p withp ranging over all polynomials of degree k ands any line segment of the complex plane problems of this type appear in connection with error bounds for the minimum residual algorithm for solving linear systems with coefficient matrixa ei di n wheren nh andd besides a few general results on p we present the explicit solution of p for segmentss which are parallel to the imaginary and symmetric to the real axis this result leads to an optimal semi iterative method for the class of matricesa i n wheren nt

optimal block iterative schemes for certain large sparse and nonsymmetric linear systems abstract the problem of determining fast iterative solutions of certain large sparse and nonsymmetric linear systems arising in applications is addressed here several iterative schemes from the accelerated overrelaxation family are considered different geometrical algorithms are used for the explicit determination of the optimal factors direct comparisons of the spectral radii of the resulting optimal schemes reveal that the optimal extrapolated accelerated gauss seidel eags is always asymptotically faster than the optimal successive overrelaxation while the optimal eags and extrapolated gauss seidel strongly compete application of the collocation method on simple boundary value problems is used to demonstrate our results numerically

region dependent optimal m stage runge kutta schemes for solving a class of nonsymmetric linear systems abstract we consider the problem of finding optimal parameters of m stage runge kutta rk schemes for solving a class of nonsymmetric system of linear equations for each m and a given system we find a compact complex region which has a simply connected complement and contains all eigenvalues of the system matrix a specially structured nonlinear minimax problem over the region is formulated to find the optimal parameters we compare four different regions to test the efficiency of the related rk schemes some analytical results are given for a 1 stage scheme a numerical scheme based on a modified version of the projected lagrangian method is introduced to solve the nonlinear minimax problem for higher stage numbers m

a condition for the superiority of the 2 2 step methods over the related chebyshev method abstract the 2 2 step iterative methods related to an optimal chebyshev method for solving a real and nonsymmetric linear system a x b are studied a condition under which the asymptotic rate of convergence of the optimal chebyshev method can be improved by a related 2 2 step method is derived the condition depends not only on the location of the extreme eigenvalues of t but also on whether the ratio of the minor axis to the major axis of the optimal ellipse is greater than the golden ratio two numerical examples are given to illustrate our results

optimal parameters for 2 cyclic aor assume email protected c n x n is a 2 cyclic consistently ordered matrix and j is denoted as its associated block jacobi iteration matrix we consider the 2 cyclic aor method for solving the consistent linear systems ax b in the case that s j 2 is either nonnegative or nonpositive we give detailed discussion and derive definite expressions on optimal parameters and spectral radius by efficient method moreover we give some numerical examples

is a chebyshev method optimal for an elliptic region also optimal for a nearly elliptic region abstract the asymptotic rate of convergence of an optimal chebyshev semiiterative method for solving a real and nonsymmetric linear system x t x c can be improved by the related 2 2 step iterative methods under certain conditions the condition for which a chebyshev method asymptotically optimal for an elliptic region is also asymptotically optimal for a nearly elliptic region is presented thus a 2 2 step method is asymptotically superior to the chebyshev method asymtotically optimal for a nearly elliptic region under certain conditions a numerical example illustrates our results

on hybrid semi iterative methods given a large sparse system of linear algebraic equations in fixed point form bf x t bf x bf c one way to solve this system is to apply a semi iterative method sim to the basic iteration method bf x m t bf x m 1 bf c it is known that if the spectrum sigma t of t is contained in some compact subset omega of the complex plane where 1 notin omega then there are asymptotically optimal sims associated with the basic iteration and omega whose asymptotic rates of convergence are best possible for the class of all matrices t with sigma t subseteq omega however for a given compact set omega an asymptotically optimal sim for omega is usually not well suited for efficient numerical computation unless this sim can be generated from a k step recurrence formula in this paper hybrid semi iterative methods are investigated which consist of two independent steps first bf x t bf x bf c is transformed into a consistent linear syst

the accelerated saor method for large linear systems abstract the paper is concerned with the chebyshev acceleration or semi iteration si and the conjugate gradient cg acceleration procedures for the saor method moreover in order to obtain more effective iteration the adaptive procedures which determine automatically the parameters required for each iteration are applied numerical results for the proposed algorithms are given and their characteristics are cleared the effectiveness and superiority to other algorithms are also considered

optimal successive overrelaxation iterative methods for p cyclic matrices we consider linear systems whose associated block jacobi matricesj p are weakly cyclic of indexp in a recent paper pierce hadjidimos and plemmons 13 proved that the block two cyclic successive overrelaxation sor iterative method is numerically more effective than the blockq cyclic sor method 2 2 finally as a counterpart to the result of pierce hadjidimos and plemmons we show that under this more general assumption the two cyclic sor method is not always superior to theq cyclic sor method 2 q p

on the equivalence of extrapolation and richardson s iteration and its applications abstract sometime ago two articles g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra appl 62 1984 241 261 appeared the same year in this journal solving the problem of optimizing the convergence factor of the stationary richardson s iteration i e x m 1 x m a ax m b m 0 1 2 for the solution of the system ax b a c n n det a 0 b c n in opfer and schober g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 when the spectrum of a was contained in a certain compact set s 1 not including the origin the optimal parameter a was determined in hadjidimos a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra all 62 1984 241 261 for the solution of ax b the consistent linear scheme x m 1 tx m c m 0 1 2 for which the spectrum of t was contained in a certain compact convex set s 2 not including the point 1 was considered and the optimal extrapolation parameter of its extrapolation scheme i e x m 1 1 i t x m c m 0 1 2 was determined because the approach in obtaining the optimal parameters in the two articles was quite different it seems it was difficult for any one to notice their equivalence and therefore to use them even in trivial cases in this article we show the equivalence of the optimal results in opfer and schober g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 and hadjidimos a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra all 62 1984 241 261 explore them further obtain new results and give applications for cases which may occur in practice finally based on observations regarding the optimal two cyclic successive overrelaxation sor and accelerated overrelaxation aor methods in two real case problems we solve the problem of optimal two cyclic aor in the more general complex case

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

on common quasi eigenvector problems abstract in this paper we study four types of common quasi eigenvector problems and two types of eigenvector problems the existence theorems for the solutions of common quasi eigenvector problems and eigenvector problems are also established

a hybrid arnoldi faber iterative method for nonsymmetric systems of linear equations we present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations say of the formax b wherea n n witha nonsingular andb n are given this hybrid method begins with a limited number of steps of the arnoldi method to obtain some information on the location of the spectrum ofa and then switches to a richardson iterative method based on faber polynomials for a polygonal domain the faber polynomials can be constructed recursively from the parameters in the schwarz christoffel mapping function in four specific numerical examples of non normal matrices we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid gmres or restarted versions of the gmres algorithm it is however sensitive as other hybrid methods also are to the amount of information on the spectrum ofa acquired during the first arnoldi phase of this procedure

towards the determination of the optimal p cyclic ssor abstract suppose that a c n n is a block p cyclic consistently ordered matrix and let b and s denote the block jacobi and the block symmetric successive overrelaxation ssor iteration matrices associated with a respectively extending previous work by hadjidimos and neumann the present authors have determined the exact regions of convergence of the ssor method in the b plane for any p 3 under the further assumption that the eigenvalues of b p are real of the same sign in the present work the investigation goes on further several questions are raised and among others the problem of the determination of the optimal regions of convergence in the spirit of niethammer and varga as well as that of the optimal relaxation factor are examined

some recent results on the modified sor theory abstract for the solution of the linear system ax i t x c 1 where t is a weakly cyclic of index p 2 matrix of a special structure the block sor method with different relaxation factors associated with the row blocks of a is considered first a well known relationship connecting the eigenvalue spectra of the block jacobi matrix t of a and its associated modified sor msor matrix is proved for all p 3 via an approach due to varga niethammer and cai next it is shown that the matrix analogue of the eigenvalue relationship holds at least for p 3 this together with the facts that the matrix analogue holds true for p 2 and also for any p 3 provided all relaxation factors coincide suggests a more general validity of the aforementioned matrix identity then based on the matrix relationship an equivalence is established between the msor method and a stationary p parameter p step iterative one for the solution of 1 so one can study convergence properties and determine optimal parameters of both methods by studying the simpler of the two that is the p step one finally some applications of the theory developed when p 2 are presented and a brief discussion concerning comparisons of the optimum msor and aor methods in a specific case is given

a survey of aor and tor methods abstract this paper makes a survey on the aor and tor methods we present the recent results about convergence and determination of the optimum parameters mainly those obtained by chinese authors

an extension of the accelerated overrelaxation aor theory for linear systems with heritian matrix abstract in this paper the convergence theory of the accelerated overrelaxation aor method for the solution of the linear system ax b where a is hermitian ah a is studied under the general splitting a d e eh where d is hermitian and e 1 2 d a s with sh s the purpose of the paper is to cover the case where d is hermitian and negative definite the case where d is hermitian and positive definite was covered in 8

an adaptive method for solving nonsymmetric linear systems involving applications of scpack consider the following large system of linear equations rm x it t x bf c eqno 0 where it is assumed that t is n by n real matrix with 1 excluded from the spectrum of t and that c is a real vector of dimension n this dissertation uses the results from function theory approximation theory and conformal mapping theory and constructs an adaptive numerical method to solve equation 0 with the aid of scpack this adaptive method periodically estimates the eigenvalue set to obtain near asymptotically optimal rates of convergence of the associated k l step semiiterative methods

iterative and semi iterative methods for computing stationary probability vectors of markov operators iterative and semi iterative methods for computing stationary prob ability vectors of markov type operators are proposed and their convergence properties are analyzed the methods studied apply to certain classes of prob lems in infinite dimensional spaces as well as to classical n x n stochastic ma trices

stationary and almost stationary iterative k l step methods for linear and nonlinear systems of equations we first discuss the solution of a fixed point equationx x of a frechet differentiable self mapping by iterative methods of the general form x m left phi left sum limits i 0 m 1 x i gamma i m 1 right sum limits j 0 m 1 x j beta j m right frac 1 beta m m m 1 2 ldots defined by two infinite nonsingular upper triangular matricesb j m andc j m with column sums 1 we show that two such methods defined byb c and tilde b tilde c respectively are in a certain sense equivalent if and only if c 1 b tilde c 1 tilde b in particular hat b c 1 and the unit matrix replacingc define an equivalent so called semiiterative method we introduce k l step methods as those whereb andc have upper bandwidthk andl 1 respectively they require storing max k l previous iterates only for stationary methodsb andc have toeplitz structure except for their first row which is chosen such that the column sum condition holds an euler method which may require to store all iterates is equivalent to a stationary k l step method if and only if the underlying conformal mapg is a rational function of the formg w w 0 k k v 0 v l 1 w 1 by choosingg withg 1 1 such that for some onto the exterior of some continuums known to contain the eigen values of the frechet derivative of one obtains a feasible procedure for designing a locally converging stationary k l step method custom made for a set of problems in the case x tx d with a linear operatort where one wants to solve a linear system of equations we show that the residual polynomials of a stationary semiiterative method are generalized faber polynomials with respect to a particular weight function using another weight function leads to what we call almost stationary methods the classical chebyshev iteration is an example of such a method we define equivalent almost stationary k l step methods and give a corresponding convergence result

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

on sor waveform relaxation methods waveform relaxation is a numerical method for solving large scale systems of ordinary differential equations on parallel computers it differs from standard iterative methods in that it computes the solution on many time levels or along a continuous time interval simultaneously this paper deals with the acceleration of the standard waveform relaxation method by successive overrelaxation sor techniques in particular different sor acceleration schemes based on multiplication with a scalar parameter or convolution with a time dependent function are described and theoretically analyzed the theory is applied to a one dimensional and two dimensional model problem and checked against results obtained by numerical experiments

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

richardson s iteration for nonsymmetric matrices abstract to solve the linear n n system 1 ax a for any nonsingular matrix a richardson s iteration 2 x j 1 x j j ax j a j 1 2 n which is applied in a cyclic manner with cycle length n is investigated where the j are free parameters the objective is to minimize the error x n 1 x where x is the solution of 1 if the spectrum of a is known to lie in a compact set s one is led to the chebyshev type approximation problem 3 min p 1 v n max z s p z where v n is the linear span of z z 2 z n if p solves 3 then the reciprocals of the zeros of p are optimal iteration parameters j it is shown that for a real problem 1 the iteration 2 can be carried out with real arithmetic alone even when there are complex j the stationary case n 1 is solved completely i e for all compact sets s the problem 3 is solved explicitly as a consequence the converging stationary iteration processes can be characterized for arbitrary closed disks s the problem 3 can be solved for all n n and a simple proof is provided the lemniscates associated with s are introduced they appear as an important tool for studying the stability of the iteration 2

polynomials by conformal mapping for the richardson iteration method for complex linear systems some methods using conformal mappings for determining polynomials for the richardson iteration scheme for complex linear systems a bf x bf b are described the polynomials can be computed fairly rapidly and give rise to iterative schemes with a rate of convergence very close to optimal the methods described are applicable when the eigenvalues of the matrix a lie in a known bounded simply connected region omega in the complex plane omega is assumed to have a smooth boundary

optimum stationary and nonstationary iterative methods for the solution of singular linear systems a number of iterative methods for the solution of the singular linear systemax b det a 0 andb in the range ofa is analyzed and studied among them are the stationaryk step the accelerated overrelaxation aor and the nonstationary second order chebyshev semi iterative ones it is proved that under certain assumptions the corresponding optimum semiconvergent schemes which present a great resemblance with their analogs for the nonsingular case can be determined finally a number of numerical examples shows how one can use the theory to obtain the optimum parameters for each applicable semiconvergent method

some eigenvector theorems proved by a fan kkm theorem abstract in this paper we use the technique of kkm mapping and a fan kkm theorem to study the existence of eigenvectors for some mappings on normed linear spaces

optimal stretched parameters for the sor iterative method noutsos d optimal stretched parameters for the sor iterative method journal of computational and applied mathematics 48 1993 293 308 linear systems whose associated block jacobi matrices j are p cyclic consistently ordered are considered in a number of papers the case of r j p c cyj pp where y p are real nonnegative numbers with 3 1 is studied by using cusped hypocycloids and also by using shortened hypocycloids in this paper the sor method is studied by using stretched hypocycloids the optimal stretched parameters and the corresponding optimal spectral radii are found and their behaviour is studied finally a comparison with the corresponding shortened and cusped sor parameters and spectral radii is also made

on the application of orthogonal polynomials to the iterative solution of linear systems of equations with indefinite or non hermitian matrices let ax b be a large linear system of equations and let the eigenvalues of the matrix a lie in one or several known simply connected regions sj j 1 1 l in the complex plane we consider the iterative solution of such linear systems by methods that make use of polynomials pn z n 0 orthogonal on the boundary of lj 1sj we show that for boundary curves such that the pn z satisfy a three term recurrence relation iterative methods based on this recurrence relation yield an optimal asymptotic rate of convergence for boundary curves for which the pn z do not satisfy a three term recurrence relation we show that n cyclic richardson iteration methods with relaxation parameters chosen as the reciprocal roots of pn z give nearly asymptotic optimal convergence for n sufficiently large our analysis suggests that in many cases the iterative methods should be based on residual polynomials pn z pn 0 instead of on the corresponding kernel polynomials qz nk 0pk 0 pk z nk 0 pk 0 2 numerical examples are presented

an efficient algorithm for the symmetric principal minor assignment problem abstract we consider the problem of reconstructing a symmetric matrix from its principal minors which has several applications in information theory and statistical modeling we develop a theory of symmetric matrices with equal corresponding principal minors based on a simple equivalent property due to oeding 2011 10 we then use this theory to provide a method for choosing a canonical representative from the class of all symmetric matrices with specified principal minors finally we provide an efficient algorithm for computing this canonical representative given its principal minors as input

skew symmetric matrices and their principal minors abstract in this article we study the following problem given a positive integer k what is the relationship between two matrices with entries in a field k and having equal corresponding principal minors of order at most k our main theorem improves a result of loewy 13 for skew symmetric matrices all of whose off diagonal entries are nonzero

on an ssor matrix relationship and its consequences a matrix relationship connecting the jacobi and the symmetric successive overrelaxation ssor matrices associated with a k cyclic consistently ordered matrix a is presented next the equivalence of the ssor method and a certain monoparametric k step one for the solution of the linear algebraic system ax b is established the aforementioned equivalence can be exploited to derive regions of convergence optimum parameters involved etc of the two iterative methods above this is done by studying the simplest of the two methods that is the monoparametric k step one to show how the idea works the case k 2 is very briefly discussed

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

on the existence of solutions of variational inequalities in banach spaces abstract in this paper we study the existence of the solution of the variational inequality tx y x 0 by applying the generalized projection operator k b b where b is a banach space with dual space b and by using the well known fankkm theorem

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

principal minors part i a method for computing all the principal minors of a matrix an order o 2 n algorithm for computing all the principal minors of an arbitrary n n complex matrix is motivated and presented offering an improvement by a factor of n 3 over direct computation the algorithm uses recursive schur complementation and submatrix extraction storing the answer in a binary order an implementation of the algorithm in matlab is also given and practical considerations are discussed and treated accordingly

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

comparison theorems for the convergence factor of iterative methods for singular matrices abstract iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix t i e by the quantity t max t 1 where t is the spectrum of t theorems are presented comparing the convergence factor of two iterative methods the comparison is based on the relationship between the matrices of the splittings a cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison although this cone is based on the unknown projection onto the null space of a matrix the characterization provided in the paper allows in specific instances the cone to be readily computable

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

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

non adaptive and adaptive saor cg algorithms abstract the paper is concerned with an improvement over the symmetric accelerated overrelaxation saor method for an iterative solution of large linear systems at first the conjugate gradient cg acceleration procedure is introduced to the saor method and the non adaptive saor cg algorithm is developed next the adaptive procedure to determine automatically the cg parameters n n and the saor parameters is constructed based on the adaptive procedure the adaptive saor cg algorithm is proposed and its characteristics are shown with numerical experiments a comparison with the optimum sor algorithm and the adaptive ssor algorithm is also given it is finally proved that the proposed adaptive saor cg algorithm is feasible and very efficient for the iterative solution

a necessary condition for the convergence of the accelerated overrelaxation aor method abstract the purpose of this note is to give a general necessary condition for the convergence of the aor method correcting and completing at the same time previous results of this type

iterative k step methods for computing possibly repulsive fixed points in banach spaces abstract we discuss iterative methods of the form xn 0 xx 1 1xn 1 kxn k n k k 1 for computing a fixed point x of a frechet differentiable self mapping of a sub set in a banach space by suitably choosing the coefficients 0 k the local convergence is established under certain assumptions on the spectrum of x this spectrum need not be contained in the unit disk however if it is convergence can often be speeded up considerably compared to picard iteration the methods are generalizations of the methods of v n kublanovskaya and w niethammer for linear systems of equations a more general type of iteration with nonstationary coefficients is considered also for the proof of their local convergence a generalization of the convergence theorems of perron ostrowski and kitchen is presented the connection with other methods in particular mann s iterative processes is also discussed

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

the superiority of a new type 2 2 step iterative method over the related chebyshev method a new type 2 2 step iterative method related to an optimal chebyshev method is developed for solving real and nonsymmetric linear systems of the form ax b it is an extension of the 2 2 step iterative method introduced in numer linear algebra appl 7 2000 169 the superiority of the new type 2 2 step iterative method over the optimal chebyshev method is derived in the case where the known 2 2 step iterative method may not improve the asymptotic rate of convergence two numerical examples are given to illustrate the results

on the convergence of the saor method and the error bounds for its acceleration given a linear system with hermitian positive definite coefficient matrix a a splitting of varga s type 1 is considered and the corresponding generalised saor scheme is presented we obtain the convergence theorem for the saor method and give the error bound for the a norm of the error vector for the saor semi iterative method hence extending the saor theory

principal minors part ii the principal minor assignment problem the inverse problem of finding a matrix with prescribed principal minors is considered a condition that implies a constructive algorithm for solving this problem will always succeed is presented the algorithm is based on reconstructing matrices from their principal submatrices and schur complements in a recursive manner consequences regarding the overdeterminancy of this inverse problem are examined leading to a faster polynomial time version of the algorithmic construction care is given in the matlab implementation of the algorithms regarding numerical stability and accuracy

