how to prove that a preconditioner cannot be superlinear in the general case of multilevel toeplitz matrices we recently proved that any multilevel circulant preconditioner is not superlinear a cluster it may provide cannot be proper the proof was based on the concept of quasi equimodular matrices although this concept does not apply for example to the sine transform matrices in this paper with a new concept of partially equimodular matrices we cover all trigonometric matrix algebras widely used in the literature we propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices at the same time we show that these preconditioners are in a certain sense the best among the sublinear preconditioners with only a general cluster for multilevel toeplitz matrices

integro sextic spline interpolation and its super convergence in this paper we use sextic b splines to construct an approximating function based on the integral values rather than the usual function values at the knots it is called integro sextic spline interpolation the super convergence eighth sixth and fourth order respectively in approximating function values second order derivative values and fourth order derivative values at the knots is mainly proved

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

spectral behavior of matrix sequences and discretized boundary value problems abstract in this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension more specifically we give a unified treatment of notions such as distribution equal distribution localization equal localization clustering and sub clustering as a case study we consider the matrix sequences arising from the finite difference fd discretization of elliptic and semielliptic boundary value problems bvps the spectral analysis is then extended to toeplitz based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular belong to l 1 and to the case of multidimensional problems the related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices

the eigenvalue distribution of products of toeplitz matrices clustering and attraction abstract we use a recent result concerning the eigenvalues of a generic non hermitian complex perturbation of a bounded hermitian sequence of matrices to prove that the asymptotic spectrum of the product of toeplitz sequences whose symbols have a real valued essentially bounded product h is described by the function h in the szego way then using mergelyan s theorem we extend the result to the more general case where h belongs to the tilli class the same technique gives us the analogous result for sequences belonging to the algebra generated by toeplitz sequences if the symbols associated with the sequences are bounded and the global symbol h belongs to the tilli class a generalization to the case of multilevel matrix valued symbols and a study of the case of laurent polynomials not necessarily belonging to the tilli class are also given

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

preconditioning strategies for hermitian indefinite toeplitz linear systems in this paper we propose and analyze preconditioning strategies for hermitian indefinite linear systems by using indefinite preconditioners under very elementary assumptions we show that the eigenvalues are real moreover in the case of multilevel toeplitz structures we prove distributional and localization results these techniques used in connection with the cg gmres bicgstab and qmr algorithms allow us to solve in an optimal way the corresponding linear systems a wide numerical experimentation confirms the efficiency of the proposed procedures

some concepts of negative dependence abstract the theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial dirichlet and the multivariate hypergeometric some conditions of negative dependence that are satisfied by these distributions and which have practical meaning are introduced preservation results for some of these concepts are derived useful inequalities for some widely used distributions are obtained results of mallows 1969 that apply to the multinomial distributions are extended to more distributions examples are listed author

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

the extension of the concept of the generating function to a class of preconditioned toeplitz matrices abstract it is well known that the generating function f l 1 of a class of hermitian toeplitz matrices a n f describes very precisely the spectrum of each matrix of the class recently this theory has been extended in various senses in three directions the non hermitian case the block case and the case of preconditioned toeplitz matrices here by using some results of the first and the second extension we deduce new properties of distribution of the spectrum of preconditioned matrices and we define the concept of generating function for these matrices moreover we discuss the consequences of these results on the analysis of convergence of preconditioned conjugate gradient methods previously defined in the literature finally we perform some numerical experiments which confirm the theoretical idea

a fast algorithm for linear complex chebyshev approximations we propose a new algorithm for finding best minimax polynomial approximations in the complex plane the algorithm is the first satisfactory generalization of the well known remez algorithm for real approximations among all available algorithms ours is the only quadratically convergent one numerical examples are presented to illustrate rapid convergence

precise large deviations for compound random sums in the presence of dependence structures in this paper we deal with the compound random sums of dependent real valued random variables with heavy tailed distributions we establish a precise large deviation result for a nonstandard renewal risk model in which innovations are extended negatively dependent real valued random variables with a common dominatedly varying distribution function their interarrival times are extended negatively dependent nonnegative random variables and the numbers of innovations caused by individual events are also extended negatively dependent positive random variables as an illustration of the obtained result we give two applications related to some insurance risk models

the local integro cubic splines and their approximation properties we show the integro cubic splines proposed by behforooz 1 can be constructed locally by using b representation of splines the approximation properties of the local splines are also considered

circulant approximation for preconditioning in stochastic automata networks stochastic automata networks sans are widely used in modeling practical systems such as queueing systems communication systems and manufacturing systems for the performance analysis purposes one needs to calculate the steady state distributions of sans usually the steady state distributions have no close form solutions and cannot be obtained efficiently by direct methods such as lu decomposition due to the huge size of the generator matrices an efficient numerical method should make use of the tensor structure of sans generator matrices the generalized conjugate gradient cg methods are possible choices though their convergence rates are slow in general to speed up the convergence rate preconditioned cg methods are considered in this paper in particular circulant based preconditioners for the sans are constructed the preconditioners presented in this paper are easy to construct and can be inverted efficiently numerical examples of practical sans are also given to illustrate the fast convergence rate of the method

superlinear convergence for pcg using band plus algebra preconditioners for toeplitz systems the paper studies fast and efficient solution algorithms for nxn symmetric ill conditioned toeplitz systems t n f x b where the generating function f is known a priori real valued nonnegative and has isolated roots of even order the preconditioner that we propose is a product of a band toeplitz matrix and matrices that belong to a certain trigonometric algebra the basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand alone preconditioner as a result we obtain a flexible preconditioner which can be applied to the system t n f x b infusing superlinear convergence to the pcg method the important feature of the proposed technique is that it can be extended to cover the 2d case i e ill conditioned block toeplitz matrices with toeplitz blocks we perform many numerical experiments whose results confirm the theoretical analysis and effectiveness of the proposed strategy

c g preconditioning for toeplitz matrices abstract we consider the problem of solving a toeplitz system of equations by conjugate gradient method when a sequence of nested toeplitz matrices is associated to a function the spectral behaviour of the matrices involved is closely related to the analytical properties of the generating function thus it is possible to devise efficient preconditioning techniques by using various functional approximation strategies this approach leads to attractive results in the case of ill conditioned matrices for which a wide class of preconditioners are proposed

on integro quartic spline interpolation in this paper we use quartic b spline to construct an approximating function to agree with the given integral values of a univariate real valued function over the same intervals it is called integro quartic spline interpolation our interpolation method is new and easy to implement moreover it can work successfully even without any boundary conditions the interpolation errors are studied the super convergence sixth order and fourth order respectively in approximating function values and second order derivative values at the knots is proved numerical examples illustrate that our method is very effective and our integro interpolating quartic spline has higher approximation ability than others

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

toeplitz circulant preconditioners for toeplitz systems and their applications to queueing networks with batch arrivals the preconditioned conjugate gradient method is employed to solve toeplitz systems t n bf x bf b where the generating functions of the n by n toeplitz matrices t n are functions with zeros in this case circulant preconditioners are known to give poor convergence whereas band toeplitz preconditioners offer only linear convergence and can handle only real valued functions with zeros of even orders we propose here preconditioners which are products of band toeplitz matrices and circulant matrices the band toeplitz matrices are used to cope with the zeros of the given generating function and the circulant matrices are used to speed up the convergence rate of the algorithm our preconditioner can handle complex valued functions with zeros of arbitrary orders we prove that the preconditioned toeplitz matrices have singular values clustered around 1 for large n we apply our preconditioners to solve the stationary probability distribution vectors of markovian queueing models with batch arrivals we show that if the number of servers is fixed independent of the queue size n then the preconditioners are invertible and the preconditioned matrices have singular values clustered around 1 for large n numerical results are given to illustrate the fast convergence of our methods

fast numerical solution of parabolic integrodifferential equations with applications in finance we numerically solve parabolic problems u t ca u 0 in 0 t times omega t infty where omega subset mathbb r is a bounded interval and ca is a strongly elliptic integrodifferential operator of order rho in 0 2 a discontinuous galerkin dg discretization in time and a wavelet discretization in space are used the densely populated matrices in the corresponding linear systems of equations are replaced by sparse ones using appropriate wavelet compression techniques the linear systems in each time step are solved by an incomplete gmres iteration under these conditions we show that the complexity of our algorithm is linear up to some logarithmic terms in the number of spatial degrees of freedom and present error estimates applications to purely discontinuous levy processes arising in finance are given

