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

transition to chaos in a shell model of turbulence abstract we study a shell model for the energy cascade in three dimensional turbulence by varying the coefficients of the non linear terms in such a way that the fundamental symmetries of navier stokes are conserved when the control parameter related to the strength of backward energy transfer is small enough the dynamical system has a stable fixed point corresponding to the kolmogorov scaling by using the bi orthogonal decomposition the transition to chaos is shown to follow the ruelle takens scenario for 0 3953 there exists a strange attractor which remains close to the kolmogorov fixed point the intermittency of the chaotic evolution and of the scaling can be described by an intermittent one dimensional map we introduce a modified shell model which has a good scaling behaviour also in the infrared region we study the multifractal properties of this model for large number of shells and for values of slightly above the chaotic transition in this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range

chaotic behavior in shell models and shell maps we study the chaotic behavior of the goy shell model by measuring the variation of the maximal lyapunov exponent with the parameter e which determines the nature of the second invariant the generalized helicity invariant after a hopf bifurcation we observe a critical point at e c 0 38704 above which the maximal lyapunov exponent grows nearly linearly for high values of e the evolution becomes regular again which can be explained by a simple analytic argument a model with few shells shows two transitions to simplify the model substantially we introduce a shell map which exhibits similar properties as the goy model

a sharp lower bound for the hausdorff dimension of the global attractors of the 2d navier stokes equations for a special class of the navier stokes equations on the two dimensional torus we give a lower bound in the formg2 3 whereg is the grashof number for the hausdorff dimension of its global attractor which is optimal up to a logarithmic term

regularity of inviscid shell models of turbulence in this paper we continue the analytical study of the sabra shell model of energy turbulent cascade we prove the global existence of weak solutions of the inviscid sabra shell model and show that these solutions are unique for some short interval of time in addition we prove that the solutions conserve energy provided that the components of the solution satisfy ensuremath mid u n ensuremath mid ensuremath le c k n ensuremath 1 3 sqrt n phantom rule 0 2em 0ex mathrm log n 1 ensuremath 1 for some positive absolute constant c which is the analog of the onsager s conjecture for the euler s equations moreover we give a beal kato majda type criterion for the blow up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the two dimensional parameters regime

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

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

cyclic group and knapsack facets any integer program may be relaxed to a group problem we define the master cyclic group problem and several master knapsack problems show the relationship between the problems and give several classes of facet defining inequalities for each problem as well as a set of mappings that take facets from one type of master polyhedra to another

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

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

random multipliers numerically stabilize gaussian and block gaussian elimination proofs and an extension to low rank approximation abstract we study two applications of standard gaussian random multipliers at first we prove that with a probability close to 1 such a multiplier is expected to numerically stabilize gaussian elimination with no pivoting as well as block gaussian elimination then by extending our analysis we prove that such a multiplier is also expected to support low rank approximation of a matrix without customary oversampling our test results are in good accordance with this formal study the results remain similar when we replace gaussian multipliers with random circulant or toeplitz multipliers which involve fewer random parameters and enable faster multiplication we formally support the observed efficiency of random structured multipliers applied to approximation but we still continue our research in the case of elimination we specify a narrow class of unitary inputs for which gaussian elimination with no pivoting is numerically unstable and then prove that with a probability close to 1 a gaussian random circulant multiplier does not fix numerical stability problems for such inputs we also prove that the power of the random circulant preprocessing increases if we also include random permutations

cyclic group blocking polyhedra gomory linear algebra appl 2 451 558 1969 gave a subadditive characterization of the facets of the group polyhedron although there are exponentially many facets see gomory and johnson in math program 3 359 389 1972 example 4 6 and exponentially many vertices for the group polyhedron for the master cyclic group problem gomory s characterization of the non trivial facets has polynomially many subadditive inequalities in fact of order g 2 for a finite abelian group g we reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non trivial facets the system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed our minimal representation leads to a characterization of the vertices of length three gomory et al math program 96 321 339 2003 introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets we develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples hunsaker 2003 gave a knapsack shooting experiment which we use to find a simple pattern for the most hit knapsack facets

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

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

a 3 slope theorem for the infinite relaxation in the plane in this paper we consider the infinite relaxation of the corner polyhedron with 2 rows for the 1 row case gomory and johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme the celebrated 2 slope theorem despite increased interest in understanding the multiple row setting no generalization of this theorem was known for this case we present an extension of the 2 slope theorem for the case of 2 rows by showing that minimal 3 slope functions satisfying an additional regularity condition are facets and hence extreme moreover we show that this regularity condition is necessary unveiling a structure which is only present in the multi row setting

improved shell model of turbulence we introduce a shell model of turbulence that exhibits improved properties in comparison to the standard and very popular gledzer ohkitani and yamada goy model the nonlinear coupling is chosen to minimize correlations between different shells in particular the second order correlation function is diagonal in the shell index and the third order correlation exists only between three consecutive shells spurious oscillations in the scaling regime which are an annoying feature of the goy model are eliminated by our choice of nonlinear coupling we demonstrate that the model exhibits multiscaling similar to the goy model the scaling exponents are shown to be independent of the viscous mechanism as is expected for navier stokes turbulence and other shell models these properties of the model make it optimal for further attempts to achieve understanding of multiscaling in nonlinear dynamics

