new homotopic factorization and symmetrization techniques for newton s and newton structured iteration we modify our earlier homotopic continuation process for iterative matrix inversion to improve the choice of the initial approximate inverse at every homotopic step this enables us to control the condition of the auxiliary matrices and to accelerate the convergence of the iteration the algorithm also extends our earlier approach where we factorized the input matrix into the product of better conditioned factors we accelerate our new algorithm in the case of a toeplitz like input comment on the extensions to the computations with other structured matrices such as cauchy pick and banded matrices and propose new techniques for the compression of the displacement of hermitian matrices

asymptotic behavior of trajectories for some nonautonomous almost periodic processes abstract in this paper asymptotics are studied for some almost periodic processes on a complete metric space x d 1 it is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t this property does not hold for general almost periodic contractive processes 2 a compactness result is obtained for weakly almost periodic complete trajectories of some possibly nonlinear processes in a uniformly convex banach space 3 the existence of almost periodic trajectories is studied for affine processes in a uniformly convex banach space these results are applicable to some evolution equations of the form du dt a t u t f t where t is almost periodic r v uniformly convex banach space and a t is a periodic time dependent m accretive operator in v

randomized preprocessing versus pivoting abstract it is known that without pivoting gaussian elimination can run significantly faster particularly for matrices that have structures of toeplitz or hankel types but becomes numerically unsafe the known remedies take their toll e g symmetrization squares the condition number of the input matrix can we fix the problem without such a punishment taking this challenge we combine randomized preconditioning techniques with iterative refinement and prove that this combination is expected to make pivoting free gaussian elimination numerically safe while keeping it fast for matrices having structures of toeplitz or hankel types transition to gaussian elimination with no pivoting decreases arithmetic time bound from cubic to nearly linear and our tests show dramatic decrease of the cpu time as well

iterative inversion of structured matrices iterative processes for the inversion of structured matrices can be further improved by using a technique for compression and refinement via the least squares computation we review such processes and elaborate upon incorporation of this technique into the known frameworks

iterative methods for linear complementarity problems with upperbounds on primary variables this paper is concerned with iterative methods for the linear complementarity problem lcp of findingx and y in rn such thatc dx y 0 b x 0 andxt c dx y yt b x 0 whenb 0 this is the lcp m q withm l 0d t which is in turn equivalent to a linear variational inequality over a rectangle this type of problem arises for example from quadratic programming qp problems with upper and lower bounds ifd is symmetric and from multicommodity market equilibrium problems with institutional price controls imposed ifd is not necessarily symmetric iterative methods for the lcp m q with symmetricm are well developed and studied using qp applications for nonsymmetric cases withm being either an h matrix with positive diagonals or a z matrix there exists a robust iterative method with guaranteed convergence this paper extends this algorithm so that the lcp m q withm l 0d t which is neither symmetric nor an h matrix with positive diagonals nor a z matrix can be processed when onlyd notm satisfies such properties the case whered is nonsymmetric is explicitly discussed

weighted complementarity problems a new paradigm for computing equilibria this paper introduces the notion of a weighted complementarity problem wcp which consists of finding a pair of vectors x s belonging to the intersection of a manifold with a cone such that their product in a certain algebra x circ s equals a given weight vector w when w is the zero vector then wcp reduces to a complementarity problem the motivation for introducing the more general notion of a wcp lies in the fact that several equilibrium problems in economics can be formulated in a natural way as wcps moreover those formulations lend themselves to the development of highly efficient algorithms for solving the corresponding equilibrium problems for example fisher s competitive market equilibrium model can be formulated as a wcp that can be efficiently solved by interior point methods moreover it is shown that the quadratic programming and weighted centering problem which generalizes the notion of a linear programming and weighted centering problem recently proposed by anstreicher

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

minimal sets of almost periodic motions les conditions de nemytskii stepanov et deysach sell pour un mouvement x sont equivalentes a la presque periodicite asymptotique de x r la presque periodicite asymptotique de x r n est pas necessaire afin que x soit un ensemble minimal de mouvements presque periodiques

asymptotic almost periodicity and motions of semigroups of operators abstract in this survey we present an overview of recent work in resolving certain questions that arise in a qualitative study of solutions to the abstract cauchy problem focusing on motions of c 0 semigroups of operators and their asymptotic behavior we address 1 the characterization of asymptotically almost periodic motions and 2 the characterization of those asymptotically almost periodic functions for which this property carries over to their integral moreover we take up the problem raised by nemytskii and stepanov as to when the limit set of a motion of a dynamical system is a minimal set of almost periodic motions

sign solvable linear complementarity problems abstract this paper presents a connection between qualitative matrix theory and linear complementarity problems lcps an lcp is said to be sign solvable if the set of the sign patterns of the solutions is uniquely determined by the sign patterns of the given coefficients we provide a characterization for sign solvable lcps such that the coefficient matrix has nonzero diagonals which can be tested in polynomial time this characterization leads to an efficient combinatorial algorithm to find the sign pattern of a solution for these lcps the algorithm runs in o time where is the number of the nonzero coefficients

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

finite termination of a newton type algorithm for a class of affine variational inequality problems abstract many optimization problems can be reformulated as a system of equations one may use the generalized newton method or the smoothing newton method to solve the reformulated equations so that a solution of the original problem can be found such methods have been powerful tools to solve many optimization problems in the literature in this paper we propose a newton type algorithm for solving a class of monotone affine variational inequality problems avips for short in the proposed algorithm the techniques based on both the generalized newton method and the smoothing newton method are used in particular we show that the algorithm can find an exact solution of the avip in a finite number of iterations under an assumption that the solution set of the avip is nonempty preliminary numerical results are reported

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

superlinear primal dual affine scaling algorithms for lcp we describe an interior point algorithm for monotone linear complementarity problems in which primal dual affine scaling is used to generate the search directions the algorithm is shown to have global and superlinear convergence with q order up to but not including two the technique is shown to be consistent with a potential reduction algorithm yielding the first potential reduction algorithm that is both globally and superlinearly convergent

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

on effectively computing the analytic center of the solution set by primal dual interior point methods the computation of the analytic center of the solution set can be important in linear programming applications where it is desirable to obtain a solution that is not near the relative boundary of the solution set in this work we discuss the effective computation of the analytic center solution by the use of primal dual interior point methods a primal dual interior point algorithm designed for effectively computing the analytic center solution is proposed and theory and numerical results are presented

the modulus based nonsmooth newton s method for solving linear complementarity problems as applying the nonsmooth newton s method to the equivalent reformulation of the linear complementarity problem a modulus based nonsmooth newton s method is established and its locally quadratical convergence conditions are presented in the implementation local one step convergence is discussed by properly choosing the initial vector and the generalized jacobian and a mixed algorithm is given for finding an initial vector numerical experiments show that the proposed methods are efficient and accelerate the convergence performance of the modulus based matrix splitting iteration methods

root finding by expansion with independent constraints elimination methods are highly effective for the solution of linear and nonlinear systems of equations but reversal of the elimination principle can be beneficial as well competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms that is their convergence from the start a well known example is the dual linear and nonlinear programming which enhances the power of optimization algorithms we believe that this is just an ad hoc application of general principle of expansion with independent constraints it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization at the end of this paper we comment on other applications and extensions of this principle presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros such as weierstrass durand kerner s root finder which reduces its root finding task to viete s vieta s system of n polynomial equations with n unknowns we adjust this root finder to the approximation of a single zero of p preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear together with computing a zero of a polynomial p the algorithm deflates this polynomial as by product and then could be reapplied to the quotient to approximate the next zero of p alternatively by using m processors that exchange no data one can concurrently approximate up to m zeros of p our tests confirm the efficiency of the proposed algorithms technically our root finding boils down to computations with structured matrices polynomials and partial fraction decompositions our study of these links can be of independent interest e g as by product we express the inverse of a sylvester matrix via its last column thus extending the celebrated result of gohberg and sementsul 1972 22 from toeplitz to sylvester matrix inverses

modified sms method for computing outer inverses of toeplitz matrices we introduce a new algorithm based on the successive matrix squaring sms method this algorithm uses the strategy of e displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square toeplitz matrix using the idea of displacement theory which decreases the memory space requirements as well as the computational cost our method tends to be very effective for toeplitz matrices

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

weak asymptotic almost periodicity for semigroups of operators abstract we address the question as to when a motion or almost orbit of a strongly continuous semigroup of operators in a banach space is weakly asymptotically almost periodic our results on existence and representation are applied to the study of weak stability properties of operator semigroups and the problem of existence of almost periodic motions for a uniformly bounded co semigroups of linear operators and b nonlinear contraction semigroups

asymptotically almost periodic solutions of stochastic functional differential equations abstract in this paper we investigate a class of stochastic functional differential equations of the form dx t ax t f t x t x t dt g t x t x t dw t our main results concern the existence and exponential stability of quadratic mean asymptotically almost periodic mild solutions an example is given to illustrate our results

homotopic residual correction processes we present and analyze homotopic continuation residual correction algorithms for the computation of matrix inverses for complex indefinite hermitian input matrices our homotopic methods substantially accelerate the known nonhomotopic algorithms unlike the nonhomotopic case our algorithms require no pre estimation of the smallest singular value of an input matrix furthermore we guarantee rapid convergence to the inverses of well conditioned structured matrices even where no good initial approximation is available in particular we yield the inverse of a well conditioned n x n matrix with a structure of toeplitz hankel type in o n log 3 n flops for a large class of input matrices our methods can be extended to computing numerically the generalized inverses our numerical experiments confirm the validity of our analysis and the efficiency of the presented algorithms for well conditioned input matrices and furnished us with the proper values of the parameters that define our algorithms

a non modulus linear method for solving the linear complementarity problem abstract in this paper a non modulus linear method for solving the linear complementarity problem is established by using the sign patterns of the solution of the equivalent modulus equation in the proposed method the efficient numerical algorithms for solving the linear equations can be applied to the large sparse problems numerical examples show that the new method is valid

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

superfast algorithms for cauchy like matrix computations and extensions abstract an effective algorithm of m morf ph d thesis department of electrical engineering stanford university stanford ca 1974 in proceedings of the ieee international conference on assp ieee computer society press silver spring md 1980 pp 954 959 r r bitmead and b d o anderson linear algebra appl 34 1980 103 116 computes the solution x t 1 b to a strongly nonsingular toeplitz or toeplitz like linear system t x b a short displacement generator for the inverse t 1 of t and det t we extend this algorithm to the similar computations with n n cauchy and cauchy like matrices recursive triangular factorization of such a matrix can be computed by our algorithm at the cost of executing o nr 2 log 3 n arithmetic operations where r is the scaling rank of the input cauchy like matrix c r 1 if c is a cauchy matrix consequently the same cost bound applies to the computation of the determinant of c a short scaling generator of c 1 and the solution to a nonsingular linear system of n equations with such a matrix c our algorithm does not use the reduction to toeplitz like computations we also relax the assumptions of strong nonsingularity and even nonsingularity of the input not only for the computations in the field of complex or real numbers but even where the algorithm runs in an arbitrary field we achieve this by using randomization and we also show a certain improvement of the respective algorithm by kaltofen for toeplitz like computations in an arbitrary field our subject has close correlation to rational tangential matrix interpolation under passivity condition e g to nevanlinna pick tangential interpolation problems and has further impact on the decoding of algebraic codes

randomized preprocessing of homogeneous linear systems of equations abstract our randomized preprocessing enables pivoting free and orthogonalization free solution of homogeneous linear systems of equations in the case of toeplitz inputs we decrease the estimated solution time from quadratic to nearly linear and our tests show dramatic decrease of the cpu time as well we prove numerical stability of our approach and extend it to solving nonsingular linear systems inversion and generalized moore penrose inversion of general and structured matrices by means of newton s iteration approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank matrix eigen solving and root finding for polynomial and secular equations and for polynomial systems of equations some by products and extensions of our study can be of independent technical intersest e g our extensions of the sherman morrison woodbury formula for matrix inversion our estimates for the condition number of randomized matrix products and preprocessing via augmentation

on s asymptotically ω periodic functions on banach spaces and applications abstract this paper is devoted to the study of the class of continuous and bounded functions f 0 x for which exists 0 such that lim t f t f t 0 in the sequel called s asymptotically periodic functions we discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically periodic functions we also study the existence of s asymptotically periodic mild solutions of the first order abstract cauchy problem in banach spaces

superfast tikhonov regularization of toeplitz systems toeplitz structured linear systems arise often in practical engineering problems correspondingly a number of algorithms have been developed that exploit toeplitz structure to gain computational efficiency when solving these systems the earliest fast algorithms for toeplitz systems required o n 2 operations while more recent superfast algorithms reduce the cost to o n log n 2 or below in this work we present a superfast algorithm for tikhonov regularization of toeplitz systems using an extension and transformation technique our algorithm translates a tikhonov regularized toeplitz system into a type of specialized polynomial problem known as tangential interpolation under this formulation we can compute the solution in only o n log n 2 operations we use numerical simulations to demonstrate our algorithm s complexity and verify that it returns stable solutions

primal dual affine scaling interior point methods for linear complementarity problems a first order affine scaling method and two m th order affine scaling methods for solving monotone linear complementarity problems lcps are presented all three methods produce iterates in a wide neighborhood of the central path the first order method has o nl 2 lognl 2 log lognl 2 iteration complexity if the lcp admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with q order two if m omega log sqrt n l then both higher order methods have o sqrt n l iteration complexity the q order of convergence of one of the methods is m 1 for problems that admit a strict complementarity solution while the q order of convergence of the other method is m 1 2 for general monotone lcps

structured matrices and newton s iteration unified approach abstract recent progress in the study of structured matrices shows advantages of unifying the treatment of various classes of such matrices we recall some fundamental techniques for such a unification and then specify it in full details for newton s iteration which rapidly improves an initial approximation to the inverse matrix by performing two matrix multiplications per recursive step the iteration is particularly suitable for n n structured matrices represented with o n entries of their short generators rather than with their own n 2 entries based on such a representation matrix operations are performed much more rapidly and use much less memory space a major problem is to control the length of the generators which tends to grow quite rapidly in the iterative process two known methods solve this problem for toeplitz like and cauchy like matrices we extend both methods to a more general class of structured matrices and estimate the convergence rate as well as the computational complexity some novel techniques are introduced in this study in particular for the estimation of the norms of the inverse displacement operators

superlinearly convergent o sqrt n l iteration interior point algorithms for linear programming and the monotone linear complementarity problem zhang and tapia have recently developed an interior point algorithm for the linear programming problem that has the property that the duality gap converges to zero q quadratically in the nondegenerate case and if the iterates converge q superlinearly in the degenerate case their algorithm also solves the integer model for lp in o nl iterations where l is the input length of the lp this paper presents an algorithm which maintains the quadratic superlinear convergence property but solves lp in o sqrt n l iterations in the integer model also the algorithm is generalized into an algorithm which solves lcp the lcp algorithm which is similar to an adaptive path following algorithm proposed by mizuno yoshise and kikuchi has similar local convergence properties if we assume that the iterates converge to a strictly complementary solution

convergence of interior point algorithms for the monotone linear complementarity problem the literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values but very little is known about the convergence of the iterate sequences this paper studies the horizontal linear complementarity problem and derives general convergence properties for algorithms based on newton iterations this problem provides a simple and general framework for most existing primal dual interior point methods the conclusion is that most of the published algorithms of this kind generate convergent sequences in many cases whenever the convergence is not too fast in a certain sense the sequences converge to the analytic center of the optimal face

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere