root refinement for real polynomials using quadratic interval refinement we consider the problem of approximating all real roots of a square free polynomial f with real coefficients given isolating intervals for the real roots and an arbitrary positive integer l the task is to approximate each root to l bits after the binary point abbott has proposed the quadratic interval refinement method qir for short which is a variant of regula falsi combining the secant method and bisection we formulate a variant of qir denoted aqir approximate qir that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively it returns a certified result for any given real polynomial whose roots are all simple in addition we propose several techniques to improve the asymptotic complexity of qir we prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial the size and the separation of the roots that is parameters exclusively related to the geometric location of the roots for integer coefficients our variant improves in theory and practice the variant with exact integer arithmetic combining our approach with multipoint evaluation we obtain near optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

on approximate zeros and rootfinding algorithms for a complex polynomial on donne des criteres pour qu un nombre complexe soit une racine approchee d un polynome f par la methode de newton ou d euler d ordre k on construit des familles d algorithmes surement convergents

new resultant inequalities and complex polynomial factorization the author deduces some new probabilistic estimates on the distances between the zeros of a polynomial p x by using some properties of the discriminant of p x and applies these estimates to improve the fastest deterministic algorithm for approximating polynomial factorization over the complex field namely given a natural n positive epsilon such that log 1 epsilon o n log n and the complex coefficients of a polynomial p x sum n i 0 p ix i such that p n ne 0 sum i p i le 1 a factorization of p x within the error norm epsilon is computed as a product of factors of degrees at most n 2 by using o log 2n time and n 3 processors under the pram arithmetic model of parallel computing or by using o n 2 log 2 n arithmetic operations the algorithm is randomized of las vegas type allowing a failure with a probability at most delta for any positive delta 1 such that log 1 delta o log n except for a narrow class of polynomials p x these results can be also obtained for epsilon such that log 1 epsilon o n 2 log n

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

a fast parallel algorithm for determining all roots of a polynomial with real roots given a polynomial p z of degree n with m bit integer coefficients and an integer mu the problem of determining all its roots with error less than 2 mu is considered it is shown that this problem is in the class nc if p z has all real roots some very interesting properties of a sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem using newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision this algorithm determines good approximations to the linear factors of p z

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

recursive algorithm without extra function evaluations for the jacobian matrix of vieta s polynomial system with applications this paper describes a simple recursion without extra function calculating for jacobian matrix of vieta s polynomial system and its applications in simultaneously approximating all the zeros of a polynomial with complex coefficient

a bibliography on roots of polynomials abstract a categorized bibliography on roots of polynomials is presented covering the period from the earliest times until late 1990 the actual bibliography is on a computer diskette accompanying this issue together with programs designed to print parts of the main work such as entries belonging to a certain category

algebraic complexity of computing polynomial zeros abstract using the power sum techniques of turan we evaluate all the complex zeros of an n th degree univariate polynomial with relative errors using o n 2 log n n log n log 1 arithmetic operations o n log n log 1 operations suffice to approximate with relative error a single complex zero using turan s method and all the zeros if they are real using graeffe s method in all cases n processors suffice for cn times parallel acceleration where c is a positive constant incorporating turan s techniques into another more recent algorithm gives a single complex zero with absolute error using o log 2 n log log 1 arithmetic parallel steps n 2 processors which places the problem in nc and also gives all the complex zeros in o n 2 log n log log 1 arithmetic operations where 1 is the absolutely largest zero computations with o log n 1 binary bits support the latter estimates for the arithmetic complexity which leads to a simple proof of the current best estimate for the boolean circuit complexity bit operation complexity of computing all the complex zeros of a polynomial announced by a schonhage in 1982 but not proven yet

the price of anarchy under nonlinear and asymmetric costs in this paper we characterize the price of anarchy i e the inefficiency between user and system optimal solutions when costs are nonseparable asymmetric and nonlinear generalizing earlier work that has addressed the price of anarchy under separable costs the results in this paper apply primarily to nonatomic games such as the traffic equilibrium problem but also in competitive multiperiod pricing and competitive supply chain settings the bounds established in this paper are tight and explicitly account for the degree of asymmetry and nonlinearity of the cost function we first provide a proof method for problems with a positive definite jacobian matrix subsequently we use ideas from semidefinite optimization in order to account for problems with a positive semidefinite jacobian matrix where the first approach does not apply this latter connection also provides a different application of semidefinite optimization

on the worst case arithmetic complexity of approximating zeros of systems of polynomials let d 1 cdots d n be positive integers let mathcal p denote the set of systems of polynomials f mathbb c n to mathbb c n that have only finitely many zeros including those at infinity and that satisfy degree f i d i for all i let 0 varepsilon leqq r it is shown for fixed d 1 cdots d n that with respect to a certain model of computation the worst case computational complexity of obtaining varepsilon approximations to at least those zeros xi satisfying xi leqq r for arbitrary f in mathcal p is theta log log r varepsilon that is to say both upper and lower bounds are proved an algorithm for proving the upper bound is introduced the number of operations required by this algorithm is o left n mathcal d 4 log mathcal d log log r varepsilon n 2 mathcal d 4 left begin array 20 c 1 sigma d i n end array right 4 right quad text where mathcal d prod limits i 1

remark on algorithms to find roots of polynomials the problem of finding the roots of a polynomial is equivalent to finding the eigenvalues of an upper hessenberg matrix which can be done with the qr algorithm it is shown that the qr algorithm has considerable advantages over other standard algorithms to find the roots of a polynomial

polynomial root finding algorithmsand branched covers a family of root finding algorithms is constructed that combines knowledge of the branched covering structure of a polynomial with a path lifting algorithm for finding individual roots in particular the family includes an algorithm that computes an epsilon factorization of a polynomial of degree d that has an arithmetic complexity of order d log d 2 log epsilon d 2 log d 2 at the present time this complexity is the best known in terms of the degree

on the complexity of polynomial zeros the parallel complexity of the simultaneous approximation to all the zeros of a polynomial is investigated by modifying and analyzing an algorithm given by householder it is possible to obtain a priori bounds to the number of iterations sufficient to yield a given accuracy and to the number of digits required in the finite arithmetic more classes of polynomials for which the simultaneous approximation to all the zeros can be carried out in polylogarithmic time are found some cases of polynomials customarily considered hard are easily solved the root finding problem for a polynomial of degree n having zeros z i i 1 cdots n is mathcal nc reduced to finding a polynomial a z such that a z i 1 a z i leq 1 1 n c where c is a constant

method for finding multiple roots of polynomials conventional numerical methods for finding multiple roots of polynomials are inaccurate the accuracy is unsatisfactory because the derivatives of the polynomial in the intermediate steps of the associated root finding procedures are eliminated engineering applications require that this problem be solved this work presents an easy to implement method that theoretically completely resolves the multiple root issue the proposed method adopts the euclidean algorithm to obtain the greatest common divisor gcd of a polynomial and its fast derivative the gcd may be approximate because of computational inaccuracy the multiple roots are then deflated into simple ones and then determined by conventional root finding methods the multiplicities of the roots are accordingly calculated a detailed derivation and test examples are provided to demonstrate the efficiency of this method

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

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

pattern formation and nonlocal logistic growth logistic growth process with nonlocal interactions is considered in one dimension spontaneous breakdown of translational invariance is shown to take place at some parameter region and the bifurcation regime is identified for short and long range interactions domain walls between regions of different order parameter are expressed as soliton solutions of the reduced dynamics for nearest neighbor interactions the analytic results are confirmed by numerical simulations

a parallel algorithm for the nonsymmetric eigenvalue problem this paper describes a parallel algorithm for computing the eigenvalues and eigenvectors of a nonsymmetric matrix the algorithm is based on a divide and conquer procedure and uses an iterative refinement technique

generalizations of laguerre s method higher order methods laguerre s method is an efficient and reliable method for finding zeros of polynomials and certain other functions a new derivation and motivation of laguerre s method is given which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized hermite or hyperosculatory interpolation members of this new class share with laguerre s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized hermite interpolation methods of order 4 and 3 303 are investigated and numerical results indicate that for large 100 times 100 eigenvalue problems the method of order 3 303 is as efficient and reliable as laguerre s method

the base size of a primitive diagonal group abstract a base b for a finite permutation group g acting on a set is a subset of with the property that only the identity of g can fix every point of b we prove that a primitive diagonal group g has a base of size 2 unless the top group of g is the alternating or symmetric group acting naturally in which case the minimal base size of g is determined up to two possible values we also prove that the minimal base size of g satisfies a well known conjecture of pyber moreover we prove that if the top group of g does not contain the alternating group then the proportion of pairs of points that are bases for g tends to 1 as g tends to infinity a similar result for the case when the degree of the top group is fixed is given

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

enclosing all zeros of an analytic function a rigorous approach we present a method to find all zeros of an analytic function in a rectangular domain the approach is based on finding guaranteed enclosures rather than approximations of the zeros well isolated simple zeros are determined fast and with high accuracy clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high order derivatives of the function we illustrate the proposed method through five examples of varying levels of complexity

estimating the extremal eigenvalues of a symmetric matrix abstract lower and upper bounds on the absolute values of the eigenvalues of an n n real symmetric matrix a are given by trace a m 1 m for both negative and positive even m the bounds are within a factor of 2 from the eigenvalues already for m log 2 n we present algorithms for computing trace a m by means of the inversion of some auxiliary matrices of the form i a and we estimate the solution cost for the important special classes of matrices toeplitz and toeplitz like banded and sparse having small separator families the cost is substantially lower than in the approach based on the powering of a the resulting computation of the eigenvalue bounds is deterministic it does not depend on the choice of an auxiliary vector as is the case for the power and inverse power methods

deterministic improvement of complex polynomial factorization based on the properties of the associated resultant abstract we show an improvement of the known best deterministic approach to factorization of a univariate polynomial p x over the complex field previously such an improvement was achieved by means of a random perturbation of the coefficients of the input polynomial we now apply some new techniques which lead to a similar result without any randomization the approach is unusual in exploiting some properties of the resultant of p x and p x which is completely distinct from the techniques usually applied in order to compute complex polynomial factorization

on the complexity of real root isolation using continued fractions we present algorithmic complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers one motivation is to explain the method s good performance in practice we derive an expected complexity bound of email protected b d 6 d email protected 2 where d is the polynomial degree and t bounds the coefficient bit size using a standard bound on the expected bit size of the integers in the continued fraction expansion thus matching the current worst case complexity bound for real root isolation by exact methods sturm descartes and bernstein subdivision moreover using a homothetic transformation we improve the expected complexity bound to email protected b d email protected we compute the multiplicities within the same complexity and extend the algorithm to non square free polynomials finally we present an open source c implementation in the algebraic library synaps and illustrate its completeness and efficiency as compared to some other available software for this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

an infinite family of bounds on zeros of analytic functions and relationship to smale s bound smale s analysis of newton s iteration function induce a lower bound on the gap between two distinct zeros of a given complex valued analytic function f z in this paper we make use of a fundamental family of iteration functions b m z m 2 to derive an infinite family of lower bounds on the above gap however even for m 2 where b 2 z coincides with newton s our lower bound is more than twice as good as smale s bound or its improved version given by blum cucker shub and smale when f z is a complex polynomial of degree n for small m the corresponding bound is computable in o n log n arithmetic operations for quadratic polynomials as m increases the lower bounds converge to the actual gap we show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of f z in particular using the latter result we show that given a complex polynomial f z a n z n a 0 a n a o 0 for each m 2 we can compute upper and lower bounds u m and l m such that the roots of f z lie in the annulus z l m z u m in particular l 2 1 2 max a k a 0 1 k k 1 n u 2 2 max a n k a n 1 k k 1 n and l3 5 1 2 max a 1 a k 1 a 0 a k a 2 0 1 k k 2 n 1 u 3 5 1 2 max a n 1 a n k 1 a n o n k a 2 n 1 k k 2 n 1 where a 1 a n 1 0 an application of the latter bounds is within weyl s classical quad tree algorithm for computing all roots of a given complex polynomial

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

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

factoring multivariate polynomials via partial differential equations a new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic it is based on a simple partial differential equation that gives a system of linear equations as in berlekamp s and niederreiter s algorithms for factoring univariate polynomials the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd s and by factoring univariate polynomials over the ground field the new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields local fields number fields and the complex number field the theory of the new method allows an effective hilbert irreducibility theorem thus an efficient reduction of polynomials from multivariate to bivariate

schur aggregation for linear systems and determinants we apply our recent preconditioning techniques to the solution of linear systems of equations and computing determinants we combine these techniques with the sherman morrison woodbury formula its new variations aggregation iterative refinement and advanced algorithms that rapidly compute sums and products either error free or with the desired high accuracy our theoretical and experimental study shows the power of this approach

qr factoring to compute the gcd of univariate approximate polynomials we present a stable and practical algorithm that uses qr factors of the sylvester matrix to compute the greatest common divisor gcd of univariate approximate polynomials over spl ropf x or spl copf x an approximate polynomial is a polynomial with coefficients that are not known with certainty the algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle by splitting and reversal if necessary the algorithm has been tested on thousands of examples including pairs of polynomials of up to degree 1000 and is now distributed as the program qrgcd in the snap package of maple 9

fast and efficient parallel evaluation of the zeros of a polynomial having only real zeros abstract given a polynomial p x of degree n with integer coefficients between 2 m and 2 m it suffices to use o log 2 n log b parallel arithmetic steps and n 2 processors in order to compute all the zeros of p x with absolute errors at most 2 h provided that all the zeros are real and b m h the algorithm combines the recent techniques and results of 1 3 in order to improve the previous record processor bound of 1 at least on the factor of n preserving its upper bound on parallel time

how to find all roots of complex polynomials by newton s method we investigate newton s method to find roots of polynomials of fixed degree d appropriately normalized we construct a finite set of points such that for every root of every such polynomial at least one of these points will converge to this root under newton s map the cardinality of such a set can be as small as 1 11 d log2d if all the roots of the polynomial are real it can be 1 30 d

sequential and parallel complexity of approximate evaluation of polynomial zeros our new sequential and parallel algorithms establish new record upper bounds on both arithmetic and boolean complexity of approximating to complex polynomial zeros o n2 log b log n arithmetic operations or o n log n log bn parallel steps and n log b log bn processors suffice in order to approximate with absolute errors 2m b to all the complex zeros of an nth degree polynomial p x whose coefficients have mod 2m if we only need such an approximation to a single zero of p x then o n log b log n arithmetic operations or o log2 n log bn steps and n log n log b log bn processors suffice which places the latter problem in nc that is in the class of problems that can be solved using polylogarithmic parallel time and a polynomial number of processors those estimates are reached in computations with o bn binary bits where the polynomial has integer coefficients we also reach the sequential boolean time bounds o bn3 log bn log log bn for approximating to all the zeros very minor improvement of the bound announced in 1982 by schonhage and o bn2log log n log bn log log bn for approximating to a single zero among further implications are the improvements of the known algorithms and complexity estimates for computing matrix eigenvalues for polynomial factorization over the field of complex numbers and for solving systems of polynomial equations the computations rely on recursive application of turan s proximity test of 1968 on its more recent extensions to root radii computations on contour integration via fast fourier transform fft within geometric constructions for search and exclusion and for the final minor improvements of the complexity bounds on the recursive factorization of p x over discs on the complex plane via numerical integration and newton s iterations

real algebraic numbers and polynomial systems of small degree based on precomputed sturm habicht sequences discriminants and invariants we classify isolate with rational points and compare the real roots of polynomials of degree up to 4 in particular we express all isolating points as rational functions of the input polynomial coefficients although the roots are algebraic numbers and can be expressed by radicals such representation involves some roots of complex numbers this is inefficient and hard to handle in applications in geometric computing and quantifier elimination we also define rational isolating points between the roots of the quintic we combine these results with a simple version of rational univariate representation to isolate all common real roots of a bivariate system of rational polynomials of total degree 2 and to compute the multiplicity of these roots we present our software within library synaps and perform experiments and comparisons with several public domain implementations our package is 2 10 times faster than numerical methods and exact subdivision based methods including software with intrinsic filtering