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

a k 1 slope theorem for the k dimensional infinite group relaxation we prove that any minimal valid function for the k dimensional innite group relaxation that is piecewise linear with at most k 1 slopes and does not factor through a linear map with non trivial kernel is extreme this generalizes a theorem of gomory and johnson for k 1 and cornu ejols and molinaro for k 2

analytic study of shell models of turbulence abstract in this paper we study analytically the viscous sabra shell model of the energy turbulent cascade we prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom specifically a finite dimensional global attractor and globally invariant inertial manifolds moreover we establish the existence of an exponentially decaying energy dissipation range for sufficiently smooth forcing

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

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

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

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

fast computation of determinants of bezout matrices and application to curve implicitization when using bivariate polynomial interpolation for computing the implicit equation of a rational plane algebraic curve given by its parametric equations the generation of the interpolation data is the most costly of the two stages of the process in this work a new way of generating those interpolation data with less computational cost is presented the method is based on an efficient computation of the determinants of certain constant bezout matrices

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

sharp lower bounds for the dimension of the global attractor of the sabra shell model of turbulence in this work we derive lower bounds for the hausdorff and fractal dimensions of the global attractor of the sabra shell model of turbulence in different regimes of parameters we show that for a particular choice of the forcing term and for sufficiently small viscosity term the sabra shell model has a global attractor of large hausdorff and fractal dimensions proportional to log 1 for all values of the governing parameter e except for e 1 the obtained lower bounds are sharp matching the upper bounds for the dimension of the global attractor obtained in our previous work moreover the complexity of the dynamics of the shell model increases as the viscosity tends to zero and we describe a precise scenario of successive bifurcations for different parameters regimes in the three dimensional regime of parameters this scenario changes when the parameter e becomes sufficiently close to 0 or to 1 we also show that in the two dimensional regime of parameters for a certain non zero forcing term the long term dynamics of the model becomes trivial for every value of the viscosity

worst case comparison of valid inequalities for the tsp we consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst case improvement resulting from their addition to the subtour polyhedron for example we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10 9 the corresponding factor for the class of clique tree inequalities is 8 7 while it is 4 3 for the path configuration inequalities

improved parallel computations with toeplitz like and hankel like matrices abstract the known fast algorithms for computations with general toeplitz hankel toeplitz like and hankel like matrices are inherently sequential we develop some new techniques in order to devise fast parallel algorithms for such computations including the evaluation of krylov sequences for such matrices traces of their power sums characteristic polynomials and generalized inverses this has further extensions to computing the solution or a least squares solution to a linear system of equations with such a matrix and to several polynomial computations such as computing the gcd lcm pade approximation and extended euclidean scheme for two polynomials as well as to computing the minimum span of a linear recurrence sequence the algorithms can be applied over any field of constants with the resulting advantages of using modular arithmetic the algorithms consist of simple computational blocks mostly reduced to fast fourier transforms and have potential practical value we also develop the techniques for extending all our results to the case of matrices representable as the sums of toeplitz like and hankel like matrices

a new approach to fast polynomial interpolation and multipoint evaluation abstract the fastest known algorithms for the problems of polynomial evaluation and multipoint interpolation are devastatingly unstable numerically because of their recursive use of polynomial divisions we apply a completely distinct approach to compute approximate solutions to both problems equally fast but with improved numerical stability our approach relies on new techniques so far not used in this area we reduce the problems to vandermonde matrix computations and then exploit some recent methods for improving computations with structured matrices

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

asymptotic formulas for the lyapunov spectrum of fully developed shell model turbulence the scaling behavior of the lyapunov spectrum of a chaotic shell model for three dimensional turbulence is studied in detail first we characterize the localization property of the lyapunov vectors in wave number space by using numerical results by combining this localization property with kolmogorov s dimensional argument we deduce explicitly the asymptotic scaling law for the lyapunov spectrum which in turn is shown to agree well with the numerical results this shell model is an example of high dimensional chaotic systems for which an asymptotic scaling law is obtained for the lyapunov spectrum implications of the present results for the navier stokes turbulence are discussed in particular we conjecture that the distribution of lyapunov exponents is not singular at null exponent

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

parametrization of newton s iteration for computations with structured matrices and applications abstract we apply a new parametrized version of newton s iteration in order to compute over any field f of constants the solution or at least squares solution to linear system b x v with an n n toeplitz or toeplitz like matrix b as well as the determinant of b and the coefficients of its characteristic polynomial det i b dramatically improving the processor efficiency of the known fast parallel algorithms our algorithms together with some previously known and some recent results of 1 5 as well as with our new techniques for computing polynomial god s and lcm s imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials and with both structured and general matrices

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

scaling and linear response in the goy turbulence model the goy model is a model for turbulence in which two conserved quantities cascade up and down a linear array of shells when the viscosity parameter small nu greek is small the model has a qualitative behavior which is similar to the kolmogorov theories of turbulence here a static solution to the model is examined and a linear stability analysis is performed to obtain response eigenvalues and eigenfunctions both the static behavior and the linear response show an inertial range with a relatively simple scaling structure our main results are i the response frequencies cover a wide range of scales with ratios which can be understood in terms of the frequency scaling properties of the model ii even small viscosities play a crucial role in determining the model s eigenvalue spectrum iii as a parameter within the model is varied it shows a phase transition in which there is an abrupt change in many eigenvalues from stable to unstable values iv the abrupt change is determined by the model s conservation laws and symmetries

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

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

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

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

t space and cutting planes in this paper we show how knowledge about t space translates directly into cutting planes for general integer programming problems after providing background on corner polyhedra and on t space this paper examines t space in some detail it gives a variety of constructions for t space facets all of which translate into cutting planes and introduces continuous families of facets in view of the great variety of possible facets no one of which can be dominated either by any other or by any combination of the others a measure of merit is introduced to provide guidance on their usefulness t spaces based on higher dimensional groups are discussed briefly as is the idea of going beyond cutting planes to iterated approximations of corner polyhedra

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

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

the worst case analysis of strong knapsack facets in this paper we identify strong facet defining inequalities for the master knapsack polytope our computational experiments for small master knapsack problems show that 1 k facets for small values of k k le 4 k 4 are strong facets for the knapsack polytope we show that this finding is robust by proving that the removal of these facets from the master knapsack polytope significantly weakens the resulting relaxation in the worst case we show that the 1 k facets for k 1 k 1 are the strongest in that their removal from the master knapsack polytope weakens the relaxation by a factor of 3 2 in the worst case we then show that the 1 k facets with k 3 k 3 or 4 are the next strongest we also show that the strength of the 1 k facets weakens as k grows and that the 1 k facets with k even are stronger than the 1 k facets with k odd

corner polyhedra and their connection with cutting planes we review the necessary background on corner polyhedra and use this to show how knowledge about corner polyhedra and subadditive functions translates into a great variety of cutting planes for general integer programming problems experiments are described that indicate the dominance of a relatively small number of the facets of corner polyhedra this has implications for their value as cutting planes

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

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

stochastic attractors for shell phenomenological models of turbulence recently it has been proposed that the navier stokes equations and a relevant linear advection model have the same long time statistical properties in particular they have the same scaling exponents of their structure functions this assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model the sabra shell model of turbulence and its corresponding linear advection counterpart model this relationship has been established through a homotopy like coefficient which bridges continuously between the two systems that is for 1 one obtains the full nonlinear model and the corresponding linear advection model is achieved for 0 in this paper we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise we prove the continuous dependence of the solutions with respect to the parameter moreover we show the existence of a finite dimensional random attractor for each value of and establish the upper semicontinuity property of this random attractors with respect to the parameter this property is proved by a pathwise argument our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long time statistical properties of the nonlinear and linear models

numerically safe gaussian elimination with no pivoting abstract gaussian elimination with no pivoting and block gaussian elimination are attractive alternatives to the customary but communication intensive gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely that is run into neither division by 0 nor numerical problems empirically safety and numerical safety of genp have been consistently observed in a number of papers where an input matrix was pre processed with various structured multipliers chosen ad hoc our present paper provides missing formal support for this empirical observation and explains why it was elusive so far namely we prove that genp is numerically unsafe for a specific class of input matrices in spite of its pre processing with some well known and well tested structured multipliers but we also prove that genp and bge are safe and numerically safe for the average input matrix pre processed with any nonsingular and well conditioned multiplier this should embolden search for sparse and structured multipliers and we list and test some new classes of them we also seek randomized pre processing that universally that is for all input matrices supports i safe genp and bge with probability 1 and or ii numerically safe genp and bge with a probability close to 1 we achieve goal i with a gaussian structured multiplier and goal ii with a gaussian unstructured multiplier and alternatively with gaussian structured augmentation we consistently confirm all these formal results with our tests of genp for benchmark inputs we have extended our approach to other fundamental matrix computations and keep working on further extensions

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

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed