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

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

multivariate design of experiments for engineering dimensional analysis we consider the design of dimensional analysis experiments when there is more than a single response we first give a brief overview of dimensional analysis experiments and the dimensional analysis

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

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

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

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

orthogonal multiwavelets of multiplicity four abstract we consider solutions of a system of refinement equations with a 4 1 function vector and three nonzero 4 4 coefficient matrices we give explicit expressions of coefficient matrices such that the refinement function vector and the corresponding wavelet vector have properties of short support 0 2 symmetry or antisymmetry and orthogonality the properties of convergence of the subdivision scheme approximation order and smoothness of the refinement functions are also discussed

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 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

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

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

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

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

matrix refinement type equations abstract taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the fourier transform of an l 1 solution of the following random matrix refinement type equation f x det l c f l x m p d where m r p is a random vector and l r p p c r q q are random matrices moreover the dimension of the space of all l 1 solutions of this refinement type equation is examined

a matrix valued wavelet kl like expansion for wide sense stationary random processes matrix valued wavelet series expansions for wide sense stationary processes are studied in this paper the expansion coefficients a are uncorrelated matrix random process which is a property similar to that of a matrix karhunen loe spl grave ve mkl expansion unlike the mkl expansion however the matrix wavelet expansion does not require the solution of the eigen equation this expansion also has advantages over the fourier series which is often used as an approximation to the mkl expansion in that it completely eliminates correlation the basis functions of this expansion can be obtained easily from wavelets of the matrix valued lemarie spl acute meyer type and the power spectral density of the process

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

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

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

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

model robust linear optimal designs motivated by a pioblem concerning the estimation of uranium content in calibration standards a new design criterion is developed that generalizes linear optimality to a situation in which a priori the exact form of the regression model need not be known an equivalence theorem is given some properties of the new designs are delineated and iterative methods for design construction are presented finally by using methods developed within optimal designs are computed in a number of situations including that of the motivating example

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

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

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

refinement of reproducing kernels we continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel hilbert space associated with the refinement kernel contains that with the original kernel as a subspace to motivate this study we first develop a refinement kernel method for learning which gives an efficient algorithm for updating a learning predictor several characterizations of refinement kernels are then presented it is shown that a nontrivial refinement kernel for a given kernel always exists if the input space has an infinite cardinal number refinement kernels for translation invariant kernels and hilbert schmidt kernels are investigated various concrete examples are provided

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

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

determinants and the volumes of parallelotopes and zonotopes abstract the minkowski sum of edges corresponding to the column vectors of a matrix a with real entries is the same as the image of a unit cube under the linear transformation defined by a with respect to the standard bases the geometric object obtained in this way is a zonotope z a if the columns of the matrix are linearly independent the object is a parallelotope p a in the first section we derive formulas for the volume of p a in various ways as a t a as the square root of the sum of the squares of the maximal minors of a and as the product of the lengths of the edges of p a times the square root of the determinant of the matrix of cosines of angles between pairs of edges in the second section we use the volume formulas to derive real case versions of several well known determinantal inequalities those of hadamard fischer koteljanskii fan and szasz involving principal minors of a positive definite hermitian matrix in the last section we consider zonotopes obtain a new proof of the decomposition of a zonotope into its generating parallelotopes and obtain a volume formula for z a

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

the coordinate exchange algorithm for constructing exact optimal experimental designs we describe the cyclic coordinate exchange algorithm for constructing d optimal and linear optimal experimental designs the algorithm uses a variant of the gauss southwell cyclic coordinate descent algorithm within the k exchange algorithm to achieve substantive reductions in required computing among its advantages are the following candidate sets which grow exponentially in the number of factors need not be explicitly constructed or enumerated convex design spaces or mixed convex by discrete design spaces are handled directly without the need for sophisticated nonlinear programming routines or candidate set adjustment for design problems having 10 or more factors the reductions in execution time are typically two or more orders of magnitude when compared to standard candidateset based procedures such as k exchange yet the designs produced exhibit no loss of efficiency

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

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

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

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

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

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

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

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

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

experimental design for engineering dimensional analysis dimensional analysis da is a fundamental method in the engineering and physical sciences for analytically reducing the number of experimental variables affecting a given phenomenon prior to experimentation two powerful advantages associated with the method relative to standard design of experiment doe approaches are a a priori dimension reduction and b scalability of results the latter advantage permits the experimenter to effectively extrapolate results to similar experimental systems of differing scale unfortunately da experiments are underused because very few statisticians are familiar with them in this article we first provide an overview of da and give basic recommendations for designing da experiments next we consider various risks associated with the da approach the foremost among them is the possibility that the analyst might omit a key explanatory variable leading to an incorrect da model when this happens the da model will fail and experimentation will be largely wasted to

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

an algorithm for matrix extension and wavelet construction this paper gives a practical method of extending an n x r matrix p z r n with laurent polynomial entries in one complex variable z to a square matrix also with laurent polynomial entries if p z has orthonormal columns when z is restricted to the torus t it can be extended to a paraunitary matrix if p z has rank r for each z t it can be extended to a matrix with nonvanishing determinant on t the method is easily implemented in the computer it is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter

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

an algorithm for constructing symmetric orthogonal multiwavelets by matrix symmetric extension symmetric extensions of paraunitary matrix is studied starting from a given m band compactly supported orthogonal multiscaling function with some symmetric property the problem to construct a corresponding m band multiwavelet which have the symmetric property can be reduced to the symmetric paraunitary extension of a given matrix it can be shown that such matrix extension is always solvable and an algorithmic approach to solve the symmetric paraunitary extension problem is presented this leads to a practical method for the construction of compactly supported symmetric orthogonal multiwavelets from a given multiscaling functions with an arbitrary integer dilation factor m 1 as an application some examples are given in particular the well known chui lian multiwavelets is recovered via the method

a new prefilter design for discrete multiwavelet transforms in conventional wavelet transforms prefiltering is not necessary due to the lowpass property of a scaling function this is no longer true for multiwavelet transforms a few research papers on the design of prefilters have appeared but the existing prefilters are usually not orthogonal which often causes problems in coding moreover the condition on the prefilters was imposed based on the first step discrete multiwavelet decomposition we propose a new prefilter design that combines the ideas of the conventional wavelet transforms and multiwavelet transforms the prefilters are orthogonal but nonmaximally decimated they are derived from a very natural calculation of multiwavelet transform coefficients in this new prefilter design multiple step discrete multiwavelet decomposition is taken into account our numerical examples by taking care of the redundant prefiltering indicate that the energy compaction ratio with the geronimo hardin massopust 1994 2 wavelet transform and our new prefiltering is better than the one with daubechies d sub 4 wavelet transform