polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers it is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences given sufficiently many initial bits of an algebraic number its minimal polynomial can be reconstructed and therefore the further bits of the algebraic number can be computed this also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients all algorithms work in polynomial time

linear transformations that are tridiagonal with respect to both eigenbases of a leonard pair let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let x denote the set of linear transformations x v v such that the matrix representing x with respect to the basis i is tridiagonal and the matrix representing x with respect to the basis ii is tridiagonal we show that x is spanned by i a a aa a a and these elements form a basis for x provided the dimension of v is at least 3

nearly defective complex eigenvalues in bessel matrices abstract it is known that an n n bessel matrix t n exhibits simple well separated complex eigenvalues which are getting more and more ill conditioned for increasing n the aim of this paper is to study the behavior of such eigenvalues which are simple in theory but in practice become defective using finite precision to this end we provide a sharp estimate of the distance of t n to the nearest defective matrix when n increases

bounds for the first positive zero of a mixed bessel function abstract the present article is concerned with lower and upper bounds of the first positive zero of the function h z j z zj z where j gn z is the ordinary bessel function of order 1 and j z is the derivative of j z a lower bound found here improves and extends the range of validity of the order of a lower bound found in a previous work 8 also two upper bounds given here improve a previously known upper bound 8 in the particular case 0 these bounds lead to lower and upper bounds for the first positive zero j 1 of j z which improve well known bounds in the literature

hessenberg pairs of linear transformations abstract let k denote a field and v denote a nonzero finite dimensional vector space over k we consider an ordered pair of linear transformations a v v and a v v that satisfy i iii below 1 i each of a a is diagonalizable on v 2 ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v 0 v 1 v i 1 0 i d where v 1 0 v d 1 0 3 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v 0 v 1 v i 1 0 i where v 1 0 v 1 0 we call such a pair a hessenberg pair on v in this paper we obtain some characterizations of hessenberg pairs we also explain how hessenberg pairs are related to tridiagonal pairs

displacement structure for hankel vandermonde and related derived matrices abstract we introduce some generalized concepts of displacement structure for structured matrices obtained as products and inverses of toeplitz hankel and vandermonde matrices the toeplitz case has already been studied at some length and the corresponding matrices have been called near toeplitz or toeplitz like or toeplitz derived we focus on hankel and vandermonde like matrices and in particular show how the appropriately defined displacement structure yields fast triangular and orthogonal factorization algorithms for such matrices the main contribution of this paper is presenting a unified framework rather than obtaining the fastest algorithm for each special matrix

an improved fast multipole algorithm for potential fields a new version of the fast multipole method fmm for potential fields is presented we introduce a new representation of potentials in which most translation operators are diagonal as a result for double precision calculations in two dimensions we obtain an improvement of a factor of two to four in speed compared to previously published algorithms the improvement is expected to be much greater in three dimensions the performance of the method is illustrated with several numerical examples

orthogonal polynomials and gaussian quadrature rules related to oscillatory weight functions in this paper we consider polynomials orthogonal with respect to an oscillatory weight function w x xeim x on 1 1 where m is an integer the existence of such polynomials as well as several of their properties three term recurrence relation differential equation etc are proved we also consider related quadrature rules and give applications of such quadrature rules to some classes of integrals involving highly oscillatory integrands

lossless inverse scattering digital filters and estimation theory methods to construct rational solutions of the lossless inverse scattering lis problem for one port passive digital systems are described the first method is recursive and can be viewed as a generalization of the celebrated schur algorithm the second method is global and leads to a parametrization of what we call fundamental solutions from which all lis solutions may be constructed quite a few classical problems in estimation theory and network theory may be viewed as special cases of the lis problem with each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process

the classification of leonard triples of qracah type abstract let k denote an algebraically closed field let v denote a vector space over k with finite positive dimension by a leonard triple on v we mean an ordered triple of linear transformations in end v such that for each of these transformations there exists a basis of v with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal there is a family of leonard triples said to have qracah type this is the most general type of leonard triple we classify the leonard triples of qracah type up to isomorphism we show that any leonard triple of qracah type satisfies the z 3 symmetric askey wilson relations

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

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

confluent polynomial vandermonde like matrices displacement structures inversion formulas and fast algorithm in the present paper we use the displacement structure approach to introduce a new class of what are called confluent polynomial vandermonde like matrices which generalize the ordinary simple nodes polynomial vandermonde matrices studied earlier by different authors the displacement structure approach leads to the explicit inversion formulas for confluent polynomial vandermonde like matrices and fast algorithms for inversion and for solving the associated linear systems

a fast parallel bjorck pereyra type algorithm for solving cauchy linear equations abstract we propose a new fast o n 2 parallel algorithm for solving cauchy systems of linear equations we perform an a priori rounding error analysis and obtain componentwise bounds for the forward backward and residual errors these bounds indicate that for the class of totally positive cauchy matrices the new algorithm is forward and backward stable producing a remarkably high relative accuracy in particular hilbert linear systems often considered to be too ill conditioned to be attacked can be rapidly solved with high precision the results indicate a close resemblance between the numerical properties of cauchy matrices and the much studied vandermonde matrices in fact our proposed cauchy solver is an analog of the well known bjorck pereyra algorithm for vandermonde systems as a by product we obtain favorably backward error bounds for the original bjorck pereyra algorithm several computed examples illustrate a connection of high relative accuracy to the concepts of effective well conditioning and total positivity

how to construct random functions a constructive theory of randomness for functions based on computational complexity is developed and a pseudorandom function generator is presented this generator is a deterministic polynomial time algorithm that transforms pairs g r where g is any one way function and r is a random k bit string to polynomial time computable functions r 1 2 k 1 2 k these r s cannot be distinguished from random functions by any probabilistic polynomial time algorithm that asks and receives the value of a function at arguments of its choice the result has applications in cryptography random constructions and complexity theory

accurate computation of the zeros of the generalized bessel polynomials a general method for approximating polynomial solutions of second order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized bessel polynomials and an algorithm is derived for simultaneously finding their zeros then a comparison with several alternative algorithms is carried out it shows that the computational problem of approximating the zeros of the generalized bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper

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

normalized leonard pairs and askey wilson relations abstract let v denote a vector space with finite positive dimension and let a a denote a leonard pair on v as is known the linear transformations a a satisfy the askey wilson relations a 2 a aa a a a 2 aa a a a a 2 a i a 2 a a aa aa 2 a a aa a a 2 a i for some scalars the scalar sequence is unique if the dimension of v is at least 4 if c c t t are scalars and t t are not zero then ta c t a c is a leonard pair on v as well these affine transformations can be used to bring the leonard pair or its askey wilson relations into a convenient form this paper presents convenient normalizations of leonard pairs by the affine transformations and exhibits explicit askey wilson relations satisfied by them

on the hardness of approximating shortest integer relations among rational numbers given x rn an integer relation for x is a non trivial vector m zn with inner product m x 0 in this paper we prove the following unless every np language is recognizable in deterministic quasi polynomial time i e in time o npoly log n the l shortest integer relation for a given vector x qn cannot be approximated in polynomial time within a factor of 2log0 5 n where is an arbitrarily small positive constant this result is quasi complementary to positive results derived from lattice basis reduction a variant of the well known l3 algorithm approximates for a vector x qn the l2 shortest integer relation within a factor of 2n2 in polynomial time our proof relies on recent advances in the theory of probabilistically checkable proofs in particular on a reduction from 2 prover 1 round interactive proof systems the same inapproximability result is valid for finding the l shortest integer solution for a homogeneous linear system of equations over q

diagonal pivoting for partially reconstructible cauchy like matrices with applications to toeplitz like linear equations and to boundary rational matrix interpolation problems abstract in an earlier paper we exploited the displacement structure of cauchy like matrices to derive for them a fast o n 2 implementation of gaussian elimination with partial pivoting one application is to the rapid and numerically accurate solution of linear systems with toeplitz like coefficient matrices based on the fact that the latter can be transformed into cauchy like matrices by using the fast fourier sine or cosine transform however symmetry is lost in the process and the algorithm given is not optimal for hermitian coefficient matrices in this paper we present a new fast o n 2 implementation of symmetric gaussian elimination with partial diagonal pivoting for hermitian cauchy like matrices and show how to transform hermitian toeplitz like matrices to hermitian cauchy like matrices obtaining algorithms that are twice as fast as those in the earlier work numerical experiments indicate that in order to obtain not only fast but also numerically accurate methods it is advantageous to explore the important case in which the corresponding displacement operators have nontrivial kernels this situation gives rise to what we call partially reconstructible matrices which are introduced and studied in the present paper we extend the transformation technique and the generalized schur algorithms i e fast displacement based implementations of gaussian elimination to partially reconstructible matrices we show by a variety of computed examples that the incorporation of diagonal pivoting methods leads to high accuracy we focused in this paper on the design of new numerically reliable algorithms for hermitian toeplitz like matrices however the proposed algorithms have other important applications in particular we briefly describe how they recursively solve a boundary interpolation problem for j unitary rational matrix functions

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

leonard triples from leonard pairs constructed from the standard basis of the lie algebra sl2 let k denote an algebraically closed field of characteristic zero and d 3 denote an integer an ordered pair of matrices a a is a leonard pair on the vector space kd 1 if we can find invertible matrices s1 and s2 md 1 k such that i s1 1as1 is diagonal and s1 1a s1 is irreducible tridiagonal and ii s2 1a s2 is diagonal and s2 1as2 is irreducible tridiagonal we extend this concept to three matrices we say an ordered triple of matrices a a and a is a leonard triple on kd 1 if we can find invertible matrices s1 s2 and s3 such that i s1 1as1 is diagonal and both s1 1a s1 and s1 1a s1 are irreducible tridiagonal ii s2 1a s2 is diagonal and both s2 1as2 and s2 1a s2 are irreducible tridiagonal and iii s3 1a s3 is diagonal and both s3 1as3 and s3 1a s3 are irreducible tridiagonal let a 0d010d 120 10d0anda diag d d 2 d be d 1 d 1 matrices then a a is a leonard pair on kd 1 we determine all the matrices a such that a a a form a leonard triple on kd 1

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

perturbation of the coefficients in the recurrence relation of a class of polynomials let p n x n 0 be a system of polynomials satisfying the recurrence relation p 1 x 0 p 0 x 1 p n 1 x h n p n 1 x c n p n x xp n x where h n c n are real sequences and h n 0 n 0 1 2 the co recursive polynomials p n x n 0 satisfy the same recurrence relation except for n 1 where p 1 x x c 0 0 it is well known that the problem of d the zeros of p n x is equivalent to the problem of determining the eigenvalues of a generalized eigenvalue problem tf af where t and a are symmetric matrices in this paper the problem of determining the zeros of the co recursive polynomials is reduced to a perturbation problem of the operators t and a perturbed by perturbations of rank one a function 1 2 k is found k 1 2 n whose zeros are the zeros of p n x k are the zeros of the polynomial p n x of degree n for 0 this function unifies many results concerning interlacing between the zeros of p n x and p n x for 0 moreover we obtain from this function similar results in the unstudied case 0

splitting full matrix algebras over algebraic number fields abstract let k be a fixed algebraic number field and let a be an associative algebra over k given by structure constants such that a m n k holds for some positive integer n suppose that n is bounded then an isomorphism a m n k can be constructed by a polynomial time ff algorithm an ff algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields as a consequence we obtain a polynomial time ff algorithm to compute isomorphisms of central simple algebras of bounded degree over k

modular leonard triples abstract let k denote a field and let v denote a vector space over k of finite positive dimension an ordered triple a a a of linear operators on v is said to be a leonard triple whenever for each b a a a there exists a basis of v with respect to which the matrix representing b is diagonal and the matrices representing the other two operators are irreducible tridiagonal a leonard triple a a a is said to be modular whenever for each b a a a there exists an antiautomorphism of end v which fixes b and swaps the other two operators we classify the modular leonard triples up to isomorphism

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

sensitivity of eigenvalues of an unsymmetric tridiagonal matrix several relative eigenvalue condition numbers that exploit tridiagonal form are derived some of them use triangular factorizations instead of the matrix entries and so they shed light on when eigenvalues are less sensitive to perturbations of factored forms than to perturbations of the matrix entries a novel empirical condition number is used to show when perturbations are so large that the eigenvalue response is not linear some interesting examples are examined in detail

eigenvalue condition numbers zero structured versus traditional we discuss questions of eigenvalue conditioning we study in some depth relationships between the classical theory of conditioning and the theory of the zero structured conditioning and we derive from the existing theory formulae for the mathematical objects involved then an algorithm to compare the zero structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented numerical tests are reported to highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero structure patterned matrices toeplitz and hankel will be investigated in a forthcoming paper eigenvalue patterned condition numbers toeplitz and hankel cases tech rep 3 mathematics department university of rome la sapienza 2005

on asymptotic zero distribution of laguerre and generalized bessel polynomials with varying parameters abstract we study the asymptotic zero distribution of laguerre l n n and generalized bessel b n n polynomials with the parameter n varying in such a way that the limit of n n exists our approach is based on a non hermitian orthogonality satisfied by these sequences of polynomials in the cases that remain open we formulate the corresponding conjectures

integer relations among algebraic numbers a vector m m 1 m n z n 0 is called an integer relation for the real numbers 1 n if i m i 0 holds we present an algorithm that when given algebraic numbers 1 n and a parameter e either finds an integer relation for 1 a n or proves that no relation of euclidean length shorter than 1 e exists each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation

two linear transformations each tridiagonal with respect to an eigenbasis of the other the td d canonical form and the lb ub canonical form abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider an ordered pair of linear transformations a v v and b v v which satisfy both i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we call such a pair a leonard pair on v we introduce two canonical forms for leonard pairs we call these the td d canonical form and the lb ub canonical form in the td d canonical form the leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix subject to a certain normalization in the lb ub canonical form the leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix subject to a certain normalization we describe the two canonical forms in detail as an application we obtain the following results given square matrices a b over k with a tridiagonal and b diagonal we display a necessary and sufficient condition for a b to represent a leonard pair given square matrices a b over k with a lower bidiagonal and b upper bidiagonal we display a necessary and sufficient condition for a b to represent a leonard pair we briefly discuss how leonard pairs correspond to the q racah polynomials and some related polynomials in the askey scheme we present some open problems concerning leonard pairs

transformation techniques for toeplitz and toeplitz plus hankel matrices ii algorithms abstract this paper is a continuation of g heinig a bojanczyk linear algebra appl 254 1997 193 226 where transformations mapping toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices were studied in the present paper fast algorithms for lu factorization and inversion of generalized cauchy matrices are discussed it is shown that the combination of transformation pivoting techniques leads to algorithms for indefinite toeplitz and toeplitz plus hankel matrices that are more stable than the classical ones special attention is paid to the symmetric and hermitian cases

a globally convergent method for simultaneously finding polynomial roots a new method for the simultaneous approximation of all the roots of a polynomial is given the method converges for almost every initial approximation the set of the exceptional starting points being a closed set of measure zero at least if all the polynomial roots are real and simple the method exhibits quadratic convergence not only to simple but also to multiple roots

algebraic optimization the fermat weber location problem the fermat weber location problem is to find a point in n that minimizes the sum of the weighted euclidean distances fromm given points in n in this work we discuss some relevant complexity and algorithmic issues first using tarski s theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one dimensional polynomial secondly we exhibit an explicit solution to the strong separation problem associated with the fermat weber model this separation result shows that ane approximation solution can be constructed in polynomial time using the standard ellipsoid method

totally bipartite leonard pairs and totally bipartite leonard triples of q racah type abstract let k denote an algebraically closed field of characteristic zero let v denote a vector space over k with finite positive dimension a leonard pair on v is an ordered pair of linear transformations in end v such that for each of these transformations there exists a basis for v with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal whenever these tridiagonal matrices are bipartite the leonard pair is said to be totally bipartite the notion of a leonard triple and the corresponding notion of totally bipartite are similarly defined in this paper we first classify up to isomorphism the totally bipartite leonard pairs the classification reveals that these leonard pairs are of the q racah krawtchouk or bannai ito type then we determine all leonard triples extended from given totally bipartite leonard pairs of q racah type finally we classify up to isomorphism the totally bipartite leonard triples of q racah type

on the zeros of a class of polynomials including the generalized bessel polynomials abstract it is proved that the real part of any zero of the polynomial pn 1 x of degree n which is defined by an 1pn 1 x anpn 1 x bnpn x xcnpn x p0 x 0 p1 x 1 is negative in the case bn 0 cn 0 a consequence of this result is that the zeros of the bessel polynomials as well as the zeros of the generalized bessel polynomials for a 2 have negative real parts moreover an estimate of the real part which can be easily found improves a well known inequality in the case a 3n 1

computing the complex zeros of special functions the complex zeros of solutions of second order linear odes lie over certain curves in the complex plane called anti stokes lines we consider the liouville green wkb approximation for linear homogeneous second order odes w prime prime z a z w z 0 with a z meromorphic and describe the qualitative properties of the approximate anti stokes lines based on this qualitative description we construct a fourth order method which efficiently computes the complex zeros of solutions of second order odes following the path of the approximate anti stokes lines we illustrate the method with the computation of the zeros of parabolic cylinder functions bessel functions and generalized bessel polynomials and describe specific algorithms for the computation of these zeros

leonard triples extended from a given totally almost bipartite leonard pair of bannai ito type let kk denote a field of characteristic zero and let d denote an integer at least 3 let a 1 d 02d 10102d202d 13 d 3d 10d 20dd 1 and a diag 1 d 2d 1 7 5 3 1 a diag 1 d 2d 1 7 5 3 1 be two matrices in matd 1 k matd 1 k then a a a a is a totally almost bipartite leonard pair on kd 1kd 1 of bannai ito type in this paper we determine all the matrices ae matd 1 k ae matd 1 k such that a a aea a ae form a leonard triple on kd 1kd 1

deformation techniques to solve generalised pham systems in heintz et al electron j sadio 1 1 1998 37 castro et al found comput math 2003 to appear and pardo proceedings eaca 2000 2000 pp 25 51 the authors have shown that universal solving procedures require exponential running time roughly speaking a universal solving procedure takes as input a system of multivariate polynomial equations and outputs complete symbolic information on the solution variety here we introduce a nonuniversal solving procedure adapted to generalised pham systems the aim is to compute partial information of the variety defined by the input system the algorithm is based on an homotopic deformation and on a non archimedean lifting procedure from a non singular zero of the homotopy curve the complexity of the procedure is also stated and it depends on some intrinsic quantity called the deformation degree of the given input system

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

the gaussian toeplitz matrix an analytical expression for the ll t decomposition for the gaussian toeplitz matrix with elements t ij 1 2 pi 1 2 sigma exp i j 2 2 sigma 2 is derived an exact expression for the determinant and bounds on the eigenvalues follows an analytical expression for the inverse t 1 is also derived

introduction to leonard pairs in this survey paper we give an elementary introduction to the theory of leonard pairs a leonard pair is defined as follows let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and b v v that satisfy conditions i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing b is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing b is irreducible tridiagonal we give several examples of leonard pairs using these we illustrate how leonard pairs arise in representation theory combinatorics and the theory of orthogonal polynomials

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

pivoting and backward stability of fast algorithms for solving cauchy linear equations abstract three fast o n2 algorithms for solving cauchy linear systems of equations are proposed a rounding error analysis indicates that the backward stability of these new cauchy solvers is similar to that of gaussian elimination thus suggesting to employ various pivoting techniques to achieve a favorable backward stability it is shown that cauchy structure allows one to achieve in o n2 operations partial pivoting ordering of the rows and several other judicious orderings in advance without actually performing the elimination the analysis also shows that for the important class of totally positive cauchy matrices it is advantageous to avoid pivoting which yields a remarkable backward stability of the suggested algorithms it is shown that vandermonde and chebyshev vandermonde matrices can be efficiently transformed into cauchy matrices using discrete fourier cosine or sine transforms this allows us to use the proposed algorithms for cauchy matrices for rapid and accurate solution of vandermonde and chebyshev vandermonde linear systems the analytical results are illustrated by computed examples

optimal trigonometric preconditioners for nonsymmetric toeplitz systems abstract this paper is concerned with the solution of systems of linear equations tnxn bn where tn n n denotes a sequence of nonsingular nonsymmetric toeplitz matrices arising from a generating function of the wiener class we present a technique for the fast construction of optimal trigonometric preconditioners mn mn t ntn of the corresponding normal equation which can be extended to toeplitz least squares problems in a straightforward way moreover we prove that the spectrum of the preconditioned matrix mn1t ntn is clustered at 1 such that the pcg method applied to the normal equation converges superlinearly numerical tests confirm the theoretical expectations

the prolate matrix abstract the purpose of this note is to describe some properties of a particular positive definite toeplitz matrix of arbitrary order which we call the prolate matrix this matrix arises naturally in signal processing and it is extremely ill conditioned moreover its eigenvalues exhibit rather unusual behavior making it a very useful test matrix for computational algorithms