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

successive overrelaxation sor and related methods abstract covering the last half of the 20th century we present some of the basic and well known results for the sor theory and related methods as well as some that are not as well known most of the earlier results can be found in the excellent books by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 young iterative solution of large linear systems academic press new york 1971 and berman and plemmons nonnegative matrices in the mathematical sciences siam philadelphia pa 1994 while some of the most recent ones are given in the bibliography of this paper in this survey both the point and the block sor methods are considered for the solution of a linear system of the form ax b where a c n n and b c n 0 some general results concerning the sor and related methods are given and also some more specific ones in cases where a happens to possess some further property e g positive definiteness l m h matrix property p cyclic consistently ordered property etc

modified successive overrelaxation msor and equivalent 2 step iterative methods for collocation matrices abstract we consider a class of consistently ordered matrices which arise from the discretization of boundary value problems bvps when the finite element collocation method with hermite elements is used through a recently derived equivalence relationship for the asymptotic rates of convergence of the modified successive overrelaxation msor and a certain 2 step iterative method we determine the optimum values for the parameters of the msor method as it pertains to collocation matrices a geometrical algorithm which utilizes capturing ellipse arguments has been successfully used the fast convergence properties of the optimum msor method are revealed after its comparison to several well known iterative schemes numerical examples which include the solution of poisson s equation are used to verify our results

on a matrix identity connecting iteration operators associated with a p cyclic matrix the successive overrelaxation sor and the symmetric sor ssor iteration matrices are connected with the jacobi iteration matrix in case these operators are associated with a 9 p 9 generalized consistently ordered matrix through certain matrix identities the validity of these identities has been proved in the last couple of years very recently an analogous matrix identity was shown to hold for the modified sor msor and the jacobi iteration matrices in the particular cases p 9 2 l 3 1 and 3 2 it is the main objective of this paper to extend the validity of this identity to cover an entire class or pairs p 9 the identity in question is not only of theoretical interest but of practical importance too since it can be used to show the equivalence of the msor and a class of p step iterative methods for the solution of a linear system whose matrix coefficient possesses the 9 p q generalized consistently ordered property

theorems of stein rosenberg type iii the singular case abstract in the theory of iterative methods the classical stein rosenberg theorem can be viewed as giving the simultaneous convergence or divergence of the extrapolated jacobi jor matrix j and the successive overrelaxation sor matrix in the case when the jacobi matrix j 1 is nonnegative as has been established recently by buoni and varga necessary and sufficient conditions for the simultaneous convergence or divergence of j and have been established which do not depend on the assumption that j 1 is nonnegative our aim here is to extend these results to the singular case using the notion of semiconvergence in particular for a real singular matrix a with nonpositive off diagonal entries we find conditions theorem 3 4 which imply that j and simultaneously semiconverge for all in the real interval 0 1

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

k step iterative methods for solving nonlinear systems of equations let n n be frechet differentiable and let the equation 1 x phi left x right have at least one fixed point we considerk step stationary iterative methods 2 y m mu 0 phi left y m 1 right mu 1 y m 1 mu k y m k m geqq k with 0 1 k 1 using results for an affine mapping n n it is proven that 2 may converge locally even in cases where the usual iterationx m x m 1 belonging to 1 diverges these results are extended to nonstationary methods of type 2 and to cyclic mappings

semiconvergence of extrapolated iterative methods for singular linear systems abstract in this paper we discuss convergence of the extrapolated iterative methods for solving singular linear systems a general principle of extrapolation is presented the semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coefficient matrix a is a singular m matrix with property c and an irreducible singular m matrix respectively since the generalized block jor and aor methods are respectively the extrapolated methods of the generalized block jacobi and sor methods so the semiconvergence of the generalized block jor and aor methods for solving general singular systems are proved furthermore the semiconvergence of the extrapolated power method the block jor aor and sor methods for solving markov chains are discussed

the relationship between the jacobi and the successive overrelaxation sor matrices of a k cyclic matrix abstract let a be a k l l generalized consistently ordered matrix with t and l its associated jacobi and sor matrices whose eigenvalues and satisfy the well known relationship 1 k k k k 1 for a subclass of the above matrices a we prove that the matrix analogue of the previous relationship holds exploiting the matrix relationship we show that the sor method is equivalent to a certain monoparametric k step iterative one when used for the solution of the fixed point problem x tx c