flattening and analytic continuation of affinoid morphisms remarks on a paper of gardener and schoutens we give an example of an affinoid curve without analytic continuation we use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow ups thus the global rigid analogue of hironaka s complex analytic flattening theorem given by t gardener and h schoutens in theorem 2 3 of flattening and subanalytic sets in rigid analytic geometry proc london math soc 3 83 2001 681 707 is not true since this is a key step in the proof of the affinoid elimination theorem loc cit theorem 3 12 that proof contains a serious gap we also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map this clarifies the relationship among several natural categories of rigid subanalytic sets

absolute equal distribution of families of finite sets let u uin i 1dn n n0 and v vin i 1dn n n0 where u1n u2n udn n v1n v2n vdn n n n0 and limn dn let f be a set of continuous real valued functions on r then u and v are equally distributed with respect to f if i 1dn f uin f vin o dn f f or absolutely equally distributed with respect to f if i 1dn f uin f vin o dn f f we show that these definitions are equivalent if f f c r limx f x andlimx f x exist finite and we give sufficient conditions for u and v to be absolutely equally distributed with respect to f f fisboundedanduniformlycontinuousonr and f f c r fisbounded

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

clusters preconditioners convergence abstract we present a technique that relates the existence of a cluster of singular values to the existence of a cluster of eigenvalues we also analyze some geometrical properties of eigenvalue clusters and their clusterization speed

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

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

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

the spectrum of circulant like preconditioners for some general linear multistep formulas for linear boundary value problems the spectrum of the eigenvalues the conditioning and other related properties of circulant like matrices used to build up block preconditioners for the nonsymmetric algebraic linear equations of time step integrators for linear boundary value problems are analyzed moreover results concerning the entries of a class of toeplitz matrices related to the latter are proposed generalizations of implicit linear multistep formulas in boundary value form are considered in more detail it is proven that there exists a new class of approximations which are well conditioned and whose eigenvalues have positive and bounded real and bounded imaginary part moreover it is observed that preconditioners based on other circulant like approximations which are well suited for hermitian linear systems can be severely ill conditioned even if the matrices of the nonpreconditioned system are well conditioned

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

circulant preconditioners for complex toeplitz matrices the solution of n by n complex toeplitz systems a n x b by the preconditioned conjugate gradient method is studied the preconditioner c n is the circulant matrix that minimizes b n a n f over all circulant matrices b n the authors prove that if the generating function of a n is a 2 mu periodic continuous complex valued function without any zeros then the spectrum of the normalized preconditioned matrix c n 1 a n c n 1 a n will be clustered around one hence they show that if the condition number of a n is of o n alpha the conjugate gradient method when applied to solving the normalized preconditioned system converges in at most o alpha log n 1 steps thus the total complexity of the algorithm is o alpha n log 2 n 1 log n

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

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

the positive semidefiniteness of partitioned matrices abstract the positive semidefiniteness of a partitioned matrix is characterized in terms of its submatrices the result is applied to a variety of problems concerning lowner ordered matrices which need not be partitioned themselves

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

the determinacy of common factors in large item domains bounds are here developed for the multiple correlation of common factors with the items whose factors they are it is then easy to see under broad but not completely general conditions the circumstances under which an infinite item domain does or does not perfectly determine selected subsets of its common factors

regression among factor scores structural equation models with latent variables are sometimes estimated using an intuitive three step approach here denoted factor score regression consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest in this simple example estimation of the structural parameter proceeds as follows first common factor models areseparately estimated for each latent variable second factor scores areseparately assigned to each latent variable based on the estimates third ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter we investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables it is demonstrated that the conventional approach to factor score regression performs very badly revised factor score regression using regression factor scores for the explanatory latent variables and bartlett scores for the response latent variables produces consistent estimators for all parameters

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

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

on the distribution of the singular values of toeplitz matrices in 1920 g szego proved a basic result concerning the distribution of the eigenvalues n j of the toeplitz sections tn f where f l is a real valued function simple examples show that this result cannot hold in the case where f is not real valued in this note we give an extension of this theorem for the singular values of tn f when f f0 r0 with f0 real valued and r0 continuous periodic with period 2 and such that r0 1 in addition we apply the basic theorem of szego to resolve a question of c moler

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

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

a note on the regularizing preconditioning of g toeplitz sequences via g circulants for a given nonnegative integer g a matrix a n of size n is called g toeplitz if its entries obey the rule a n a r g s r s 0 n 1 analogously a matrix a n again of size n is called g circulant if a n a r g s mod n r s 0 n 1 in a recent work we studied the asymptotic properties in terms of spectral distribution of both g circulant and g toeplitz sequences in the case where a k can be interpreted as the sequence of fourier coefficients of an integrable function f over the domain here we are interested in the preconditioning problem which is well understood and widely studied in the last three decades in the classical toeplitz case i e for g 1 in particular we consider the generalized case with g 2 and the nontrivial result is that the preconditioned sequence p n p n 1 a n where p n is the sequence of preconditioner cannot be clustered at 1 so that the case of g 1 is exceptional however while a standard preconditioning cannot be achieved the result has a potential positive implication since there exist choices of g circulant sequences which can be used as basic preconditioning sequences for the corresponding g toeplitz structures generalizations to the block and multilevel case are also considered where g is a vector with nonnegative integer entries a few numerical experiments related to a specific application in signal restoration are presented and critically discussed

a classification scheme for regularizing preconditioners with application to toeplitz systems preconditioning techniques for linear systems are widely used in order to speed up the convergence of iterative methods if the linear system is generated by the discretization of an ill posed problem preconditioning may lead to wrong results since components related to noise on input data are amplified using basic concepts from the theory of inverse problems we identify a class of preconditioners which acts as a regularizing tool in this paper we study relationships between this class and previously known circulant preconditioners for ill conditioned hermitian toeplitz systems in particular we deal with the low pass filtered optimal preconditioners and with a recent family of superoptimal preconditioners we go on to describe a set of preconditioners endowed with particular regularization properties whose effectiveness is supported by several numerical tests

a unifying approach to abstract matrix algebra preconditioning in earlier papers tyrtyshnikov 42 and the first author 14 considered the analysis of clustering properties of the spectra of specific toeplitz preconditioned matrices obtained by means of the best known matrix algebras here we generalize this technique to a generic banach algebra of matrices by devising general preconditioners related to convergent approximation processes 36 finally as case study we focus our attention on the tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric toeplitz systems can be constructed in this class

implications of indeterminate factor error covariances for factor construction prediction and determinacy the assumptions of the model for factor analysis do not exclude a class of indeterminate covariances between factors and error variables grayson 2003 the construction of all factors of the model for factor analysis is generalized to incorporate indeterminate factor error covariances a necessary and sufficient condition is given for indeterminate factor error covariances to be arbitrarily small for mean square convergence of the regression predictor of factor scores and for the existence of a unique determinate factor and error variable the determinate factor and error variable are uncorrelated and satisfy the defining assumptions of factor analysis several examples are given to illustrate the results

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

flattening and subanalytic sets in rigid analytic geometry let k be an algebraically closed field endowed with a complete non archimedean norm with valuation ring r let f colon y to x be a map of k affinoid varieties in this paper we study the analytic structure of the image f y subset x such an image is a typical example of a subanalytic set we show that the subanalytic sets are precisely the mathbf d semianalytic sets where mathbf d is the truncated division function first introduced by denef and van den dries this result is most conveniently stated as a quantifier elimination result for the valuation ring r in an analytic expansion of the language of valued rings to prove this we establish a flattening theorem for affinoid varieties in the style of hironaka which allows a reduction to the study of subanalytic sets arising from flat maps that is we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups the case of a flat map is then dealt with by a small extension of a result of raynaud and gruson showing that the image of a flat map of affinoid varieties is open in the grothendieck topology using embedded resolution of singularities we derive in the zero characteristic case a uniformization theorem for subanalytic sets a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres as a corollary we obtain the fact that any subanalytic set in the plane r2 is semianalytic 2000 mathematical subject classification 32p05 32b20 13c11 12j25 03c10

conditions for factor in determinacy in factor analysis the subject of factor indeterminacy has a vast history in factor analysis guttman 1955 lederman 1938 wilson 1928 it has lead to strong differences in opinion steiger 1979 the current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables five conditions are given which rigorously define indeterminacy it is shown that un observable factors are in determinate specifically the indeterminacy proof by guttman is extended to heywood cases the results are illustrated by two examples and implications for indeterminacy are discussed

superoptimal preconditioned conjugate gradient iteration for image deblurring we study the superoptimal frobenius operators in the two level circulant algebra we consider two specific viewpoints 1 the regularizing properties in imaging and 2 the computational effort in connection with the preconditioned conjugate gradient method some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed

rigid subanalytic subsets of the line and the plane working over an algebraically closed complete non archimedean non trivially valued field we show that any subanalytic subset of the line is semialgebraic and that any subanalytic subset of the plane is semianalytic

circulant preconditioners with unbounded inverses abstract the eigenvalue and singular value distributions for matrices s 1 n a n and c 1 n a n are examined where a n s n and c n are toeplitz matrices simple circulants and optimal circulants generated by the fourier expansion of some function f recently it has been proven that a cluster at 1 exists whenever f is from the wiener class and strictly positive both restrictions are now weakened a proof is given for the case when f may take the zero value and hence the circulants are to have unbounded inverses the main requirements on f are that it belong to l 2 and be in some sense sparsely vanishing specifically if f is nonnegative and circulants s n or c n are positive definite then the eigenvalues of s 1 n a n or c 1 n a n are clustered at 1 if f is complex valued and s n or c n are nonsingular then the singular values of s 1 n a n or c 1 n a n are clustered at 1 as well also proposed and studied are the improved circulants it is shown that improved simple circulants can be much more advantageous than optimal circulants this depends crucially on the smoothness properties of f further clustering on theorems are given that pertain to multilevel toeplitz matrices preconditioned by multilevel simple and optimal circulants

on the construction of all factors of the model for factor analysis a construction method is given for all factors that satisfy the assumptions of the model for factor analysis including partially determined factors where certain error variances are zero various criteria for the seriousness of indeterminacy are related it is shown that green s 1976 conjecture holds for a linear factor predictor the mean squared error of prediction is constant over all possible factors a simple and general geometric interpretation of factor indeterminacy is given on the basis of the distance between multiple factors it is illustrated that variable elimination can have a large effect on the seriousness of factor indeterminacy a simulation study reveals that if the mean square error of factor prediction equals 5 then two thirds of the persons are correctly selected by the best linear factor predictor

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

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

on bilinear forms in gaussian random variables and toeplitz matrices we improve a result of szeg on the asymptotic behaviour of the trace of products of toeplitz matrices

a note on sufficient conditions that a common factor will be determinate in an infinite domain of variables it is proved for the common factor model withr common factors that under certain condition s which maintain the distinctiveness of each common factor a given common factor will be determinate if there exists an unlimited number of variables in the model each having an absolute correlation with the factor greater than some arbitrarily small positive quantity

which circulant preconditioner is better the eigenvalue clustering of matrices s n 1 a n and c n 1 a n is experimentally studied where a n s n and c n respectively are toeplitz matrices strang and optimal circulant preconditioners generated by the fourier expansion of a function f x some illustrations are given to show how the clustering depends on the smoothness of f x and which preconditioner is preferable an original technique for experimental exploration of the clustering rate is presented this technique is based on the bisection idea and on the toeplitz decomposition of a three matrix product cac where a is a toeplitz matrix and c is a circulant in particular it is proved that the toeplitz displacement rank of cac is not greater than 4 provided that c and a are symmetric

absolute equal distribution of the spectra of hermitian matrices abstract if x if x x if x if x let a n b n p n where b n and p n are n n hermitian matrices we show that if p n f 2 o n then for any a i 1 n f mid i a n f mid i b n o n if f c eigenvalues numbered in nondecreasing order we consider the special case where p n are real hankel matrices we also show that if rank p n o n then a holds for every and f c combining these results yields a result concerning c n b n e n r n where e n 2 f o n and rank r n o n we also consider the case where the conditions on e n are stated in terms of schatten p norms finally we show that if t n are hermitian toeplitz matrices generated by f c with minimum m f and maximum m f 2 i 1 n n in 2 i n n 1 i n and n is a permutation of 1 2 n such that f n 1 n f n 2 n f n n n then i 1 n f i t n f f n i n o n if f c m f m f

superoptimal approximation for unbounded symbols abstract the superoptimal frobenius approximation of toeplitz matrices is considered in connection with the case of unbounded symbols in particular we use the superoptimal approximation as preconditioner for the cg method when a fisher hartwig singularity is present in the symbol with special regard to systems coming from times series and financial applications a theoretical discussion concerning classical circulant preconditioners and a numerical comparison with the strang and with the optimal approximations are presented particularly with reference to the presence of noise

on the relationship between item response theory and factor analysis of discretized variables equivalence of marginal likelihood of the two parameter normal ogive model in item response theory irt and factor analysis of dichotomized variables fa was formally proved the basic result on the dichotomous variables was extended to multicategory cases both ordered and unordered categorical data pair comparison data arising from multiple judgment sampling were discussed as a special case of the unordered categorical data a taxonomy of data for the irt and fa models was also attempted

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

the spectrum of a family of circulant preconditioned toeplitz systems the solutions of symmetric positive definite toeplitz systems ax b are studied by the preconditioned conjugate gradient method the preconditioner is the circulant matrix c that minimizes the frobenius norm c a f t chan an optimal circulant preconditioner for toeplitz systems ucla department of mathematics cam report 87 06 june 1987 the convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of c 1 a for toeplitz matrix a with entries which are fourier coefficients of a positive function in the wiener class this paper establishes the invertibility of c finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix c 1 a as the dimension increases and proves that they are clustered around 1