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

first order transitions in spherical models finite size scaling finite size behavior near the first order phase boundary of ferromagnetic spherical models is investigated for block and cylinder shaped systems ind dimensions the bulk thermodynamic singularities are rounded and asymptotically for large size obey appropriate scaling laws both short range interactions and long range couplings decaying like 1 rd with 0 are analyzed the short range results agree precisely with a recently developed scaling theory foro n symmetric systems in the limitn more generally the scaling functions are universal depending only on explicit aspects of the shape and interactions enter only in the spin wave or goldstone mode contributions which appear technically as corrections to scaling an appendix analyzes the truncation error in the approximation by many fold sums of multivariate integrals with integrands diverging like jaj j2 as 0

the circulant operator in the banach algebra of matrices abstract we study an operator c which maps every n by n matrix a n to a circulant matrix c a n that minimizes the frobenius norm a n c n f over all n by n circulant matrices c n the circulant matrix c a n called the optimal circulant preconditioner has proved to be a good preconditioner for a general class of toeplitz systems in this paper we give different formulations of the operator discuss its algebraic and geometric properties and compute its operator norms in different banach algebras of matrices using these results we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of i c n 1 a n f over all nonsingular circulant matrices c n

finite size effects in a field theoretic model with long range exchange interaction we present a systematic approach to the calculation of finite size fs effects for ano n field theoretic model with both short range sr and long range lr exchange interactions the lr exchange interaction decays at large distances as 1 rd 2 2 0 0 renormalization group calculations ind du e are performed for a system with a fully finite block geometry under periodic boundary conditions we calculate the fs shift of the critical temperature and the fs renormalized coupling constant of the model to one loop order the universal scaling variable is obtained and the fs scaling hypothesis is verified

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

on finite size scaling in the presence of dangerous irrelevant variables a scaling hypothesis on finite size scaling in the presence of a dangerous irrelevant variable is formulated for systems with long range interaction and general geometryld d d a characteristic length which obeys a universal finite size scaling relation is defined the general conjectures are based on exact results for the mean spherical model with inverse power law interaction

finite size scaling investigations in the quantum varphi4 model with long range interaction in this paper we study in detail the critical behaviour of the n quantum 4 model with long range interaction decaying with distances r by a power law as r d in the large n limit the zero temperature critical behaviour is discussed its alteration by finite temperature and or finite sizes in the space is studied the scaling behaviours are studied in different regimes depending upon whether the finite temperature or the finite sizes of the system lead a number of results for the correlation length critical amplitudes and the finite size shift for different dimensionalities between the lower d 3 2 critical dimensions are calculated

preconditioning toeplitz plus diagonal linear systems using the sherman morrison woodbury formula in order to solve the toeplitz plus diagonal linear systems arising from image restorations efficiently we propose a sparse approximate inverse preconditioner based on the sherman morrison woodbury formula the preconditioner can be constructed through an incomplete factorization combined with some dropping strategies when the preconditioner is applied to the conjugate gradient method for solving the toeplitz plus diagonal linear systems numerical results show that our preconditioning method is more effective than other existing ones

t 0 finite size scaling for a quantum system with long range interaction abstract the general idea of finite size scaling are tested by the example of the pure quantum version of an exactly solvable model of structural phase transition with long range interaction decaying at large distances r as r d where d is the space dimensionality and 0 the finite size shift of the quantum parameter playing the same role as that of the temperature for classical systems is calculated for the general geometry ld d d with 0 0 2

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

finite size scaling for the mean spherical model with inverse power law interaction a new analytical technique based on integral transformations with mittag leffler type kernels is used to derive the finite size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential the asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied

a probabilistic view on finite size scaling in infinitely coordinated spherical models abstract it is shown that the finite size scaling functions of the infinitely coordinated spherical model are closely related to the limit probability distribution of a triangular array of properly normalized block spin variables the triangular array is defined with the aid of a two parameter family of gibbs distributions which approach the critical point together with the increase of the size of the system our consideration confirms the modification of finite size for infinitely coordinated systems suggested by botet jullien and pfeuty and extends it to the nonzero field case the explicit results rigorously derived here manifest the ensemble dependence of the relevant statistical distributionss which in turn produce different finite size scaling functions we demonstrate how the finite size scaling functions of the free energy can be determined from the relations between its first derivatives with respect to the temprature and magnetic field and expectation values of block spin variables whose asymptotic behavior is given by the smooth limit distribution a general conjecture pertaining to other models is stated too

morphological classification of brains via high dimensional shape transformations and machine learning methods a high dimensional shape transformation posed in a mass preserving framework is used as a morphological signature of a brain image population differences with complex spatial patterns are then determined by applying a nonlinear support vector machine svm pattern classification method to the morphological signatures significant reduction of the dimensionality of the morphological signatures is achieved via wavelet decomposition and feature reduction methods applying the method to mr images with simulated atrophy shows that the method can correctly detect subtle and spatially complex atrophy even when the simulated atrophy represents only a 5 variation from the original image applying this method to actual mr images shows that brains can be correctly determined to be male or female with a successful classification rate of 97 using the leave one out method this proposed method also shows a high classification rate for old adults age classification even under difficult test scenarios the main characteristic of the proposed methodology is that by applying multivariate pattern classification methods it can detect subtle and spatially complex patterns of morphological group differences which are often not detectable by voxel based morphometric methods because these methods analyze morphological measurements voxel by voxel and do not consider the entirety of the data simultaneously

finite size crossover in systems with slab geometry abstract a renormalization group study of the finite size dimensional crossover is carried out with the help pf e 4 d and e0 3 d expansion techniques the finite size crossover and the invariance relation for the length scale transformation are proven up to the two loop approximation the formal equivalence between the finite size crossover in classical systems and the quantum to classical dimensional crossover in certain quantum statistical models is emphasized and exploited the finite size corrections to the fluctuation shift of the critical temperature and the width of the critical region are investigated it is shown that the shift exponent describing the fractional rounding of the critical temperature obeys the relation d 2 where d is the dimensionality of the system

circulant preconditioners constructed from kernels circulant preconditioners for hermitian toeplitz systems are considered from the viewpoint of function theory it is shown that some well known circulant preconditioners can be derived from convoluting the generating function f of the toeplitz matrix with famous kernels like the dirichlet and the fejer kernels several circulant preconditioners are then constructed using this approach finally it is proven that if the convolution product converges to f uniformly then the circulant preconditioned toeplitz systems will have a clustered spectrum

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

band preconditioners for block toeplitz toeplitz block systems abstract preconditioned conjugate gradient methods are employed to solve symmetric positive definite m by m block toeplitz with n by n toeplitz block systems am nx b where am n are generated by 2 periodic nonnegative functions with zeros serra has proposed using band block toeplitz with band toeplitz block matrices bm n with their external and internal bandwidths independent of m and n as preconditioners serra showed that if the hessians of the generating function at the zeros are positive definite then the condition number of b 1m nam n is uniformly bounded by a constant independent of m and n whereas the condition number of am n tends to infinity as m and n tend to infinity in this paper we provide a method for deriving band preconditioners for block toeplitz toeplitz block matrices numerical examples are given to illustrate the performance of the method

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 the limit gibbs states of the spherical model the effect of a class of hamiltonian perturbations vanishing in the thermodynamic limit on the limit gibbs states of the spherical and mean spherical models is studied the perturbation term is taken in the form of interaction energy with uniform magnetic field of strength h0n alpha where h0 in r1 and alpha 0 are parameters and n is the number of particles for fixed temperatures below the critical temperature in the absence of constant external magnetic fields and at alpha 1 the authors obtain convex sets of different mixed gibbs states parametrised by h0 a natural one parameter generalisation of the kac thompson transformation kernel which relates the states of the mean spherical model to the states of the spherical model is found when 0 alpha 1 and h0 not 0 or alpha 1 and h0 or infinity this kernel becomes a delta function even below the critical temperature then the states in both ensembles coincide with each other and with one of the two depending on the sign of h0 extreme points the case of alpha 1 is found to lead to the well known results corresponding to the absence of perturbation h0 0

finite size effects in the approximating hamiltonian method abstract the husimi temperley mean spherical model in which each two particles interact with equal strength is considered this model is shown to be equivalent to a d dimensional model with periodic boundary conditions and interaction potential j r where j r r d as r 0 being a parameter in the limit 0 it is found that the approximating hamiltonian method yields singular finite size scaling functions both in the neighbourhood of the critical point and near a first order phase transition a modification of this method is suggested which allows for all the essential configurations and reproduces the exact finite size scaling near a first order phase transition

fft based preconditioners for toeplitz block least squares problems discretized two dimensional deconvolution problems arising e g in image restoration and seismic tomography can be formulated as least squares computations min b tx 2 where t is often a large scale rectangular toeplitz block matrix the authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant block and related preconditioners constructed from the blocks of the rectangular matrix t preconditioning with such matrices allows efficient implementation using the one dimensional or two dimensional fast fourier transform fft two block preconditioners related to those proposed by t chan and j olkin for square nonsingular toeplitz block systems are derived and analyzed it is shown that for important classes of t the singular values of the preconditioned matrix are clustered around one this extends the authors earlier work on preconditioners for toeplitz least squares iterations for one dimensiona

structured condition numbers of structured tikhonov regularization problem and their estimations both structured componentwise and structured normwise perturbation analysis of the tikhonov regularization are presented the structured matrices under consideration include toeplitz hankel vandermonde and cauchy matrices structured normwise mixed and componentwise condition numbers for the tikhonov regularization are introduced and their explicit expressions are derived for the general linear structure based on the derived expressions we prove structured condition numbers are smaller than their corresponding unstructured counterparts by means of the power method and small sample statistical condition estimation sce fast condition estimation algorithms are proposed our estimation methods can be integrated into tikhonov regularization algorithms that use the generalized singular value decomposition gsvd for large scale linear structured tikhonov regularization problems we show how to incorporate the sce into the preconditioned conjugate gradient pcg method to get the posterior error estimations the structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones and the proposed condition estimation algorithms are reliable also an image restoration example is tested to show the effectiveness of the sce for large scale linear structured tikhonov regularization problems

toeplitz preconditioners for hermitian toeplitz systems abstract we propose a new type of preconditioners for hermitian positive definite toeplitz systems a n x b where a n are assumed to be generated by functions f that are positive and 2 periodic our approach is to precondition a n by the toeplitz matrix a n generated by 1 f we prove that the resulting preconditioned matrix a n a n will have clustered spectrum when a n cannot be formed efficiently we use quadrature rules and convolution products to construct nearby approximations to a n we show that the resulting approximations are toeplitz matrices which can be written as sums of circulant matrices as a side result we prove that any toeplitz matrix can be written as a sum of circulant matrices we then show that our toeplitz preconditioners t n are generalizations of circulant preconditioners and the way they are constructed is similar to the approach used in the additive schwarz method for elliptic problems we finally prove that the preconditioned systems t n a n will have clustered spectra around 1

bttb preconditioners for bttb least squares problems abstract in this paper we consider solving the least squares problem min x b t x 2 by using preconditioned conjugate gradient pcg methods where t is a large rectangular matrix which consists of several square block toeplitz toeplitz block bttb matrices and b is a column vector we propose a bttb preconditioner to speed up the pcg method and prove that the bttb preconditioner is a good preconditioner we then discuss the construction of the bttb preconditioner numerical examples including image restoration problems are given to illustrate the efficiency of our bttb preconditioner numerical results show that our bttb preconditioner is more efficient than the well known level 1 and level 2 circulant preconditioners

finite size scaling for systems with long range interactions abstract the present review is devoted to the fundamental problems of finite size scaling due to the presence of long range interactions the attention is focused on the precise formulation of critical finite size scaling in the case that the bulk correlation length is identically infinite the hypotheses based on the notion of a finite size scaling length are formulated in a way which treats the cases of short and long range interactions on equal grounds as well as the standard and modified above the upper critical dimensionality finite size scaling the general conjectures are tested on the exactly solvable mean spherical model with power law interaction special attention is paid to the adequate mathematical techniques the formulation of finite size scaling at first order phase transitions is also discussed since the finite size scaling length enters into the definition of the relevant scaled external field variable provided the system has d 0 infinite dimensions the review closes with a summary and short discussion of some open problems

modified finite size scaling for anharmonic crystals with quantum fluctuations abstract the modified finite size scaling for dimensions d d where d is the upper critical dimensionality is verified in the closed vicinity of both the classical and the quantum multicritical points by the example of a quantum model for an anharmonic cyrstal confined to a fully finite block geometry under periodic boundary conditions unified scaling equations corresponding to the two kinds of correlation lengths related with the fixed dimensionless temperature t when the quantum parameter is varied and with fixed when t is varied are obtained

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

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

finite size scaling for the correlation function of the spherical model with long range interactions finite size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry ld d d periodic boundary conditions and long range interactions decaying like r d at large distances r with arbitrary real parameters 0 2 d 2 d 0 the analytical technique used is based on integral transformations with kernels of mittag leffler type it makes it possible to easily generalize a number of results available only for short range interactions or in special cases of the ld d d geometry the effective correlation length is identified at arbitrary temperatures from the finite size large distance asymptotic behavior of the pair correlation function it is explicitly shown that the finite size scaling functions are not singular in the e expansion when e 0 neither at d e nor at d 2 e moreover the applicability of the e expansion is tested in the specific finite size case of d e the definition of the scaled fiel

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

inverse toeplitz preconditioners for ill posed problems abstract it has been shown recently that iterative regularization using conjugate gradient type methods for image restoration problems can be effectively preconditioned with circulant approximations here it is shown that the theoretical properties of this approach are not restricted to circulant matrices specifically a toeplitz approximate inverse preconditioning scheme for discrete ill posed problems is considered it is proved that the preconditioned system approximates the prolate matrix and that this property implies that fast convergence of conjugate gradient type methods can be expected in addition it is shown that these results can be generalized to two dimensional problems an image restoration application is used to demonstrate the properties of the preconditioner

an investigation of finite size scaling for systems with long range interaction the spherical model a method is suggested for the derivation of finite size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr d whered is the space dimensionality and 0 it allows for a unified treatment of short range 2 and long range 2 interaction the asymptotic analysis is illustrated by the mean spherical model of general geometryld d d subject to periodic boundary conditions the fisher privman equation of state is generalized to arbitrary real values ofd 0 d it is shown that thee expansion may be used to study the breakdown of standard finite size scaling at the borderline dimensionalities

classical dipoles on a finite triangular lattice the spherical model approximation abstract in the framework of the spherical model it is shown that a phase transition at finite temperature to a ferromagnetic phase in a two dimensional system with dipole dipole and exchange interactions on a lattice with arbitrary rhombicity angle exists if the ferromagnetic ground state is not continuously degenerate in spin direction when the degeneracy exists the system exhibits essential singularities in the thermodynamic functions at t 0 finite size effects in such systems are investigated in the example of the triangular latice under periodic boundary conditions the existence of only one relevant variable determining the crossover from infinite to finite size behavior is found as t 0 is approached despite that there are two characteristic lengths determining the scale of variation of the correlation function the validity of the general finite size scalling relations in terms of this variable is established