integro quintic splines and their approximation properties quite recently behforooz 2009 2 introduced a new approach to construct integro quintic splines that requires seven additional end conditions in this paper we propose five end conditions that allows to simplify the construction of this spline the proposed algorithm is very easy to implement the convergence in approximating function values and its derivatives up to five order is proved numerical results illustrate the efficiency and higher approximation ability of the method

preconditioners for ill conditioned toeplitz systems constructed from positive kernels in this paper we are interested in the iterative solution of ill conditioned toeplitz systems generated by continuous nonnegative real valued functions f with a finite number of zeros we construct new w circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f kn with a suitable positive reproducing kernel kn by the restriction to positive kernels we obtain positive definite preconditioners moreover if f has only zeros of even order le 2s then we can prove that the property int pi pi t 2k k n t mbox d t le c n 2k k 0 hspace 1 5pt ldots s of the kernel is necessary and sufficient to ensure the convergence of the pcg method in a number of iteration steps independent of the dimension n of the system our theoretical results were confirmed by numerical tests

spectral behavior of preconditioned non hermitian multilevel block toeplitz matrices with matrix valued symbol this note is devoted to preconditioning strategies for non hermitian multilevel block toeplitz linear systems associated with a multivariate lebesgue integrable matrix valued symbol in particular we consider special preconditioned matrices where the preconditioner has a band multilevel block toeplitz structure and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices in this respect our main result is as follows let i k k let m s be the linear space of complex s i s matrices and let f g i k m s be functions whose components f ij g ij i k c i j 1 s belong to l consider the matrices t n 1 g t n f where n n 1 n k varies in n k and t n f t n g are the multilevel block toeplitz matrices of size n 1 n k s generated by f g then t n 1 g t n f n n k g 1 f i e the family of matrices t n 1 g t n f n n k has a global asymptotic spectral distribution described by the function g 1 f provided g possesses certain properties which ensure in particular the invertibility of t n g for all n and the following topological conditions are met the essential range of g 1 f defined as the union of the essential ranges of the eigenvalue functions j g 1 f j 1 s does not disconnect the complex plane and has empty interior this result generalizes the one obtained by donatelli neytcheva serra capizzano in a previous work concerning the non preconditioned case g 1 the last part of this note is devoted to numerical experiments which confirm the theoretical analysis and suggest the choice of optimal gmres preconditioning techniques to be used for the considered linear systems

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

tri diagonal preconditioner for pricing options the value of a contingent claim under a jump diffusion process satisfies a partial integro differential equation pide we localize and discretize this pide in space by the central difference formula and in time by the second order backward differentiation formula the resulting system t nx b in general is a nonsymmetric toeplitz system we then solve this system by the normalized preconditioned conjugate gradient method a tri diagonal preconditioner l n is considered we prove that under certain conditions all the eigenvalues of the normalized preconditioned matrix l n 1t n l n 1t n are clustered around one which implies a superlinear convergence rate numerical results exemplify our theoretical analysis

spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d spectral analysis of coupled pdes and of their schur complements via generalized locally toeplitz sequences in 2d

how to choose the best iterative strategy for symmetric toeplitz systems in the last decade many efficient iterative solvers for n x n symmetric also hermitian or generic toeplitz systems have been devised r h chan and g strang siam j sci stat comput 10 1989 pp 104 119 t f chan siam j sci stat comput 9 1988 pp 766 771 d bini and f di benedetto proceedings 2nd spaa conference 1990 pp 220 223 d bini and p favati siam j matrix anal appl 14 1993 pp 500 507 t huckle siam j matrix anal appl 13 1992 pp 767 777 f di benedetto g fiorentino and s serra comput math appl 26 1993 pp 35 45 s serra siam j matrix anal appl 17 1996 pp 1007 1019 g fiorentino and s serra calcolo 28 1991 pp 283 305 s serra siam j matrix anal appl 20 1998 pp 31 44 in all these methods it is assumed that the considered systems have the form an f x b where the symbol f the generating function is an l1 function and the entries of an f along the kth diagonal coincide with the kth fourier coefficient of f the proof of convergence and sometimes even the definition of these solvers are derived by using some symbolic or analytic properties of f while this function is often unknown in real applications in this paper by means of the knowledge of the coefficients of an f only we propose and discuss an algorithm to economically determine the minimal features of f allowing us to choose and define the best iterative strategy as a consequence of the analysis of the proposed procedure we find some results on the structure of quasi algebra of the set a n f f in l 1 finally we exhibit several numerical experiments which confirm the effectiveness of the proposed idea

pricing options in jump diffusion models an extrapolation approach we propose a new computational method for the valuation of options in jump diffusion models the option value function for european and barrier options satisfies a partial integrodifferential equation pide this pide is commonly integrated in time by implicit explicit imex time discretization schemes where the differential diffusion term is treated implicitly while the integral jump term is treated explicitly in particular the popular imex euler scheme is first order accurate in time second order accuracy in time can be achieved by using the imex midpoint scheme in contrast to the above approaches we propose a new high order time discretization scheme for the pide based on the extrapolation approach to the solution of odes that also treats the diffusion term implicitly and the jump term explicitly the scheme is simple to implement can be added to any pide solver based on the imex euler scheme and is remarkably fast and accurate we demonstrate our approach on the examples of merton s and kou s jump diffusion models the diffusion extended variance gamma model as well as the two dimensional duffie pan singleton model with correlated and contemporaneous jumps in the stock price and its volatility by way of example pricing a one year double barrier option in kou s jump diffusion model our scheme attains accuracy of 10 5 in 72 time steps in 0 05 seconds in contrast it takes the first order imex euler scheme more than 1 3 million time steps in 873 seconds and the second order imex midpoint scheme 768 time steps in 0 49 seconds to attain the same accuracy our scheme is also well suited for bermudan options combining simplicity of implementation and remarkable gains in computational efficiency we expect this method to be very attractive to financial engineering modelers

the glt class as a generalized fourier analysis and applications recently the class of generalized locally toeplitz glt sequences has been introduced as a generalization both of classical toeplitz sequences and of variable coefficient differential operators and for every sequence of the class it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function the symbol that can be easily identified this generalizes the notion of a symbol for differential operators discrete and continuous or for toeplitz sequences for which it is identified through the fourier coefficients and is related to the classical fourier analysis the glt class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations and products in this paper we prove that the considered class is closed under inversion as well when the sequence which is inverted shows a sparsely vanishing symbol sparsely vanishing symbol a symbol which vanishes at most in a set of zero lebesgue measure furthermore we show that the glt class virtually includes any finite difference or finite element discretization of pdes and based on this we demonstrate that our results on glt sequences can be used in a pde setting in various directions 1 as a generalized fourier analysis for the study of iterative and semi iterative methods when dealing with variable coefficients non rectangular domains non uniform gridding or triangulations 2 in order to provide a tool for the stability analysis of pde numerical schemes e g a necessary von neumann criterium for variable coefficient systems of pdes is obtained uniformly with respect to the boundary conditions 3 for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case etc the final part of the paper deals indeed with problems 1 3 and other possible directions in which the glt analysis can be conveniently employed

on compact representations for the solutions of linear difference equations with variable coefficients a comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients of both n th and unbounded order is presented the equivalence between their celebrated combinatorial and determinantal representations is considered a corresponding representation is proposed using determined nested sums of their variable coefficients it makes explicit all the sum of products involved in the previous representations of such solutions some basic applications are also illustrated

superlinear pcg methods for symmetric toeplitz systems in this paper we deal with the solution by means of preconditioned conjugate gradient pcg methods of n x n symmetric toeplitz systems a n f x b with nonnegative generating function f here the function f is assumed to be continuous and strictly positive or is assumed to have isolated zeros of even order in the first case we use as preconditioner the natural and the optimal approximation of a n f proposed by bini and di benedetto and we prove that the related pcg method has a superlinear rate of convergence and a total arithmetic cost of o n log n ops under the second hypothesis we cannot guarantee that the natural matrix is positive definite while for the optimal we show that in the ill conditioned case this can be really a bad choice consequently we define a new matrix for preconditioning the given system then by applying the sherman morrison woodbury inversion formula to the preconditioned system we introduce a small constant number of subsidiary systems which can be solved again by means of the previous pcg method finally we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis

analysis of preconditioning techniques for ill conditioned toeplitz matrices several preconditioning techniques for solving toeplitz systems are known in literature but their convergence features are completely understood only in the well conditioned case we study the application of tau circulant and hartley preconditioners to ill conditioned toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

an explicit formula for the inverse of band triangular toeplitz matrix abstract in order to estimate the condition number of the preconditioned matrix proposed in f r lin w k ching inverse toeplitz preconditioners for hermitian toeplitz systems numer linear algebra appl 12 2005 221 229 we study the inverse of band triangular toeplitz matrix we derive an explicit formula for the entries of the inverse of band lower triangular toeplitz matrix by means of divided difference and use the formula to estimate the condition number of the preconditioned matrices in particular we prove that the minimal eigenvalue of preconditioned matrix is well separated from the origin

approximation by integro cubic splines years ago after i presented a paper on spline functions at oxford university professor powell criticized us for using most of the time the function values rather than the integral values on constructing of the spline functions his comments and his request became the main motivation for this work in this paper we assume that on each subinterval of the spline interval a b the integral value of the function y y x is known by using these values rather than the function values at the knots we introduce a class of new types of interpolatory cubic splines to approximate the function y y x the selection of the end conditions for our integro cubic splines will be discussed the numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation

on local integro quartic splines in this paper we propose a new local construction of integro quartic splines which successfully works without any additional end conditions and without solving any system of equations in fact we obtain analytical formulae for values of the integro spline and up to its third order derivatives values at the knots numerical experiments show our method is easy to implement and effective as well

extreme singular values and eigenvalues of non hermitian block toeplitz matrices abstract in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices tn f generated by a p variate lebesgue integrable matrix valued function f q p c h k q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of tn f by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of tn f second order ergodic formulas and localization of eigenvalues of preconditioned matrices tn 1 g tn f are presented some examples of applications to the preconditioning of these results are also discussed

matrix algebra preconditioners for multilevel toeplitz matrices are not superlinear let f be a d variate 2 periodic continuous function and let tn f n n n1 n d be the multiindexed sequence of multilevel n n toeplitz matrices n n n i ni generated by f let a an n be a sequence of matrix algebras simultaneously diagonalized by unitary transforms we show that there exist infinitely many linearly independent trigonometric polynomials and continuous nonpolynomial functions f such that rank tn f pn o n n d 1 n 1 i for any matrix sequence p pn a this implies that no superlinear matrix algebra preconditioner exists in the multilevel toeplitz case the above mentioned result improves the analysis of the author and e tyrtyshnikov siam j matrix anal appl 21 2 1999 431 where the same was proved under the assumption that the involved algebras are of circulant type 2002 elsevier science inc all rights reserved

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

a finite difference scheme for option pricing in jump diffusion and exponential levy models we present a finite difference method for solving parabolic partial integro differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a levy process or more generally a time inhomogeneous jump diffusion process we discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the levy measure we propose an explicit implicit finite difference scheme which can be used to price european and barrier options in such models we study stability and convergence of the scheme proposed and under additional conditions provide estimates on the rate of convergence numerical tests are performed with smooth and nonsmooth initial conditions

a korovkin based approximation of multilevel toeplitz matrices with rectangular unstructured blocks via multilevel trigonometric matrix spaces in this paper we are concerned with the approximation of multilevel toeplitz matrices generated by multivariate rectangular matrix valued continuous also l2 functions f i p rightarrow cal c s times t with i pi pi we first define a class cal a t of multilevel trigonometric matrix spaces with unstructured s times t blocks second we introduce a multi sequence of linear approximation operators cal p n x n n 1 n 2 ldots n p acting on complex matrices of dimensions n 1 cdots n p s times n 1 cdots n p t n j in cal n and giving the minimizer of the distance of x from a fixed representative of cal a t with regard to the frobenius norm then by stating and proving new versions of the korovkin theorem for f ranging in the banach space bf c i p cal c s times t cdot infty and by linking these theorems to the operators cal p n we prove the goodness of the proposed approximation in terms of the strong and weak equidistribution of the eigenvalues singular values of a n f and cal p n a n f the discussed results provide a very uniform tool for treating the preconditioning problem with applications to a wide variety of structures as analyzed in the twin paper s serra capizzano calcolo in press

fast band toeplitz preconditioners for hermitian toeplitz systems this paper considers the solutions of hermitian toeplitz systems where the toeplitz matrices are generated by nonnegative functions f the preconditioned conjugate gradient method with well known circulant preconditioners fails in the case when f has zeros this paper employs toeplitz matrices of fixed bandwidth as preconditioners their generating functions g are trigonometric polynomials of fixed degree and are determined by minimizing the maximum relative error f g f infty it is shown that the condition number of systems preconditioned by the band toeplitz matrices are o 1 for f with or without zeros when f is positive the preconditioned systems converge at the same rate as other well known circulant preconditioned systems an a priori bound of the number of iterations required for convergence is also given

matrix algebra preconditioners for multilevel toeplitz systems do not insure optimal convergence rate in the last decades several matrix algebra optimal and superlinear preconditioners those assuring a strong clustering at the unity have been proposed for the solution of polynomially ill conditioned toeplitz linear systems the corresponding generalizations for multilevel structures are neither optimal nor superlinear see e g contemp math 281 2001 193 concerning the notion of superlinearity it has been recently shown that the proper clustering cannot be obtained in general see linear algebra appl 343 344 2002 303 siam j matrix anal appl 22 1 1999 431 math comput 72 2003 1305 in this paper by exploiting a proof technique previously proposed by the authors see contemp math 323 2003 313 we prove that the spectral equivalence and the essential spectral equivalence up to a constant number of diverging eigenvalues are impossible too in conclusion optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel multigrid techniques

some results on complex toeplitz eigenvalues we consider the asymptotics of the eigenvalues of toeplitz matrices generated by a complex valued symbol f which is essentially bounded we prove that if the essential range of f has empty interior and does not separate the complex plane then the corresponding toeplitz eigenvalues have the canonical distribution in the sense of widom the assumptions on the symbol only concern the topological structure of its essential range and are completely independent of its smoothness properties we also prove some clustering properties of eigenvalues and some new distribution results

symbol approach in a signal restoration problem involving block toeplitz matrices we consider a special type of signal restoration problem where some of the sampling data are not available the formulation related to samples of the function and its derivative leads to a possibly large linear system associated to a nonsymmetric block toeplitz matrix which can be equipped with a 2x2 matrix valued symbol the aim of the paper is to study the eigenvalues of the matrix we first identify in detail the symbol and its analytical features then by using recent results on the eigenvalue distribution of block toeplitz matrix sequences we formally describe the cluster sets and the asymptotic spectral distribution of the matrix sequences related to our problem the localization areas the extremal behavior and the conditioning are only observed numerically but their behavior is strongly related to the analytical properties of the symbol even though a rigorous proof is still missing in the block case

circulant preconditioners for stochastic automata networks stochastic automata networks sans are widely used in modeling communication systems manufacturing systems and computer systems the san approach gives a more compact and efficient representation of the network when compared to the stochastic petri nets approach to find the steady state distribution of sans it requires solutions of linear systems involving the generator matrices of the sans very often direct methods such as the lu decomposition are inefficient because of the huge size of the generator matrices an efficient algorithm should make use of the structure of the matrices iterative methods such as the conjugate gradient methods are possible choices however their convergence rates are slow in general and preconditioning is required we note that the milu and minv based preconditioners are not appropriate because of their expensive construction cost in this paper we consider preconditioners obtained by circulant approximations of sans they have low construction cost and can be inverted efficiently we prove that if only one of the automata is large in size compared to the others then the preconditioned system of the normal equations will converge very fast numerical results for three different sans solved by cgs are given to illustrate the fast convergence of our method

interpolation by integro quintic splines recently behforooz 1 has introduced a new approach to construct cubic splines by using the integral values rather than the usual function values at the knots also he has established different sets of end conditions for cubic and quintic splines by using the integral values see behforooz 2 4 in this paper we will use the same techniques of 1 to construct integro quintic splines although by using the integral values we expected to face a more complicated process for our construction it turned out that the matrix of the system of linear equations that produces the parameters became a diagonally dominant matrix and the process became very simple the selection of the required end conditions for our integro quintic splines will be discussed the numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation