matrix decompositions using displacement rank and classes of commutative matrix algebras using the notion of displacement rank we look for a unifying approach to representations of a matrix a as sums of products of matrices belonging to commutative matrix algebras these representations are then considered in case a is the inverse of a toeplitz or a toeplitz plus hankel matrix some well known decomposition formulas for a gohberg semencul or kailath et al gader bini pan and gohberg olshevsky turn out to be special cases of the above representations new formulas for a in terms of algebras of symmetric matrices are studied and their computational aspects are discussed

a class of cubic splines obtained through minimum conditions a class of cubic spline minimizing some special functional is investigated this class is determined by the solution of a quadratic programming problem in which the minimizing function depends linearly on a parameter a 2 for a 1 2 natural splines are obtained for a 1 the spline minimizing the mean value of the third derivative is obtained it is shown that this spline has the best convergence order

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

compact fourier analysis for designing multigrid methods the convergence of multigrid methods can be analyzed based on a fourier analysis of the method or by proving certain inequalities that have to be fulfilled by the smoother and by the coarse grid correction separately here we analyze the multigrid method for the constant coefficient poisson equation with a compact fourier analysis using the formalism of multilevel toeplitz matrices and their generating functions or symbols the fourier analysis is applied for determining the smoothing factor and the overall error of the combined smoothing and coarse grid correction error reduction of a twogrid step tgs by representing the twogrid step explicitly by a symbol if the effects of the smoothing correction and the coarse grid correction are orthogonal to each other then in a twogrid step the error is removed in one step and the twogrid method can be considered as a direct solver if the coarse linear system is identical to the original matrix then the same projection and smoother again make the twogrid step a direct solver hence the multigrid cycle has to be applied only once with one smoothing step on each level and therefore the whole multigrid method can be considered as a direct solver in this paper we want to identify multigrid as a direct solver in one dimension with a coarse system derived by linear and constant interpolation and in two dimensions for a modified projection related to u trottenberg c w osterlee and a schuller multigrid academic press san diego 2001 paragraph a 2 3 by studying not only smoothing and coarse grid correction separately but the symbol of the full twogrid step as well we are furthermore able to derive better convergence estimates and information on how to find efficient combinations of projection and smoother as smoothers we consider also approximate inverse smoothers colored smoothers that take into account the different character of grid points and rank reducing smoothers that coincide with the given matrix on a large number of entries numerical examples show the effectiveness of the new approach

recursive based pcg methods for toeplitz systems with nonnegative generating functions in this paper we consider the solutions of symmetric positive definite but ill conditioned toeplitz systems an x b here we propose to solve the system by the recursive based preconditioned conjugate gradient method the idea is to use the inverse of am the principal submatrix of an with the gohberg semencul formula as a preconditioner for an the inverse of am can be generated recursively by using the formula until m is small enough the construction of the preconditioners requires only the entries of an and does not require the explicit knowledge of the generating function f of an we show that if f is a nonnegative bounded and piecewise continuous even function with a finite number of zeros of even order the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers hence the conjugate gradient method when applied to solving the preconditioned system converges very quickly numerical results are included to illustrate the effectiveness of our approach

matrix displacement decompositions and applications to toeplitz linear systems using the approach of bozzo di fiore and zellini new matrix displacement decomposition formulas are introduced it is shown how an arbitrary square matrix a can be expressed as sums of products of hessenberg algebra matrices and high level block matrices whose submatrices are hessenberg algebra matrices and have variable sizes in most cases these block factors are block diagonal matrices then these formulas are used in sequential and parallel solution of toeplitz systems

long term inventory policy making through fuzzy decision making models abstract the inventory policy meant as a replenishment rule has a considerable impact on most firms the paper considers the determination of optimal inventory policy of firms from a global viewpoint of top management the inventory is represented as a fuzzy system with the fuzzy inventory level as the output the fuzzy replenishment as the input and fuzzy demand the control problem is formulated in terms of decision making in a fuzzy environment with fuzzy constraints imposed on replenishments a fuzzy goal for preferable inventory levels to be attained and the fuzzy decision as the intersection of fuzzy constraints and the fuzzy goal at subsequent stages the planning horizon is infinite the problem is to find an optimal time invariant strategy relating the optimal replenishments to the current inventory levels maximizing the membership function of fuzzy decision the existence of such a strategy is proved and an algorithm for its determination is given the optimal time invariant strategy obtained is represented as a fuzzy conditional statement equated with a fuzzy relation which is the firm s optimal fuzzy replenishment rule

the rate of convergence of toeplitz based pcg methods for second order nonlinear boundary value problems in previous works 21 23 we proposed the use of tau 5 and band toeplitz based preconditioners for the solution of 1d and 2d boundary value problems bvp by means of the preconditioned conjugate gradient pcg methods as tau and band toeplitz linear systems can be solved 4 by using fast sine transforms 8 these methods become especially attractive in a parallel environment of computation in this paper we extend this technique to the nonlinear nonsymmetric case and in addition we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear therefore these methods work much finer than those based on separable preconditioners 18 45 on incomplete lu factorizations 36 13 27 and on circulant preconditioners 9 30 35 since the latter two techniques do not assure a linear rate of convergence on the other hand the proposed technique has a wider range of application since it can be naturally used for nonlinear nonsymmetric problems and for bvp in which the coefficients of the differential operator are not strictly positive and only piecewise smooth finally the several numerical experiments performed here and in 22 23 confirm the effectiveness of the theoretical analysis

essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices

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

a kronecker product variant of the facr method for solving the generalized poisson equation we present a fast direct method for the solution of a linear system mx y where m is a block tridiagonal toeplitzmatrix with a on the diagonal and t on the two subdiagonals a and t commute such matrices are obtained from a finite difference approximation to poisson s equation with nonconstant coefficients in one direction among others the new method is called kpcr l method and begins with l steps of cyclic reduction after which the remaining system is solved by a kronecker product method for an appropriate choice of l the asymptotic operation count for an n n grid is o n2 log2 log2 n which is faster than either cyclic reduction or the kronecker product method itself the algorithm is similar to and has the same complexity as the facr l algorithm which is a combination of cyclic reduction and fourier analysis or matrix decomposition however the facr l algorithm only reaches this complexity if a and t can be diagonalized by a fast transformation where the new method is fast for every banded a and t moreover the kpcr l method can be easily generalized to the case where a and t do not commute

optimal rank matrix algebras preconditioners abstract when a linear system ax y is solved by means of iterative methods mainly cg and gmres and the convergence rate is slow one may consider a preconditioner p and move to the preconditioned system p 1 ax p 1 y the use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate the construction of optimal rank preconditioners is strongly related to the possibility of splitting a as a p r e where e is a small perturbation and r is of low rank tyrtyshnikov 1996 1 in the present work we extend the black dot algorithm for the computation of such splitting for p circulant see oseledets and tyrtyshnikov 2006 2 to the case where p is in a for several known low complexity matrix algebras a the algorithm so obtained is particularly efficient when a is toeplitz plus hankel like we finally discuss in detail the existence and the properties of the decomposition a p r e when a is toeplitz also extending to the circulant and hartley type cases some results previously known for p circulant

a note on algebraic multigrid methods for the discrete weighted laplacian in recent contributions algebraic multigrid methods have been designed and studied from the viewpoint of spectral complementarity in this note we focus our efforts on specific applications and more precisely on large linear systems arising from the approximation of the weighted laplacian with various boundary conditions we adapt the multigrid idea to this specific setting and we present and critically discuss a wide set of numerical experiments showing the potentiality of the considered approach

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

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

generalized updating problems and computation of the eigenvalues of rational toeplitz matrices abstract algebraic and computational properties of the rank one updating of a generalized eigenvalue problem are investigated the results are applied to the computation of the eigenvalues of full toeplitz matrices related to the laurent expansion of a rational function extending a method of handy and barlow already known for the banded toeplitz case

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

spectral analysis of the anti reflective algebra abstract anti reflective boundary conditions have been studied in connection with fast deblurring algorithms in the case of d dimensional objects signals for d 1 images for d 2 here we study how under the assumption of strong symmetry of the point spread functions and under mild degree conditions the associated matrices depend on a symbol and define an algebra homomorphism furthermore the eigenvalues can be exhaustively described in terms of samplings of the symbol and other related functions and appropriate o n d log n arithmetic operations algorithms can be derived for the related computations these results in connection with the use of the anti reflective transform are of interest when employing filtering type procedures for the reconstruction of noisy and blurred objects

intersymbol interference due to linear and nonlinear distortion the paper provides a recursion model for the calculation of the probability density function pdf of intersymbol interference isi which is caused by the combined effect of linear channel dispersion and of nonlinear distortion the nonlinearity introduces statistical interdependencies between the interfering symbols and these dependencies are implicitly taken into account in a trellis structured recursion rule the results were verified by time consuming monte carlo simulation and show e g that the nonlinear characteristic of a high power amplifier hpa reduces in some cases the error probability caused by the linear dispersion surprisingly the isi due to the nonlinear amplifier is increased in the case of offset modulation

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

preconditioning of block toeplitz matrices by sine transforms the iterative solution of a block toeplitz linear system by the conjugate gradient method is analyzed the preconditioning step being solved by means of a discrete sine transform convergence properties are established and compared to the behavior of the block circulant preconditioner recently proposed in the literature as in the scalar case the new approach takes advantage if the system is ill conditioned

a korovkin type theory for finite toeplitz operators via matrix algebras preconditioned conjugate gradients pcg are widely and successfully used methods for solving a toeplitz linear system a n vec x vec b 59 9 20 5 34 62 6 10 28 45 44 46 49 frobenius optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the frobenius distance from a n the convergence features of these pcg have been naturally studied by means of the weierstrass jackson theorem 17 36 45 owing to the profound relationship between the spectral features of the matrices a n generated by the fourier coefficients of a continuous function f and the analytical properties of the symbol f itself in this paper we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f on the other hand by modifying the korovkin theorem to study the frobenius optimal preconditioning problem we provide a new and unifying tool for analyzing all frobenius optimal preconditioners in any generic matrix algebra related to trigonometric transforms finally the multilevel case is sketched and discussed by showing that a korovkin type theory also holds in a multivariate sense

fast parallel solvers for elliptic problems in an earlier work 1 we have considered the use of t 2 preconditioners for the solution of dirichlet boundary value problems by means of the preconditioned conjugate gradients methods in this paper we propose the use of toeplitz matrices for the fast parallel solution of differential problems we propose some algorithms to solve preconditioned systems which are especially suitable in parallel environments we also present several numerical results to prove the effectiveness of our technique

the solution of ill conditioned symmetric toeplitz systems via two grid and wavelet methods abstract in this paper we present a generalized multigrid method combined with wavelet filters for solving ill conditioned symmetric toeplitz systems t n x b where t n r n n is generated by nonnegative functions with zeros first we propose the construction of general cohen daubechies and feauveau cdf 9 7 biorthogonal wavelet systems so that a new class of compactly supported biorthogonal wavelet systems gcdf are achieved with specified vanishing moments for scaling functions in order to solve ill conditioned toeplitz systems by using the two grid method tgm we use the constructed gcdf wavelets to get prolongation and restriction operators as a result the proposed tgm is proved by numerical experiment to be more efficient than the classic tgm especially when t n is seriously ill conditioned for the vanishing moments being n experimental tests illustrate that the tgm with damped jacobi smoother converges when the generating function has zeros of order less than or equal to 2n n 8 besides we prove in theory that the proposed method converges for toeplitz systems that are generated by functions with zeros of order less than or equal to four

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

frequency independent and frequency dependent nonlinear models of twt amplifiers simple two parameter formulas are presented for the functions involved in the amplitude phase and the quadrature nonlinear models of a twt amplifier and are shown to fit measured data very well also a closed form expression is derived for the output signal of a twt amplifier excited by two phase modulated carriers and an expression containing a single integral is given when more than two such earriers are involved finally a frequencydependent quadrature model is proposed whose parameters are obtainable from single tone measurements

control of a deterministic system es a fuzzy environment over infinite planning horizon the multistage control of a time invariant finite state deterministic system in a fuzzy environment over infinite planning horizon is considered the fuzzy environment is meant as fuzzy constraints imposed on controls and fuzzy goals to be attained the fuzzy constraints are assumed to be state dependent the fuzzy decision is assumed to he the intersection of fuzzy constraints and fuzzy goals the problem is to find an optimal strategy i e a sequence of policies relating the control to the state maximizing the membership function of fuzzy decision for the problem a functional equation is formulated an algorithm of policy iteration type is given the existence of a lime invariant optimal strategy and the convergence of the algorithm are proved a numerical example is shown

multigrid methods for symmetric positive definite block toeplitz matrices with nonnegative generating functions in this paper we introduce a generalized multigrid method for solving linear systems bf t n m bf x bf b where bf t n m in re nm times nm is a symmetric block toeplitz matrix with symmetric toeplitz blocks we use a special choice of the projection operator whose coefficients simply depend on the generating functions associated with the proposed class of matrices this choice leads to iterative methods with convergence rates independent of the euclidean condition number kappa 2 bf t n m and of the dimension of the involved matrices the total arithmetic cost is o nm log nm for dense matrices and o nm for band matrices with band blocks in the pram model only o log nm parallel steps are required this algorithm is therefore competitive with the preconditioned conjugate gradient methods proposed by chan and jin siam j sci statist comput 13 1992 pp 1218 1235 ku and kuo siam j sci statist comput 13 1992 pp 948 966 di benedetto siam j sci comput 17 1996 and serra bit 34 1994 pp 579 594 for dense matrices and improves those results for band block toeplitz matrices with band toeplitz blocks

fuzzy dynamic programming main developments and applications abstract we survey major developments and applications of fuzzy dynamic programming which is here advocated as a promising attempt at making dynamic programming models more realistic by a relaxation of often artificial assumptions of precision as to the constraints goals states and their transitions termination time etc issues related to numerical efficiency are considered applications in r d planning health care and medicine socioeconomic regional development energy systems water and environmental systems chemical engineering etc are surveyed

analysis of a multigrid method as an iterative technique for solving linear systems a general class of iterative methods is introduced for solving symmetric positive definite linear systems these methods use two different approximations to the inverse of the matrix of the problem one of which involves the inverse of a smaller matrix it is shown that the methods of this class reduce the error by a constant factor at each step and that under ideal circumstances this constant is equal to kappa 1 kappa 1 where kappa is the ratio of the largest eigenvalue to the j 1 eigenvalue of the matrix j being the dimension of the smaller matrix involved a multigrid method is presented as an example of a method of this class and it is shown that while the multigrid method does not quite achieve this optimal rate of convergence it does reduce the error at each step by a constant factor independent of the mesh spacing h the size of this constant factor and properties of the differential equation and the discretization that affect it are also discussed

mathematical problems in error calculations for interference several treatments of interference problems are examined with a view to determine whether doubtful mathematical assumptions about the existence of probability density functions affect the conclusions most derivations can be recast so that the existence of a density function is not necessary for the conclusions to be valid

on the properties of matrices defining some classes of bvms it is known that the matrices defining the discrete problem generated by a k step boundary value method bvm have a quasi toeplitz band structure 7 in particular when the boundary conditions are skipped they become toeplitz matrices in this paper by introducing a characterization of positive definiteness for such matrices we shall prove that the toeplitz matrices which arise when using the methods in the classes of bvms known as generalized bdf and top order methods have such property

matrix algebras in optimal preconditioning the theory and the practice of optimal preconditioning in solving a linear system by iterative processes is founded on some theoretical facts understandable in terms of a class v of spaces of matrices including diagonal algebras and group matrix algebras the v structure lets us extend some known crucial results of preconditioning theory and obtain some useful information on the computability and on the efficiency of new preconditioners three preconditioners not yet considered in literature belonging to three corresponding algebras of v are analyzed in detail some experimental results are included

on the eigenvalue problem for toeplitz band matrices abstract a formula is given for the characteristic polynomial of an n th order toeplitz band matrix with bandwidth k n in terms of the zeros of a k th degree polynomial with coefficients independent of n the complexity of the formula depends on the bandwidth k and not on the order n also given is a formula for eigenvectors in terms of the same zeros and k coefficients which can be obtained by solving a k k homogeneous system

multigrid methods for anisotropic bttb systems multigrid methods are highly efficient solution techniques for large sparse multilevel toeplitz systems which are positive definite and ill conditioned in this paper we develop multigrid methods which are especially designed for anisotropic two level toeplitz bttb matrices first a method is described for systems with anisotropy along coordinate axes as a suitable combination of semicoarsening and full coarsening steps although the basic idea is known from the solution of partial differential equations we present it here in a more formal way using generating functions and their level curves this enables us not only to prove the optimal convergence of the two grid method but also to carry over the results to systems with anisotropy in other directions we introduce new coordinates in order to describe these more complicated systems in terms of generating functions this enables us to solve them with the same efficiency for the two level method we present a convergence proof in this more general case

fast direct solution methods for symmetric banded toeplitz systems based on the sine transform we present new fast direct methods for solving a large symmetric banded toeplitz system of order n with bandwidth p we make use of structured matrices which can be diagonalized by the discrete sine transform matrix sometimes called matrices a first method writes the toeplitz matrix as the sum of a matrix and a low rank matrix a second method embeds the toeplitz matrix in a larger matrix of order m the methods are similar to jain ieee trans acoust speech signal process 26 1978 121 and linzer linear algebra appl 170 1992 1 who worked with circulant matrices both algorithms consist in solving two systems and two smaller systems a system of order n can be solved in o n log n by using a discrete sine transform if n 1 has small prime factors therefore the second algorithm is preferable since we can choose m such that m 1 has small prime factors on the other hand in the second method the smaller systems can become large when m differs too much from n w hile in the first method the order is always p 1 in both methods the small systems have low displacement rank so we can use fast methods to solve them 2002 elsevier science inc all rights reserved

structure of algebras of commutative matrices commutative spaces of matrices a nk 1 akjk are studied where jk is a set of 0 1 matrices with prescribed sum s and the ak s are complex parameters the great number of commutative spaces a obeying simple prescriptions on the first two rows of s can all be described in terms of two basic algebras of matrices denoted by tn and n a relationship between the structure of spaces a and the multiplicative complexity of the bilinear forms defined by the jk s is discussed

a direct method for solving circulant tridiagonal block systems of linear equations this paper presents a modification of rojo s algorithm comput math appl 20 1990 61 to solve block circulant tridiagonal systems of linear equations which are toeplitz and hermitian this new approach gives us a general direct algorithm for solving the problem we will show how to choose a block matrix as a parameter to describe the method we employ the factorization of block toeplitz tridiagonal matrices as the product of two block toeplitz subdiagonal and superdiagonal matrices the algorithm is based on obtaining the solution of the nonlinear matrix equation email protected b c 1b finally some numerical results will be given

two level toeplitz preconditioning approximation results for matrices and functions large 2 level toeplitz systems arise in a variety of applications see e g r h chan and m ng siam rev 38 1996 pp 427 482 for which efficient numerical methods for their solution are required some successful numerical techniques need the explicit knowledge of the generating function f of the considered system t n f rm x b an assumption that usually is not fulfilled in real applications in this paper we analyze and complete the procedure proposed in d noutsas s serra capizzano and p vassalos numer linear algebra appl 12 2005 pp 231 239 for the 2 level case in such a way from the knowledge of the coefficients of t n f we determine optimal preconditioning strategies for the solution of our systems finally some numerical experiments are performed and discussed in connection with our theoretical analysis

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

sequence detection in nonlinear channels a convenient alternative to analog predistortion a new maximum likelihood sequence detection receiver for spectrally efficient linear modulations on bandlimited bandpass nonlinear channels is proposed the receiver is based on oversampling the received signal corrupted by noise and nonlinear distortion contrary to other solutions in the literature in the proposed technique there is no need for a bank of matched filters and the receiver front end reduces to a single lowpass filter for a given peak power level a performance gain can be achieved over more traditional approaches to transmission on nonlinear channels such as those based on predistortion if a moderate spectral expansion is allowed to analyze the receiver performance the concept of distance spectrum is employed since the minimum distance alone cannot account for a reliable performance evaluation both analysis and simulation are carried out for realistic narrowband nonlinear channels possibly employing reduced state sequence detection appreciable gain margins are confirmed to be possible in these realistic cases

efficient algorithms for the evaluation of the eigenvalues of block banded toeplitz matrices on presente des algorithmes de calcul des valeurs propres des matrices bandes blocs de toeplitz

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

analysis of preconditioning strategies for collocation linear systems in the past few years several authors have studied the preconditioning of collocation matrices by finite differences fds matrices arising from the associated collocation points here we discuss how to solve in an efficient way nonuniform grid fd linear systems including those related to a generic fd collocation preconditioner the main idea is based on a further step of preconditioning defined in terms of diagonal and toeplitz matrices first we identify the limit spectral distributions of the involved fd collocation matrix sequences and then we prove that the proposed toeplitz based preconditioners assure a clustering at the unity with respect to the eigenvalues in the 1d case in the 2d case the situation is different so that more appropriate strategies are discussed a wide numerical experimentation emphasizing the correctness of the theoretical results is also reported

fuzzy goal programming with different importance and priorities abstract this paper formulates fuzzy goal programming fgp incorporating different importance and preemptive priorities by using an additive model to maximize the sum of achievement degrees of all fuzzy goals in contrast to previous works the proposed approach allows the decision maker to determine a desirable achievement degree for each fuzzy goal to reflect explicitly the relative importance of these goals this approach can generate a set of achievement degrees consistent with the decision maker s expectations even though the relative importance of the goals may change furthermore in this paper we incorporate the decision maker s preemptive priority structure into a single formulation the resulting solutions satisfy both the preemptive priority structure and have the maximum achievement degrees in sum the proposed approaches effectiveness and computational superiority over the existing approaches are demonstrated and compared with examples from the literature

boundary conditions and multiple image re blurring the lbt case in this paper we are concerned with deblurring problems in the case of multiple images coming from the large binocular telescope an important example of telescope of interferometric type for this problem we are interested in checking the role of the boundary conditions in the quality of the reconstructed image in particular we will consider reflective and anti reflective boundary conditions and the re blurring idea the results of the proposed combinations are quite satisfactory when compared with classical dirichlet or periodic boundary conditions especially when increasing the number of images acquired by the lbt this behavior is confirmed by a wide numerical experimentation

on parallel computations with banded matrices we devise parallel algorithms for solving a banded linear system of equations and for computing the determinant of a banded matrix substantially improving the previous record computational complexity estimates of e our algorithms are in nc or rnc and support new record bounds on the parallel time complexity of these computations and simultaneously the bounds on their potential work the product of time and processor bounds which match within constant or logarithmic factors the record sequential time bounds for the same computations moreover these algorithms are in nc1 or rnc1 if the bandwidth is a constant

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

deblurring methods using antireflective boundary conditions in this paper we consider the numerical solution of self adjoint deblurring problems on bounded intervals for these problems it has recently been shown that appropriate modeling of the solution near the boundary of the interval may significantly improve the numerical reconstructions among the alternatives the so called antireflective boundary condition appears to be the best known choice here we develop an appropriate i e stable and efficient implementation of this model in two steps namely by i transforming the problem to homogeneous boundary values and ii using the fast sine transform to solve the transformed problem this approach allows us to incorporate regularization in a very straightforward way numerical reconstructions are superior qualitatively and quantitatively to those obtained with the reblurring method of donatelli and serra capizzano

some theorems on linear positive operators and functionals and their applications abstract in this paper we prove two theorems concerning linear positive operators and functionals in a very general sense we show how these results can be used to derive new results in a wide range of applications from approximation theory to the numerical treatment of differential equations and to the numerical analysis connected with signals and images

convergence analysis of two grid methods for elliptic toeplitz and pdes matrix sequences summary the solution of large toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers 13 14 22 the technique was modified in 6 36 and a rigorous proof of convergence of the tgm two grid method was given in the special case where the generating function has only a zero at x 0 0 of order at most two here by extending the latter approach we perform a complete analysis of convergence of the tgm under the sole assumption that f is nonnegative and with a zero at x 0 0 of finite order an extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic pdes with nonconstant coefficients and of any order is then discussed

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 and efficient parallel solution of sparse linear systems this paper presents a parallel algorithm for the solution of a linear system a bf x bf b with a sparse n times n symmetric positive definite matrix a associated with the graph g a that has n vertices and has an edge for each nonzero entry of a if g a has an s n separator family and a known s n separator tree then the algorithm requires only o log 3 n time and e m s n log n processors for the evaluation of the solution vector bf x a 1 bf b where e is the number of edges in g a and m n is the number of processors sufficient for multiplying two n times n rational matrices in time o log n furthermore for this computational cost the algorithm computes a recursive factorization of a such that the solution of any other linear system a bf x bf b with the same matrix a requires only o log 2 n time and e log n s n 2 processors

a v cycle multigrid for multilevel matrix algebras proof of optimality we analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant tau hartley or are of toeplitz type in the case of matrix algebra linear systems we prove that the convergence rate is independent of the system dimension even in presence of asymptotical ill conditioning this happens iff f takes the zero value more precisely if the d level coefficient matrix has partial dimension nr at level r with r 1 ldots d then the size of the system is n varvec n prod r 1 d n r varvec n n 1 ldots n d and o n n operations are required by the considered v cycle multigrid in order to compute the solution within a fixed accuracy since the total arithmetic cost is asymptotically equivalent to the one of a matrix vector product the proposed method is optimal some numerical experiments concerning linear systems arising in 2d and 3d applications are considered and discussed

infinite series of interference variables with cantor type distributions the sum of an infinite series of weighted binary random variables arises in communications problems involving intersymbol and adjacent channel interference if the weighting decays asymptotically at least exponentially and if the decay is not too slow the sum has an unusual distribution which has neither a density nor a discrete mass function and therefore cannot be manipulated with usual techniques the distribution of the sum is given and the calculus for dealing with the distribution is presented it is shown that these cantor type random variables arise in a range of digital communications models and exact explicit expressions for performance measures such as the probability of error may be obtained several examples are given

some algorithms for solving special tridiagonal block toeplitz linear systems this paper is focused on different methods and algorithms for solving tridiagonal block toeplitz systems of linear equations we consider the el sayed method ph d thesis 1996 for such systems and propose several modifications that lead to different algorithms which we discuss in detail our algorithms are then compared with some classical techniques as far as implementation time is concerned number of operations and storage comments and conclusions for computing efficiency of the proposed new algorithms are given numerical experiments corroborating the theoretical results are also presented

solution of toeplitz normal equations by sine transform based preconditioning abstract the normal equations constructed by a toeplitz matrix are studied in order to find a suitable preconditioner related to the discrete sine transform new results are given about the structure of the product of two toeplitz matrices which allow the cgn method to achieve a superlinear rate of convergence this preconditioner outperforms the circulant one for the iterative solution of toeplitz least squares problems such strategy can also be applied to nonsymmetric linear systems a block generalization is discussed

computing preconditioners via subspace projection we give an alternative method for computing the matrix argmina pf where f is the frobenius norm and p is taken from the set of square matrices simultaneously diagonalized by a given orthogonal trigonometric transform specific examples are given

korovkin tests approximation and ergodic theory we consider sequences of s k n x t k n matrices a n f with a block structure spectrally distributed as an l 1 p variate s x t matrix valued function f and for any n we suppose that a n is a linear and positive operator for every fixed n we approximate the matrix a n f in a suitable linear space m n of s k n x t k n matrices by minimizing the frobenius norm of a n f x n when x n ranges over m n the minimizer x n is denoted by p k n a n f we show that only a simple korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts 1 the sequence p k n a n f is distributed as f 2 the sequence a n f p k n a n f is distributed as the constant function 0 i e is spectrally clustered at zero the first result is an ergodic one which can be used for solving numerical approximation theory problems the second has a natural interpretation in the theory of the preconditioning associated to cg like algorithms

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

multigrid methods for multilevel circulant matrices we introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form f x j n where f is continuous and independent of n n 1 ldots n d and x j n equiv 2 pi j n 2 pi j 1 n 1 ldots 2 pi j d n d 0 le j r le n r 1 the interest of the proposed technique pertains to the multilevel banded case where the total cost is optimal i e o n arithmetic operations ops n prod r 1 d n r instead of o n log n ops arising from the use of ffts in fact multilevel banded circulants are used as preconditioners for elliptic and parabolic pdes with dirichlet or periodic boundary conditions and for some two dimensional image restoration problems where the point spread function psf is numerically banded so that the overall cost is reduced from o k varepsilon n n log n to o k varepsilon n n where k varepsilon n is the number of pcg iterations to reach the solution within an accuracy of varepsilon several numerical experiments concerning one rank regularized circulant discretization of elliptic 2q differential operators over one dimensional and two dimensional square domains with mixed boundary conditions are performed and discussed

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

multi iterative methods abstract a general class of iterative methods is introduced for solving positive definite linear systems ax b these methods use two or more different iterative techniques and each of them reduces the error by a constant factor in a different subspace of r n the multigrid as an example of a multi iterative method combines these partial results and we reach a reduction of the error by at each step where is the number of applications of the first scheme and gt 1 is a constant independent of n in the second part we consider the prolongation operator p with a special antidiagonal structure and we show that the multigrid method for toeplitz matrices 1 is an example of a method of this class in a third part we extend the multigrid method to block toeplitz matrices obtaining a block antidiagonal structure for toeplitz and block toeplitz cases the choice of p which maximizes convergence speed simply depends on the coefficients of matrix a the numerical results on 1 d and 2 d elliptic differential problems confirm the effectiveness of this multi iterative method in fact its spectral radius is very small and independent of the mesh spacing h

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

fast discrete transforms by means of eigenpolynomials abstract let a a ij be an n n matrix consider the discrete transform u a u and associate with the j th column of a the eigenpolynomial a j x n 1 i 0 a ij x j the properties of eigenpolynomials play an important role in the case where a is a matrix of eigenvectors of a toeplitz matrix 1 2 here we consider the cases where a is the matrix defining the discrete fourier transform dft the discrete hartley transform dht the discrete sine transform dst and the discrete cosine transform dct in its two versions of 3 and 4 for each eigenpolynomial of each transform we explicitly determine all its zeros we use eigenpolynomials as a unifying tool for describing the decimation in frequency dif versions of the main known algorithms for dft moreover by using the information about the zeros of the eigenpolynomials we devise new algorithms for dft dht dst and dct which reach or improve in the case of dst and dct the record complexity bounds without requiring the use of the algorithm for complex multiplication with three multiplications and three additions 5 and of the implicit preconditioning

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

a new algebra of toeplitz plus hankel matrices and applications we introduce a new simultaneously diagonalizable real algebra a of symmetrical centrosymmetrical matrices having a toeplitz plus hankel structure we give the corresponding orthonormal basis of eigenvectors which are alternately symmetrical and skewsymmetrical vectors an application is the construction of a symmetrical toeplitz plus centrosymmetrical hankel matrix of equal row sums having a prescribed real spectrum this matrix can be used as the starting matrix for symmetrical centrosymmetrical isospectral flows in particular for the isospectral flow corresponding to the construction of a regular toeplitz matrix having prescribed eigenvalues moreover if a is a noise representation of an unknown matrix in a of rank k then we give a procedure to approximate a by a matrix in a of rank k

optimal preconditioning for image deblurring with anti reflective boundary conditions abstract inspired by the theoretical results on optimal preconditioning stated by ng chan and tang in the framework of reflective boundary conditions bcs in this paper we present analogous results for anti reflective bcs here a key technical difficulty is represented by the non orthogonal character of the anti reflective transform and indeed the proof proposed by ng chan and tang does not work nevertheless in both cases the optimal preconditioner is the blurring matrix associated to the symmetrized point spread function psf the geometrical idea on which our proof is based is very simple and general so it may be useful in the future to prove theoretical results for new proposed bcs numerical tests show that the optimal preconditioning strategy is effective when using both preconditioned conjugate gradient methods and recently introduced nonstationary preconditioned iterations

the conditioning of fd matrix sequences coming from semi elliptic differential equations abstract in this paper we are concerned with the study of spectral properties of the sequence of matrices a n a coming from the discretization using centered finite differences of minimal order of elliptic or semielliptic differential operators l a u of the form 1 d d x a x d d x u x f x on 0 1 dirichlet b c on where the nonnegative bounded coefficient function a x of the differential operator may have some isolated zeros in more precisely we state and prove the explicit form of the inverse of a n a and some formulas concerning the relations between the orders of zeros of a x and the asymptotic behavior of the minimal eigenvalue condition number of the related matrices as a conclusion and in connection with our theoretical findings first we extend the analysis to higher order semi elliptic differential operators and then we present various numerical experiments showing that similar results must hold true in 2d as well

korovkin theorems and linear positive gram matrix algebra approximations of toeplitz matrices abstract in this paper we are concerned with the approximation of toeplitz matrices generated by continuous 2 periodic functions f i r with i for this purpose we define a class of matrix algebras related to suitable choices of gram functions where we look for good preconditioners in particular we construct these preconditioners through linear operators and linear positive operators lpos approximating in some sense the function f then by making use of some matrix versions 39 42 41 of the korovkin and weierstrass theorems we analyze the convergence features of old and new preconditioners finally among the given results are adapted in order to deal with l1 generating functions and the related preconditioners are compared with the ones devised by using band toeplitz matrices 6 36

centrosymmetric isospectral flows and some inverse eigenvalue problems this paper is concerned with the solution of some structured inverse eigenvalue problems in the class of centrosymmetric matrices for this aim isospectral flows evolving in the space of centrosymmetric matrices are considered to numerically construct a symmetric toeplitz matrix or a persymmetric hankel matrix from prescribed eigenvalues we establish a link between the two problems and we investigate the use of simultaneously diagonalizable algebra based on sine transform linear algebra appl 52 53 1983 992 to choose the starting centrosymmetric matrices for the isospectral flows some numerical tests show that our approach can tackle both problems when the solvability is guaranteed and it can give good insights when the existence of the solution is not guaranteed

band toeplitz preconditioners for block toeplitz systems abstract we consider the solutions of block toeplitz systems with toeplitz blocks by the preconditioned conjugate gradient pcg method here the block toeplitz matrices are generated by nonnegative functions f x y we use band toeplitz matrices as preconditioners the generating functions g x y of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing f g f we prove that the condition number of the preconditioned system is o 1 an a priori bound on the number of iterations for convergence is obtained

on algebras of toeplitz plus hankel matrices we characterize a wide class of maximal algebras of toeplitz plus hankel matrices by exploiting properties of displacement operators

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

a direct method for solving block circulant banded system of linear equations abstract a direct method to solve block circulant banded system of linear equations is presented in the paper the method applies the kernel solution of linear equations with circulant banded matrix proposed by fuyong 2011 17 by using the kernel solution of linear equations with block circulant banded matrix we obtain the boundary equations of the linear equations and by solving the boundary equations of the linear systems we finally get the solution of the linear equations the proposed method has great advantage in getting solutions of linear systems with block circulant banded matrix coefficient matrices an example is presented in the paper

fast deconvolution with approximated psf by rstls with antireflective boundary conditions the problem of reconstructing signals and images from degraded ones is considered in this paper the latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image moreover the coefficient matrix is very ill conditioned requiring an additional regularization term different boundary conditions can be proposed in this paper antireflective boundary conditions are considered since both sides of the linear system have uncertainties and the coefficient matrix is highly structured the regularized structured total least squares approach seems to be the more appropriate one to compute an approximation of the true signal image with the latter approach the original problem is formulated as an highly nonconvex one and seldom can the global minimum be computed it is shown that regularized structured total least squares problems for antireflective boundary conditions can be decomposed into single variable subproblems by a discrete sine transform such subproblems are then transformed into one dimensional unimodal real valued minimization problems which can be solved globally some numerical examples show the effectiveness of the proposed approach

on the spectrum of stiffness matrices arising from isogeometric analysis we study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems motivated by the applicative interest in the fast solution of the related linear systems we are looking for a spectral characterization of the involved matrices in particular we investigate non singularity conditioning extremal behavior spectral distribution in the weyl sense as well as clustering of the eigenvalues to a certain compact subset of mathbb c c all the analysis is related to the notion of symbol in the toeplitz setting and is carried out both for the cases of 1d and 2d problems

block band toeplitz preconditioners derived from generating function approximations analysis and applications we are concerned with the study and the design of optimal preconditioners for ill conditioned toeplitz systems that arise from a priori known real valued nonnegative generating functions f x y having roots of even multiplicities our preconditioned matrix is constructed by using a trigonometric polynomial x y obtained from fourier kernel approximations or from the use of a proper interpolation scheme both of the above techniques produce a trigonometric polynomial x y which approximates the generating function f x y and hence the preconditioned matrix is forced to have clustered spectrum as x y is chosen to be a trigonometric polynomial the preconditioner is a block band toeplitz matrix with toeplitz blocks and therefore its inversion does not increase the total complexity of the pcg method preconditioning by block toeplitz matrices has been treated in the literature in several papers we compare our method with their results and we show the efficiency of our proposal through various numerical experiments