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

two double exponential gaps for automata with a limited pushdown abstract we shall consider nondeterministic and deterministic automata equipped with a limited pushdown constant height npda s and dpda s as well as their two way versions constant height 2 npda s and 2 dpda s we show two double exponential gaps for these devices namely i for complementing constant height one way npda s and ii for converting constant height 2 npda s or 2 dpda s into one way devices

on the complexities of multipoint evaluation and interpolation we compare the complexities of multipoint polynomial evaluation and interpolation we show that over a field of characteristic zero both questions have equivalent complexities up to a constant number of polynomial multiplications

on the control of stochastic systems abstract a general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the iterative method of the authors or its more recent adaptations for stochastic operator equations it is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations the method therefore is applicable to the control problem application is made to the state space equation x ax bu c where a b c are stochastic matrices corresponding to stochastic operators i e involving randomly time varying elements e g a ij t a t t f a p s it is emphasized that the processes are arbitrary stochatic processes with known statistics no assumption is made of wiener or markov behavior or of smallness of fluctuations and no closure approximations are necessary the method differs in interesting aspects from volterra series expansions used by wiener and others and has advantages over the other methods because of recent progress in the solution of the nonlinear case it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages

transformations of matrix structures work again abstract matrices with the structures of toeplitz hankel vandermonde and cauchy types are omnipresent in modern computations in sciences engineering and signal and image processing these four matrix classes have distinct features but in our 1990 paper in mathematics of computation we showed that vandermonde and hankel multipliers can be applied to transform each class into the others and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others the power of this approach was widely recognized later when novel numerically stable algorithms solved nonsingular toeplitz linear systems of equations in quadratic versus classical cubic arithmetic time based on transforming toeplitz into cauchy matrix structures more recent papers combined the same transformation with a link of the cauchy matrices to the hierarchical semiseparable matrix structure which is a specialization of matrix representations employed by the fast multipole method this produced numerically stable algorithms that approximated the solution of a nonsingular toeplitz linear system of equations in nearly linear arithmetic time we first revisit the successful method of structure transformation covering it comprehensively then we analyze the latter approximation algorithms for toeplitz linear systems and extend them to approximate multiplication of vandermonde and cauchy matrices by a vector and to approximate solution of vandermonde and cauchy linear systems of equations provided that they are nonsingular and well conditioned we decrease the arithmetic cost of the known numerical approximation algorithms for these tasks from quadratic to nearly linear and similarly for the computations with the matrices having structures that generalize the structures of vandermonde and cauchy matrices and for polynomial and rational evaluation and interpolation we also accelerate a little further the known numerical approximation algorithms for a nonsingular toeplitz or toeplitz like linear system by employing distinct transformations of matrix structures and we briefly comment on some natural research challenges particularly some promising applications of our techniques to high precision computations

new studies of randomized augmentation and additive preprocessing abstract a standard gaussian random matrix hereafter referred to just as gaussian matrix has full rank with probability 1 and is well conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from the square shape and becomes more rectangular if we append sufficiently many gaussian rows or columns to any normalized and possibly rank deficient or ill conditioned matrix then the augmented matrix has full rank with probability 1 and is well conditioned with a probability close to 1 we specify and prove these properties of augmentation and extend them to additive preprocessing that is to adding a product of two rectangular gaussian matrices by applying our randomization techniques to a matrix that has numerical rank r we accelerate the known algorithms for the approximation of its trailing singular spaces associated with all its positive singular values except for the r largest values our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace gaussian empirically the outputs of the resulting algorithms are as accurate as the outputs under gaussian preprocessing our novel duality techniques provides formal support so far missing for these empirical observations and opens door to de randomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low rank approximation of a matrix by means of random sampling

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

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

superfast divide and conquer method and perturbation analysis for structured eigenvalue solutions we present a superfast divide and conquer method for finding all the eigenvalues as well as all the eigenvectors in a structured form of a class of symmetric matrices with off diagonal ranks or n

canonical derivatives partial derivatives and finite automaton constructions let e be a regular expression our aim is to establish a theoretical relation between two well known automata recognizing the language of e namely the position automaton pe constructed by glushkov or mcnaughton and yamada and the equation automaton ee constructed by mirkin or antimirov we define the notion of c derivative for canonical derivative of a regular expression e and show that if e is linear then two brzozowski s derivatives of e are aci similar if and only if the corresponding c derivatives are identical it allows us to represent the berry sethi s set of continuations of a position by a unique c derivative called the c continuation of the position hence the definition of ce the c continuation automaton of e whose states are pairs made of a position of e and of the associated c continuation if states are viewed as positions ce is isomorphic to pe on the other hand a partial derivative as defined by antimirov is a class of c derivatives for some equivalence relation thus ce reduces to ee finally ce makes it possible to go from pe to ee while this cannot be achieved directly from the state graphs these theoretical results lead to an o e 2 space and time algorithm to compute the equation automaton where e is the size of the expression this is the complexity of the most efficient constructions yielding the position automaton while the size of the equation automaton is not greater and generally much smaller than the size of the position automaton

succinctness of regular expressions with interleaving intersection and counting in this paper we study the succinctness of regular expressions res extended with interleaving intersection and counting operators we show that in a translation from res with interleaving to standard regular expressions a double exponential size increase cannot be avoided we also consider the complexity of translations to finite automata we give a tight exponential lower bound on the translation of res with intersection to nfas and for each of the three classes of res we show that in a translation to a dfa a double exponential size increase cannot be avoided together with known results this gives a complete picture of the complexity of translating res extended with interleaving intersection or counting into standard regular expressions nfas and dfas

solving structured linear systems with large displacement rank linear systems with structures such as toeplitz vandermonde or cauchy likeness can be solved in email protected a 2n operations where n is the matrix size a is its displacement rank and email protected denotes the omission of logarithmic factors we show that for such matrices this cost can be reduced to email protected a w 1n where w is a feasible exponent for matrix multiplication over the base field the best known estimate for w is w 2 38 resulting in costs of order email protected a 1 3 8n we present consequences for hermite pade approximation and bivariate interpolation

linear complexity parallel algorithms for linear systems of equations with recursive structure abstract a method of derivation of parallel algorithms for n 1 n 1 matrices with recursive structure is presented and applied to toeplitz hankel and other toeplitz like matrices the derived algorithms executed on o n parallel processors require o n arithmetic operations per processor

acyclic automata and small expressions using multi tilde bar operators a regular expression with n occurrences of symbol can be converted into an equivalent automaton with n 1 states the so called glushkov automaton of the expression conversely it is possible to decide whether a given n 1 state automaton is a glushkov one and if so to convert it back to an equivalent regular expression of width n our goal is to extend the class of automata for which such a linear retranslation is possible we define new regular operators called multi tilde bars allowing us to simultaneously apply a multi tilde operator and a multi bar one to a list of expressions the main results are that a multi tilde bar expression of width n can be converted into an n 1 state position like automaton and that any acyclic n state automaton can be turned into an extended expression of width o n

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

the size cost of boolean operations on constant height deterministic pushdown automata we study the size cost of boolean operations on constant height deterministic pushdown automata i e on the traditional pushdown automata with a built in constant limit on the height of the pushdown we design a simulation showing that a complement can be obtained with a polynomial tradeoff for intersection and union we show an exponential simulation and prove that the exponential blow up cannot be avoided

parallel complexity of computations with general and toeplitz like matrices filled with integers and extensions computations with toeplitz and toeplitz like matrices are fundamental for many areas of algebraic and numerical computing the list of computational problems reducible to toeplitz and toeplitz like computations includes in particular the evaluation of the greatest common divisor gcd the least common multiple lcm and the resultant of two polynomials computing pade approximation and the berlekamp massey recurrence coefficients as well as numerous problems reducible to these transition to toeplitz and toeplitz like computations is currently the basis for the design of the parallel randomized nc rnc algorithms for these computational problems our main result is in constructing nearly optimal randomized parallel algorithms for toeplitz and toeplitz like computations and consequently for numerous related computational problems including the computational problems listed above where all the input values are integers and all the output values are computed exactly this includes randomized parallel algorithms for computing the rank the determinant and a basis for the null space of an n n toeplitz or toeplitz like matrix a filled with integers as well as a solution x to a linear system a x f if the system is consistent our algorithms use o log n log n log a parallel time and o n log n processors each capable of performing in unit time an arithmetic operation a comparision or a rounding of a rational number to a closest integer the cost bounds cover the cost of the verification of the correctness of the output the computations by these algorithms can be performed with the precision of o n log a bits which matches the precision required in order to represent the output except for the rank computation where the precision of the computation decreases the algorithms involve either a single random parameter or at most 2n 1 parameters the cited processor bounds are less by roughly factor n than ones supported by the known algorithms that run in polylogarithmic arithmetic time and do not use rounding to the closest integers technically we first devise new algorithms supporting our old nearly optimal complexity estimates for parallel computations with general matrices filled with integers then we decrease dramatically by roughly factor n1 376 the processor bounds required in these algorithms in the case where the input matrix is toeplitz like our algorithms exploit and combine some new techniques which may be of independent interest e g in the study of parallel and sequential computation of recursive factorization of integer matrices as well as our earlier techniques of variable diagonal relating to each other several known algebraic and numerical methods stream contraction and the truncation of displacement generators in toeplitz like computations our development and application of these techniques may be of independent

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

diagonal pivoting for partially reconstructible cauchy like matrices with applications to toeplitz like linear equations and to boundary rational matrix interpolation problems abstract in an earlier paper we exploited the displacement structure of cauchy like matrices to derive for them a fast o n 2 implementation of gaussian elimination with partial pivoting one application is to the rapid and numerically accurate solution of linear systems with toeplitz like coefficient matrices based on the fact that the latter can be transformed into cauchy like matrices by using the fast fourier sine or cosine transform however symmetry is lost in the process and the algorithm given is not optimal for hermitian coefficient matrices in this paper we present a new fast o n 2 implementation of symmetric gaussian elimination with partial diagonal pivoting for hermitian cauchy like matrices and show how to transform hermitian toeplitz like matrices to hermitian cauchy like matrices obtaining algorithms that are twice as fast as those in the earlier work numerical experiments indicate that in order to obtain not only fast but also numerically accurate methods it is advantageous to explore the important case in which the corresponding displacement operators have nontrivial kernels this situation gives rise to what we call partially reconstructible matrices which are introduced and studied in the present paper we extend the transformation technique and the generalized schur algorithms i e fast displacement based implementations of gaussian elimination to partially reconstructible matrices we show by a variety of computed examples that the incorporation of diagonal pivoting methods leads to high accuracy we focused in this paper on the design of new numerically reliable algorithms for hermitian toeplitz like matrices however the proposed algorithms have other important applications in particular we briefly describe how they recursively solve a boundary interpolation problem for j unitary rational matrix functions

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

fast algorithms for multiple evaluations of the riemann zeta function the best previously know algorithm for evaluating the riemann zeta function it with bounded and t large to moderate accuracy was based on the rienmann siegel formula and required on the order of t 1 2 operations for each value that was computed new algorithms are presented in this paper which enable one to compute any single value of it with fixed and t t t t 1 2 in o te operations on numbers of o log t bits for any e 0 for example these algorithms lead to methods for numerically verifying the riemann hypothesis for the first n zeros in what is expected to be o n 1 e operations

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

language operations with regular expressions of polynomial size this work deals with questions regarding to what extent regularity preserving language operations affect the descriptional complexity of regular expressions some language operations are identified which are feasible for regular expressions in the sense that the result of the operation can be represented as a regular expression of size polynomial in that of the operands we prove that taking language quotients in particular the prefix and suffix closures of a regular set can incur at most a quadratic blow up on the required expression size the circular shift operation can cause only a cubic increase in size and at least a quadratic bloat can be necessary in the worst case

efficient solution of linear systems of equations with recursive structure abstract inversion algorithms of order n2 for n n matrices with recursive structure are derived in a unified way several known algorithms as well as some new ones for toeplitz type hilbert type and vandermonde type matrices are given

an inversion formula and fast algorithms for cauchy vandermonde matrices abstract matrices of composed type consisting of a vandermonde and a cauchy part and their connection to rational interpolation are investigated an inversion formula is presented and fast solution algorithms with o n log2 n complexity for cauchy vandermonde systems are developed

parallel computation of polynomial gcd and some related parallel computations over abstract fields several fundamental problems of computations with polynomials and structured matrices are well known for their resistance to effective parallel solution this includes computing the gcd the 1cm and the resultant of two polynomials as well as any selected entry of either the extended euclidean scheme for these polynomials or the pade approximation table associated to a fixed power series the solution of the berlekamp massey problem of recovering the n coefficients of a linear recurrence sequence from its first 2n terms the solution of a toeplitz linear system of equations computing the ranks and the null spaces of toeplitz matrices and several other computations with toeplitz hankel vandermonde and cauchy generalized hilbert matrices and with matrices having similar structures our algorithms enable us to reach new record estimates for randomized parallel arithmetic complexity of all these computations over any field of constants that is o log n h time and o n2 log log n log n h g arithmetic processors where n is the input size g varies from 1 to 2 and h varies from 2 to 4 depending on the computational problem and on the fixed field of constants the estimates include the cost of correctness tests for the solution the results have further applications to many related computations over abstract fields and rely on some new techniques in particular on recursive pade approximation and regularization of pade approximation via randomization which might be of some independent technical interest in particular an auxiliary algorithm computes over any field the coefficients of a polynomial from the power sums of its zeros which is an important problem of algebraic coding

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

an efficient computation of generalized inverse of a matrix abstract we propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop this yields a high efficiency index for that computational task the algorithm has a relatively mild numerical instability and we stabilize it at the price of adding two extra matrix multiplications per iteration loop this implies an efficiency index that exceeds the known record for numerically stable iterations for this task which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra our numerical tests cover a variety of examples in the category of generalized inverses such as drazin case rectangular case and preconditioning of linear systems the test results are in good accordance with our formal study

from regular expressions to deterministic automata abstract the main theorem allows an elegant algorithm to be refined into an efficient one the elegant algorithm for constructing a finite automaton from a regular expression is based on derivatives of regular expressions the efficient algorithm is based on marking of regular expressions derivatives of regular expressions correspond to state transitions in finite automata when a finite automaton makes a transition under input symbol a a leading a is stripped from the remaining input correspondingly if the input string is generated by a regular expression e then the derivative of e by a generates the remaining input after a leading a is stripped brzozowski 1964 used derivatives to construct finite automata the state for expression e has a transition under a to the state for the derivative of e by a this approach extends to regular expressions with new operators including intersection and complement however explicit computation of derivatives can be expensive marking of regular expressions yields an expression with distinct input symbols following mcnaughton and yamada 1960 we attach subscripts to each input symbol in an expression ab b ba becomes a 1 b 2 b 3 b 4 a 5 conceptually the efficient algorithm constructs an automaton for the marked expression the marks on the transitions are then erased resulting in a nondeterministic automaton for the original unmarked expression this approach works for the usual operations of union concatenation and iteration however intersection and complement cannot be handled because marking and unmarking do not preserve the languages generated by regular expressions with these operators

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

gpgcd an iterative method for calculating approximate gcd of univariate polynomials we present an iterative algorithm for calculating approximate greatest common divisor gcd of univariate polynomials with the real or the complex coefficients for a given pair of polynomials and a degree our algorithm finds a pair of polynomials which has a gcd of the given degree and whose coefficients are perturbed from those in the original inputs making the perturbations as small as possible along with the gcd the problem of approximate gcd is transferred to a constrained minimization problem then solved with the so called modified newton method which is a generalization of the gradient projection method by searching the solution iteratively we demonstrate that in some test cases our algorithm calculates approximate gcd with perturbations as small as those calculated by a method based on the structured total least norm stln method and the uvgcd method while our method runs significantly faster than theirs by approximately up to 30 or 10 times respectively compared with their implementation we also show that our algorithm properly handles some ill conditioned polynomials which have a gcd with small or large leading coefficient

nearly optimal solution of rational linear systems of equations with symbolic lifting and numerical initialization abstract hensel s symbolic lifting is a highly effective method for the solution of a general or structured e g toeplitz or hankel linear system of equations with integer or rational coefficients of bounded length it can handle ill conditioned inputs for which numerical methods become costly lifting amounts to recursive multiplications by vectors of the input coefficient matrices and its precomputed inverse modulo a fixed integer s such multiplications only involve small numbers of data movements and arithmetic operations with bounded precision the known methods for precomputation of the inverse are more costly however in particular they involve more data movements as our remedy for this bottleneck stage we create an auxiliary matrix sharing its inverse modulo s with the input matrix and we readily compute this inverse by applying numerical iterative refinement which is a numerical counterpart of lifting in the case of general unstructured as well as toeplitz hankel and other popular structured inputs our hybrid algorithms involve a small number of data movements and optimal number of boolean that is bitwise operations up to a logarithmic factor we extend the algorithms to nearly optimal computation of polynomial greatest common divisors gcds least common multiples lcms and pade approximations as well as the berlekamp massey reconstruction of linear recurrences we also cover newton s lifting for matrix inversion specialize it to the case of structured input and combine it with hensel s to enhance the overall efficiency our initialization techniques work for newton s lifting as well furthermore we extend all our lifting algorithms to allow their initialization modulo powers of two thus implementing them in the binary base

localized radial basis functions based pseudo spectral method lrbf psm for nonlocal diffusion problems abstract spectral pseudo spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations in this paper we will present the use of a localized radial basis functions based pseudo spectral method lrbf psm for solving 2d nonlocal problems with radial nonlocal kernels the basic idea of the lrbf psm is to construct a set of orthogonal functions by rbfs on each overlapping sub domain from which the global solution can be obtained by extending the approximation on each sub domain to the entire domain numerical implementation indicates that the proposed lrbf psm is simple to use efficient and robust to solve various nonlocal problems

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

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

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

stabilization of a stochastic nonlinear economy abstract methodology for dynamical systems described by nonlinear stochastic differential equations is applied to economic systems particularly the phillips turnovsky model generalizing the model further to nonlinear terms and stochastic coefficients and inputs to determine expectations and correlation matrices for the variables which represent solution processes of a set of coupled equations the theory is quite general and will apply if the model is changed to use different nonlinear forms e g for taxes as a function of income or different statistics for fluctuations which need not be stationary or gaussian or small or to include retarded effects

schur aggregation for linear systems and determinants we apply our recent preconditioning techniques to the solution of linear systems of equations and computing determinants we combine these techniques with the sherman morrison woodbury formula its new variations aggregation iterative refinement and advanced algorithms that rapidly compute sums and products either error free or with the desired high accuracy our theoretical and experimental study shows the power of this approach

chebyshev type methods and preconditioning techniques recently a newton s iterative method is attracting more and more attention from various fields of science and engineering this method is generally quadratically convergent in this paper some chebyshev type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system ax b theoretic analysis and numerical experiments show that chebyshev s method is more effective than newton s one in the case of constructing approximate inverse preconditioners

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

regular expression for a language without empty word in 1 a bruggemann klein asks the following question is there a linear time algorithm transforming a regular expression e into an expression e with oliver in this paper we give a recursive definition of e which enables us to provide such an algorithm furthermore we show that e e 12 log e 1 e 12 where e is the size of e and e is the size of e

solving linear systems of equations with randomization augmentation and aggregation abstract seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization augmentation and aggregation to reduce our task to computations with well conditioned matrices of full rank our algorithms avoid pivoting and orthogonalization preserve matrix structure and sparseness and in the case of an ill conditioned input perform only a small part of the computations with high accuracy we extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned our study can be of independent technical interest we extend the known results on conditioning of random matrices to randomized preconditioning estimate the condition numbers of randomly augmented matrices and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations

efficient matrix preconditioners for black box linear algebra abstract the main idea of the black box approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials in most cases preconditioning is necessary to obtain the desired result here good preconditioners will be used to ensure geometrical algebraic properties on matrices rather than numerical ones so we do not address a condition number we offer a review of problems for which algebraic preconditioning is used provide a bestiary of preconditioning problems and discuss several preconditioner types to solve these problems we present new conditioners including conditioners to preserve low displacement rank for toeplitz like matrices we also provide new analyses of preconditioner performance and results on the relations among preconditioning problems and with linear algebra problems thus improvements are offered for the efficiency and applicability of preconditioners the focus is on linear algebra problems over finite fields but most results are valid for entries from arbitrary fields

product rules for the displacement of near toeplitz matrices abstract we consider displacements which are linear operations mapping a near toeplitz matrix into a low rank matrix the objective is to use this low rank representation in solving matrix equations in place of using the full matrix two formulas for the displacement of the product of two matrices are presented and applied applications include multiplication of toeplitz matrices inversion of near toeplitz matrices and finding the eigenvectors of a special class of matrices