chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

on the methods for solving yule walker equations the three well known fast algorithms for the solution of yule walker equations the levinson euclidean and berlekamp massey algorithms are reviewed and the relationship between each of them and the pade approximation problem is shown the algorithms are classified with reference to three criteria namely 1 the path they follow in the pade table 2 the organization of the computations one pass or two pass and 3 the auxiliary variables used this classification shows that the set of known classical algorithms is not complete and the missing variants are proposed with these variants of the berlekamp massey and euclid algorithms both forward and backward predictors can be obtained without additional cost a unified representation of the two pass algorithms is given in such a way that the application of the divide and conquer strategy becomes straightforward a general doubling algorithm which represents all the associated doubling algorithms in an exhaustive way is provided

a primer of perron frobenius theory for matrix polynomials abstract we present an extension of perron frobenius theory to the spectra and numerical ranges of perron polynomials namely matrix polynomials of the form l i m a m 1 m 1 a 1 a 0 where the coefficient matrices are entrywise nonnegative our approach relies on the companion matrix linearization first we recount the generalization of the perron frobenius theorem to perron polynomials and report some of its consequences subsequently we examine the role of l in multistep difference equations and provide a multistep version of the fundamental theorem of demography finally we extend issos results on the numerical range of nonnegative matrices to perron polynomials

max algebra and pairwise comparison matrices abstract this paper is a continuation of our 2004 paper max algebra and pairwise comparison matrices in which the max eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix this approximation was based on minimizing the maximal relative error in a later paper by dahl a different error measure was used and led to a slightly different approximating transitive matrix here some geometric properties of this approximation problem are discussed these lead among other results to a new characterization of a max eigenvector of an irreducible nonnegative matrix the case of toeplitz matrices is discussed in detail and an application to music theory that uses toeplitz symmetrically reciprocal matrices is given

pmax1 and smax properties and asymptotic stability in the max algebra abstract in this work we introduce the class of p max 1 matrices for the max algebra and derive some properties of these that echo similar results for p matrices in the conventional algebra in analogy with 1 we define the row p max 1 property and the smax property of a finite set of nonnegative matrices moreover we relate these concepts to stability questions for sets of matrices and difference inclusions in the max algebra

the stability of inversion formulas for toeplitz matrices abstract it is shown that under suitable assumptions the well known formulas for the inverse of toeplitz matrices that are due to gohberg and semencul and heinig are weakly stable i e they are numerically forward stable if the matrices that are by assumption nonsingular are actually well conditioned the same is true for another less known pair of inversion formulas that only involve the left biorthogonal szego polynomials

exponential asymptotics and boundary value problems keeping both sides happy at all orders we introduce templates for exponential asymptotic expansions that in contrast to matched asymptotic approaches enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter graphic 1 and have a single boundary layer at one end of the interval for linear equations the template is a transseries that takes the form of a sliding ladder of exponential scales for nonlinear equations the transseries template is a two dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed an exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of terms in the series beyond all orders in the nonlinear case we also demonstrate how these transseries can be resummed to generate multiple scales type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion 1 embed inline graphic 1 gif

computations with gohberg semencul type formulas for toeplitz matrices abstract the inverse of a toeplitz matrix t n can be represented in different ways by gohberg semencul type formulas as the sum of products of upper and lower triangular toeplitz matrices if we have given such a formula we can solve every equation t n b in o n log n steps there are three main questions arising with such representations of t n 1 1 which special linear equations can be solved in order to get generating vectors for gohberg semencul type formulas 2 which algorithm we want to apply to solve these questions and 3 which special gohberg semencul type formula we want to use for evaluating t n 1 b in this paper we present an elementary approach to derive all gohberg semencul type formulas then we introduce representations of t n 1 with special properties in particular we prove that there exists a gohberg semencul type formula such that the generating vectors are pairwise orthogonal finally based on the previous results we give new fast and stable algorithms for solving linear toeplitz systems that can be used to compute generating vectors for gohberg semencul type formulas in the case of a breakdown or near breakdown in the k th step the new method introduces a perturbation in the k th entry t k of t n such that the perturbed submatrix t k is well conditioned by using the sherman morrison woodbury formula it is possible to recover the original problem as soon as the corresponding submatrix t k r is again nonsingular

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

a fast numerical algorithm for a basic dual integral equation of the flapping wing in a flow of non viscous incompressible fluid abstract the present paper proposes a new approach to the classical problem of the harmonic oscillations of a thin wing in a flow of non viscous incompressible fluid the problem is reduced to a dual integral equation permitting application of numerical methods the numerical experiments are performed by using some advanced fast non stationary iterative methods with the help of the two dimensional fast fourier transform there is given a brief survey on the iterative methods to evaluate the most efficient algorithms in application to the considered problem of the flapping wing theory

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

a projection method for computing the minimum eigenvalue of a symmetric positive definite toeplitz matrix abstract a projection method for computing the minimal eigenvalue of a symmetric and positive definite toeplitz matrix is presented it generalizes and accelerates the algorithm considered in 12 w mackens h voss siam j matrix anal appl 18 1997 521 534 global and cubic convergence is proved randomly generated test problems up to dimension 1024 demonstrate the methods good global behaviour

a fast hankel solver based on an inversion formula for loewner matrices abstract we propose a new o n2 algorithm for solving complex n n linear systems that have hankel structure via ffts the hankel system is transformed into a loewner system an inversion formula enables us to calculate the inverse of the loewner matrix explicitely the parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle we present an o n2 algorithm to solve these interpolation problems one of the advantages of this algorithm is that it incorporates pivoting we have implemented our hankel solver in fortran 90 numerical examples are included they show the effectiveness of our pivoting strategy

parallel solution of symmetric positive definite systems with hyperbolic rotations abstract an algorithm based on hyperbolic rotations is presented for the solution of linear systems of equations ax b with symmetric positive definite coefficient matrix a forward elimination and backsubstitution are replaced by matrix vector multiplications rendering the method amenable to implementation on a variety of parallel and vector machines this method can be simplified and formulated without square roots if a is also toeplitz a systolic vlsi architecture implementing the resulting recurrence equations is more efficient than previously proposed pipelined toeplitz system solvers the hardware count becomes independent of the matrix size if its inverse is banded

look ahead levinson and schur algorithms for non hermitian toeplitz systems we present generalizations of the nonsymmetric levinson and schur algorithms for non hermitian toeplitz matrices with some singular or ill conditioned leading principal submatrices the underlying recurrences allow us to go from any pair of successive well conditioned leading principal submatrices to any such pair of larger order if the look ahead step size between these pairs is bounded our generalized levinson and schur recurrences require o n 2 operations and the schur recurrences can be combined with recursive doubling so that an o n log 2 n algorithm results the overhead in operations and storage of look ahead steps is very small there are various options for applying these algorithms to solving linear systems with toeplitz matrix

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

a look ahead algorithm for the solution of general hankel systems the solution of systems of linear equations with hankel coefficient matrices can be computed with onlyo n 2 arithmetic operations as compared too n 3 operations for the general cases however the classical hankel solvers require the nonsingularity of all leading principal submatrices of the hankel matrix the known extensions of these algorithms to general hankel systems can handle only exactly singular submatrices but not ill conditioned ones and hence they are numerically unstable in this paper a stable procedure for solving general nonsingular hankel systems is presented using a look ahead technique to skip over singular or ill conditioned submatrices the proposed approach is based on a look ahead variant of the nonsymmetric lanczos process that was recently developed by freund gutknecht and nachtigal we first derive a somewhat more general formulation of this look ahead lanczos algorithm in terms of formally orthogonal polynomials which then yields the look ahead hankel solver as a special case we prove some general properties of the resulting look ahead algorithm for formally orthogonal polynomials these results are then utilized in the implementation of the hankel solver we report some numerical experiments for hankel systems with ill conditioned submatrices

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

parallel and superfast algorithms for hankel systems of equations utilizing kernel structure properties a unified construction of hankel matrix inversion algorithms is presented three types of algorithms are obtained 1 o n 2 complexity levinson type 2 o n parallel complexity schur type and 3 o n log2 n complexity asymptotically fast ones all algorithms work without additional assumption like strong nonsingularity

a fast preconditioned conjugate gradient toeplitz solver abstract we give a simple factorization of an arbitrary hermitian positive definite matrix in which the factors are well conditioned hermitian and positive definite in fact given knowledge of the extreme eigenvalues of the original matrix a we can achieve an optimal improvement making the condition numbers of each of the two factors equal to the square root of the condition number of a we apply this technique to the solution of hermitian positive definite toeplitz systems large linear systems with hermitian positive definite toeplitz matrices arise in some signal processing applications we give a stable fast algorithm for solving these systems that is based on the preconditioned conjugate gradient method the algorithm exploits toeplitz structure to reduce the cost of an iteration to o n log n by applying the fast fourier transform to compute matrix vector products we use our matrix factorization as a preconditioner

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

parallel complexity of computations with general and toeplitz like matrices filled with integers and extensions computations with toeplitz and toeplitz like matrices are fundamental for many areas of algebraic and numerical computing the list of computational problems reducible to toeplitz and toeplitz like computations includes in particular the evaluation of the greatest common divisor gcd the least common multiple lcm and the resultant of two polynomials computing pade approximation and the berlekamp massey recurrence coefficients as well as numerous problems reducible to these transition to toeplitz and toeplitz like computations is currently the basis for the design of the parallel randomized nc rnc algorithms for these computational problems our main result is in constructing nearly optimal randomized parallel algorithms for toeplitz and toeplitz like computations and consequently for numerous related computational problems including the computational problems listed above where all the input values are integers and all the output values are computed exactly this includes randomized parallel algorithms for computing the rank the determinant and a basis for the null space of an n n toeplitz or toeplitz like matrix a filled with integers as well as a solution x to a linear system a x f if the system is consistent our algorithms use o log n log n log a parallel time and o n log n processors each capable of performing in unit time an arithmetic operation a comparision or a rounding of a rational number to a closest integer the cost bounds cover the cost of the verification of the correctness of the output the computations by these algorithms can be performed with the precision of o n log a bits which matches the precision required in order to represent the output except for the rank computation where the precision of the computation decreases the algorithms involve either a single random parameter or at most 2n 1 parameters the cited processor bounds are less by roughly factor n than ones supported by the known algorithms that run in polylogarithmic arithmetic time and do not use rounding to the closest integers technically we first devise new algorithms supporting our old nearly optimal complexity estimates for parallel computations with general matrices filled with integers then we decrease dramatically by roughly factor n1 376 the processor bounds required in these algorithms in the case where the input matrix is toeplitz like our algorithms exploit and combine some new techniques which may be of independent interest e g in the study of parallel and sequential computation of recursive factorization of integer matrices as well as our earlier techniques of variable diagonal relating to each other several known algebraic and numerical methods stream contraction and the truncation of displacement generators in toeplitz like computations our development and application of these techniques may be of independent

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

spectral properties of matrix polynomials in the max algebra we consider the spectral properties of matrix polynomials over the max algebra in particular we show how the perron frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra we also present a number of inequalities for the largest max eigenvalue of a matrix polynomial

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

inversion of a block lo wner matrix in this paper we give a fast algorithm to compute the parameters of an inversion formula for any nonsingular block lowner matrix the connection with matrix rational interpolation is given

high performance algorithms for toeplitz and block toeplitz matrices in this paper we present several high performance variants of the classical schur algorithm to factor various toeplitz matrices for positive definite block toeplitz matrices we show how hyperbolic householder transformations may be blocked to yield a block schur algorithm this algorithm uses blas3 primitives and makes efficient use of a memory hierarchy we present three algorithms for indefinite toeplitz matrices two of these are based on look ahead strategies and produce an exact factorization of the toeplitz matrix the third produces an inexact factorization via perturbations of singular principal miners we also present an analysis of the numerical behavior of the third algorithm and derive a bound for the number of iterations to improve the accuracy of the solution for rank deficient toeplitz least squares problems we present a variant of the generalized schur algorithm that avoids breakdown due to an exact rank deficiency in the presence of a near rank deficiency an approximate rank factorization of the toeplitz matrix is produced finally we suggest an algorithm to solve the normal equations resulting from a real toeplitz least squares problem based on transforming to cauchy like matrices this algorithm exploits both realness and symmetry in the normal equations

periodic quasi orthogonal spline bases and applications to least squares curve fitting of digital images presents a new covariant basis dubbed the quasi orthogonal q spline basis for the space of n degree periodic uniform splines with k knots this basis is obtained analogously to the b spline basis by scaling and periodically translating a single spline function of bounded support the construction hinges on an important theorem involving the asymptotic behavior in the dimension of the inverse of banded toeplitz matrices the authors show that the gram matrix for this basis is nearly diagonal hence the name quasi orthogonal the new basis is applied to the problem of approximating closed digital curves in 2d images by least squares fitting since the new spline basis is almost orthogonal the least squares solution can be approximated by decimating a convolution between a resolution dependent kernel and the given data the approximating curve is expressed as a linear combination of the new spline functions and new control points another convolution maps these control points to the classical b spline control points a generalization of the result has relevance to the solution of regularized fitting problems

approximate inverse free preconditioners for toeplitz matrices abstract in this paper we propose approximate inverse free preconditioners for solving toeplitz systems the preconditioners are constructed based on the famous gohberg semencul formula we show that if a toeplitz matrix is generated by a positive bounded function and its entries enjoys the off diagonal decay property then the eigenvalues of the preconditioned matrix are clustered around one experimental results show that the proposed preconditioners are superior to other existing preconditioners in the literature

hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one in this paper we discuss the special properties of hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one we show that the re expansions are independent of the inhomogeneity we illustrate how this leads to a symmetry breaking in the stokes constants within a pair of formal solutions of a differential equation a consequence is that stokes constants may exactly vanish in higher order equations leading to dramatic simplifications in the hyperasymptotic structures two examples are included

minimal eigenvalue of a real symmetric positive definite toeplitz matrix abstract the problem of computing the minimal eigenvalue of a real symmetric positive definite toeplitz matrix is considered algorithms for estimating such an eigenvalue which need only one or two inverses are presented the suggested algorithms are based on good initial approximations to the corresponding eigenvector which can be derived by approximating the toeplitz matrix by circulant matrices

from bareiss algorithm to the stable computation of partial correlations abstract this paper presents a new algorithm for the stable computation of sample partial correlation coefficients we start by generalizing bareiss algorithm for the solution of linear systems of equations with non symmetric toeplitz coefficient matrix to matrices that are not toeplitz and show that it computes their lu and ul factorizations for symmetric positive definite matrices b the normalized version of bareiss algorithm is the hyperbolic cholesky algorithm which computes the upper and lower triangular cholesky factors u and l of b by means of 2 2 hyperbolic rotations if b is a convariance matrix certain partial correlation coefficients may be obtained as the geometric means of the multiplier pairs in bareiss algorithm or as the hyperbolic tangents of the hyperbolic rotations the partial correlation coefficients for a given data matrix a may suffer from serious loss of numerical accuracy when computed from the covariance matrix b a t a while the data flow graph of bareiss algorithm suggests a method due to cybenko for computing partial correlation coefficients directly from the columns of a the graph of hyperbolic cholesky suggests an alternative that requires fewer operations nd exhibits a higher degree of parallelism in that method one computes the qr decomposition a qu of the data matrix and applies orthogonal rotations to transform u to l the sines of the rotation angles constitute a set of partial correlation coefficients we extend this approach to determine arbitrary partial correlation coefficients at little extra computational cost

a frequency domain state space approach to ls estimation and its application in turbo equalization a frequency domain state space approach to the least squares ls estimation for finite impulse response fir system identification is proposed by exploiting the toeplitz structure of the data matrix leading to two low complexity fft fast fourier transform based unbiased recursive estimators the proposed recursive estimators are applied to the smoothing estimation of time varying frequency selective channels in turbo equalization systems and channel estimation approaches with complexity o logl per symbol are developed where l is the number of channel taps

a direct method to solve block banded block toeplitz systems with non banded toeplitz blocks a fast solution algorithm is proposed for solving block banded block toeplitz systems with non banded toeplitz blocks the algorithm constructs the circulant transformation of a given toeplitz system and then by means of the sherman morrison woodbury formula transforms its inverse to an inverse of the original matrix the block circulant matrix with toeplitz blocks is converted to a block diagonal matrix with toeplitz blocks and the resulting toeplitz systems are solved by means of a fast toeplitz solver the computational complexity in the case one uses fast toeplitz solvers is equal to x m n k o mn 3 o k 3n 3 flops there are m block rows and m block columns in the matrix n is the order of blocks 2k 1 is the bandwidth the validity of the approach is illustrated by numerical experiments

stokes multiplier expansion in an inhomogeneous differential equation with a small parameter accurate approximations to the solutions of a second order inhomogeneous equation with a small parameter are derived using exponential asymptotics the subdominant homogeneous solutions that are sw

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

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

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

a discrete theory of irregular sampling abstract we derive algorithms for the iterative reconstruction of discrete band limited signals from their irregular samples they converge at a geometric rate and all constants are computable explicitly we also treat the related problem of interpolation of trigonometric polynomials and give estimates for the condition number of certain positive definite toeplitz matrices

design and analysis of toeplitz preconditioners the solution of symmetric positive definite toeplitz systems ax b by the preconditioned conjugate gradient pcg method was recently proposed by strang 1986 and analyzed by chan 1989 and chan and strang 1989 the convergence rate of the pcg method depends on the choice of preconditioners for the given toeplitz matrices the authors present a general approach to the design of toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions all resulting preconditioners can therefore be inverted via various fast transform algorithms with o n log n operations for a wide class of problems the pcg method converges in a finite number of iterations independent of n so that the computational complexity for solving these toeplitz systems is o n log n

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

stable row recurrences for the pade table and generically superfast lookahead solvers for non hermitian toeplitz systems abstract we present general recurrences for the pade table that allow us to skip ill conditioned pade approximants while we proceed along a row of the table in conjunction with a certain inversion formula for toeplitz matrices these recurrences form the basis for fast algorithms for solving non hermitian toeplitz systems under the assumption that the lookahead step size i e the number of successive skipped approximants remains bounded we give both o n 2 and o n log 2 n algorithms which are presumably weakly stable with little additional work still in o n 2 operations we can also obtain a decomposition of the toeplitz matrix t according to tr ld where r is upper triangular l is unit lower triangular and d is block diagonal the relation to continued fractions is also discussed

further optimized look ahead recurrences for adjacent rows in the pade table and toeplitz matrix factorizations abstract in a recent paper we introduced a new look ahead algorithm for recursively computing pade approximants this algorithm generates a subsequence of the pade approximants on two adjacent rows defined by fixed numerator degree of the pade table its two basic versions reduce to the classical levinson and schur algorithms if no look ahead is required in this paper we show that the computational overhead of the look ahead steps in the o n2 versions of the look ahead levinson and the look ahead schur type algorithm can be further reduced if the algorithms are used to solve toeplitz systems of equations tx b then the corresponding block ldu decompositions of t 1 or t respectively can be found with less computational effort than with any other look ahead algorithm available today

a modified t chan s preconditioner for toeplitz systems we present a modified t chan s preconditioner for solving toeplitz linear systems by the preconditioned conjugate gradient pcg method in this paper especially we give some results when the matrices are hermitian positive definite toeplitz matrices the operation and convergence of the pcg method are discussed numerical examples presented illustrate the effectiveness of the preconditioner obtained

on the asymptotic stability of nonnegative matrices in max algebra abstract in the max algebra system for an n n nonnegative matrix a a ij the eigenequation for max eigenvalue and corresponding max eigenvector x is a x x where a x i max 1 j n a ij x j and a is the maximum circuit geometric mean it is shown that the following conditions are mutually equivalent i a r n ii a 1 iii a lim k a k 0 where a max x 1 x 0 a x and a limsup k a k 1 k

look ahead methods for block hankel systems in this paper the updating formulas used by three look ahead methods for solving hankel systems are generalized to the square block case each of the original methods was described in the literature using the terminology of different but strongly related fields formal orthogonal polynomials pade approximants structured matrices this paper gives several of these connections generalized to the block case and shows that each viewpoint has its own merits

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

numerical experience with a superfast real toeplitz solver abstract we briefly describe the generalized schur algorithm for the superfast solution of positive definite toeplitz systems of equations and its relationship with schur s algorithm and the szego recursions we then present some experimental results obtained with our fortran implementation of this superfast toeplitz solver the algorithm displays favorable behavior in that the growth rates of the resulting residuals and errors are comparable with those of the szego recursions

inversion components of block hankel like matrices abstract the inversion problem for square matrices having the structure of a block hankel like matrix is studied examples of such matrices include hankel striped hankel layered and vector hankel matrices it is shown that the components that both determine nonsingularity and construct the inverse of such matrices are closely related to certain matrix polynomials these matrix polynomials are multidimensional generalizations of pade hermite and simultaneous pade approximants the notions of matrix pade hermite and matrix simultaneous pade systems are also introduced these are shown to provide a second set of inverse components for block hankel like matrices a recurrence relation is presented that allows for efficient computation of matrix pade hermite and matrix simultaneous pade systems as a result it is shown that the inverse components can be computed via either the matrix euclidean algorithm or a matrix berlekamp massey algorithm applied to an associated matrix power series an alternative algorithm based on this recurrence relation is also presented for a block hankel like matrix of type n 0 n 1 n k this algorithm is shown to compute the inverse components with a complexity of o k n 0 n k 2 block matrix operations although this can be higher in some pathological cases this is the same complexity as with existing algorithms this algorithm has the significant advantage however that no extra conditions are required on the input matrix other block algorithms require that certain submatrices be nonsingular similar results hold in the case of block toeplitz like matrices

a lookahead algorithm for the solution of block toeplitz systems abstract we give a weakly stable algorithm to solve a block toeplitz system of linear equations if the lookahead steps taken to compute the parameters of the inversion formula for the block toeplitz matrix are small compared to the order n of the matrix the algorithm requires o n2 floating point operations the parameters of the inversion formula are interpreted and computed in a recursive way as solutions of certain interpolation problems given the formal laurent series based on the data of the block toeplitz matrix

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

a note on a toeplitz inversion formula abstract it is shown that a formula of heinig for the inverse of a toeplitz matrix follows from a formula of the author for the inverse of a hankel matrix

on the spectrum in max algebra abstract we give new proofs of several fundamental results of spectral theory in max algebra this includes the description of the spectrum in max algebra of a given non negative matrix via local spectral radii the spectral theorem and the spectral mapping theorem in max algebra the latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non negative matrices

a fast preconditioned conjugate gradient toeplitz and toeplitz like solvers abstract for a toeplitz or toeplitz like matrix t we define a preconditioning applied to the symmetrized matrix t h t which decreases the condition number compared to the one of t h t and even the one of t this enables us to accelerate the conjugate gradient algorithm for solving toepiltz and toeplitz like linear systems thus extending the previous results of 1 restricted to the hermitian positive definite case the extension relies on some recent formulae of gohberg and olshevsky for the inverses of toeplitz like matrices