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

numerical factorization of multivariate complex polynomials one can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components the importance and nature of this problem however justify a special treatment we exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently certify the decomposition with linear traces and apply interpolation techniques to construct the irreducible factors with a random combination of differentials we lower multiplicities and reduce to the regular case estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision we apply our software to study the singularities of stewart gough platforms

recognizing the efficiency weak efficiency and inefficiency of dmus with an epsilon independent linear program abstract one of the important problems in the ccr and bbc models is determining the non archimedean infinitesimal epsilon mehrabian et al s mehrabian g r jahanshahloo m r alirezaei g r amin an assurance interval for the non archimedean epsilon in dea models operations research 48 2 2000 344 347 presented a procedure for determining overall assurance interval of e they also showed that choosing e in this interval guarantees the feasibility boundedness of the multiplier envelopment side of ccr models in this paper we show that choosing e arbitrarily in the proposed overall assurance interval may not lead to correct recognition of the efficient and inefficient dmus then we propose a new model that can recognize the efficient and inefficient dmus without dependence on different values of e

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

accelerated solution of multivariate polynomial systems of equations we propose new las vegas randomized algorithms for the solution of a square nondegenerate system of equations with well separated roots the algorithms use oc delta csttn d 2 log d log b arithmetic operations in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal to approximate all real roots of the system as well as all roots lying in a fixed n dimensional box or disc here d is an upper bound on the number of all complex roots of the system e g bezout or bernshtein bound delta is the number of real roots or the roots lying in the box or disc and epsilon 2 b is the required upper bound on the output errors for computing the normal form modulo the ideal the efficient practical algorithms of b mourrain and p trebuchet in proceedings of the international symposium on symbolic and algebraic computation acm new york 2000 pp 231 238 or j c faugere j pure appl algebra 139 1999 pp 61 88 can be applied we also yield the bound oc csttn d 2 log d on the complexity of counting the numbers of all roots in a fixed box disc and all real roots for a large class of inputs and typically in practical computations the factor delta is much smaller than d delta o d this improves by the order of magnitude the known complexity estimates of the order of at least 3n d4 d3 log b or d4 which so far are the record estimates even for the approximation of a single root of a system and for each of the cited counting problems respectively our progress relies on proposing several novel techniques in particular we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators dual space of linear forms and normal forms of polynomials in the quotient algebra furthermore our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree

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

enclosing all zeros of a system of analytic functions abstract we present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral we compare this to a simpler also rigorous method based on bisection the latter is determined to be more efficient in the examples considered this is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method

additive preconditioning eigenspaces and the inverse iteration we incorporate our recent preconditioning techniques into the classical inverse power rayleigh quotient iteration for computing matrix eigenvectors every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations due to our modification we solve a well conditioned linear system instead we prove that this modification preserves local quadratic convergence show experimentally that fast global convergence is preserved as well and yield similar results for higher order inverse iteration covering the cases of multiple and clustered eigenvalues

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

an adaptive subdivision method for root finding of univariate polynomials abstract the most common methods for root finding of polynomials are iterative this type of methods shows some drawbacks like the impossibility of parallelization the unpredictable nature of the iterations that prevents to search roots inside a pre specified region of complex plane and the moderated degree of the polynomials that they can handle this work describes a recursive root finding method it is based on the winding number of plane curves that is related to the number of zeros of a polynomial in a plane region by our previous work garcia zapata and diaz martin 2014 we find the winding number at reasonable computational cost so we can approximate the roots by recursive subdivision of the search region this subdivision approach is parallelizable with a known bound of the computational cost and the search can be restrained in contrast to the iterative methods we use error returns to identify and avoid singular curves adapting the subdivision to the roots this allows us to handle high degree polynomials without resorting to the expensive multiprecision arithmetics of other recursive methods a significant contribution is that we formally prove its correctness

galois actions on blocks and classes of finite groups abstract we consider galois actions on blocks and conjugacy classes we extend some of the results valid in the case of complex conjugation we give some counterexamples where this extension is not possible

on stable equivalences and blocks with one simple module using a stable equivalence due to rouquier we show that alperin s weight conjecture holds for any p block of a finite group with defect 2 whose brauer correspondent has a unique isomorphism class of simple modules

symmetric functions applied to decomposing solution sets of polynomial systems many polynomial systems have solution sets comprised of multiple irreducible components possibly of different dimensions a fundamental problem of numerical algebraic geometry is to decompose such a solution set using floating point numerical processes into its components prior work has shown how to generate sets of generic points guaranteed to include points from every component furthermore we have shown how monodromy can be used to efficiently predict the partition of these points by membership in the components however confirmation of this prediction required an expensive procedure of sampling each component to find an interpolating polynomial that vanishes on it this paper proves theoretically and demonstrates in practice that linear traces suffice for this verification step which gives great improvement in both computational speed and numerical stability moreover in the case that one may still wish to compute an interpolating polynomial we show how to do so more efficiently by building a structured grid of samples using divided differences and applying symmetric functions several test problems illustrate the effectiveness of the new methods

root finding with threshold circuits we show that for any constant d complex roots of degree d univariate rational or gaussian rational polynomials given by a list of coefficients in binary can be computed to a given accuracy by a uniform tc 0 algorithm a uniform family of constant depth polynomial size threshold circuits the basic idea is to compute the inverse function of the polynomial by a power series we also discuss an application to the theory v tc 0 of bounded arithmetic

picard groups for derived module categories we introduce in this paper a generalization of picard groups to derived cat egories of algebras we study general properties of those groups and we construct braid group actions on these groups for particular classes of algebras

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

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

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

a family of simultaneous zero finding methods abstract applying hansen patrick s formula for solving the single equation f z 0 to a suitable function appearing in the classical weierstrass method two one parameter families of interation functions for the simultaneous approximation of all simple and multiple zeros of a polynomial are derived it is shown that all the methods of these families have fourth order of convergence some computational aspects of the proposed methods and numerical examples are given

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 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

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

solving dea models in a single optimization stage can the non archimedean infinitesimal be replaced by a small finite epsilon abstract single stage dea models aim to assess the input or output radial efficiency of a decision making unit and potential mix inefficiency in a single optimization stage this is achieved by incorporating the sum of input and output slacks multiplied by a small theoretically non archimedean infinitesimal value epsilon in the envelopment model or equivalently by using this value as the lower bound on the input and output weights in the dual multiplier model when this approach is used it is common practice to select a very small value for epsilon this is based on the expectation that for a sufficiently small epsilon the radial efficiency and optimal slacks obtained by solving the single stage model should be approximately equal to their true values obtained by the two separate optimization stages however as well known selecting a small epsilon may lead to significant computational inaccuracies in this paper we prove that there exists a threshold value referred to as the effective bound such that if epsilon is smaller than this bound the solution to the single stage program is not approximate but precise exactly the same as in the two stage approach provided there are no computational errors

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

a computational comparison of the first nine members of a determinantal family of root finding methods abstract for each natural number m greater than one and each natural number k less than or equal to m there exists a root finding iteration function bm k defined as the ratio of two determinants that depend on the first m k derivatives of the given function this infinite family is derived in kalantari j comput appl math 126 2000 287 318 and its order of convergence is analyzed in kalantari bit 39 1999 96 109 in this paper we give a computational study of the first nine root finding methods these include newton secant and halley methods our computational results with polynomials of degree up to 30 reveal that for small degree polynomials bm k 1 is more efficient than bm k but as the degree increases bm k becomes more efficient than bm k 1 the most efficient of the nine methods is b4 4 having theoretical order of convergence equal to 1 927 newton s method which is often viewed as the method of choice is in fact the least efficient 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

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

the computation of multiple roots of a polynomial this paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise and whose exact form contains multiple roots the addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots but the algorithms presented in this paper preserve the multiplicities of the roots in particular even though the given polynomial is corrupted by noise and all computations are performed on these inexact coefficients the algorithms sew together the simple roots that originate from the same multiple root thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial the algorithms described in this paper do not require that the noise level imposed on the coefficients be known and all parameters are calculated from the given inexact coefficients examples that demonstrate the theory are presented

stability of imex runge kutta methods for delay differential equations stability of imex implicit explicit runge kutta methods applied to delay differential equations ddes is studied on the basis of the scalar test equation du dt lu t mu t t where t is a constant delay and l m are complex parameters more specifically p stability regions of the methods are defined and analyzed in the same way as in the case of the standard runge kutta methods a new imex method which possesses a superior stability property for ddes is proposed some numerical examples which confirm the results of our analysis are presented

computing dynamic output feedback laws the pole placement problem asks to find laws to feed the output of a plant governed by a linear system of differential equations back to the input of the plant so that the resulting closed loop system has a desired set of eigenvalues converting this problem into a question of enumerative geometry efficient numerical homotopy algorithms to solve this problem for general multiple input multiple output systems have been proposed recently despite the wider application range of dynamic feedback laws the realization of the output of the numerical homotopies as a machine to control the plant in the time domain has not been addressed before in this note we present symbolic numeric algorithms to turn the solution to the question of enumerative geometry into a useful control feedback machine we report on numerical experiments with our publicly available software phcpack and illustrate its application on various control problems from the literature

efficient polynomial root refiners a survey and new record efficiency estimates a typical iterative polynomial root finder begins with a relatively slow process of computing a crude but sufficiently close initial approximation to a root and then rapidly refines it the policy of using the same iterative process at both stages of computing an initial approximation and refining it however is neither necessary nor most effective the efficiency of an iteration at the former stage resists formal study and is usually decided empirically whereas formal study of the efficiency at the latter stage of refinement is not hard and is the subject of the current paper we define this local efficiency as log 1 0qd log 1 0 q 1 d q is the convergence order and d is the number of function evaluations per iteration it is inversely proportional to the number of flops involved assuming that about 2n flops are needed per evaluation of a polynomial of a degree n at a single point we extend the definition to cover the recent matrix methods for polynomial root finding as well as some methods that combine n approximations to all n roots to refine them simultaneously for the approximation of a single root of a polynomial of degree n the maximum local efficiency achieved so far is log 1 02 0 301 but we show its growth to infinity for simultaneous approximation of all n roots as n grows to infinity

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

the amended dsesc power method for polynomial root finding cardinal s matrix version of the sebastiao e silva polynomial root finder rapidly approximates the roots as the eigenvalues of the associated frobenius matrix we preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters as in cardinal s algorithm we repeatedly square the frobenius matrix in nearly linear arithmetic time per squaring which yields dramatic speedup versus the recent effective polynomial root finder based on the application of the inverse power method to the frobenius matrix

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

matrix computations and polynomial root finding with preprocessing solution of homogeneous linear systems of equations is a basic operation of matrix computations the customary algorithms rely on pivoting orthogonalization and svd but we employ randomized preprocessing instead this enables us to accelerate the solution dramatically both in terms of the estimated arithmetic cost and the observed cpu time the approach is effective in the cases of both general and structured input matrices and we extend it and its computational advantages to the solution of nonhomogeneous linear systems of equations matrix eigen solving the solution of polynomial and secular equations and approximation of a matrix by a nearby matrix that has a smaller rank or a fixed structure e g of the toeplitz or hankel type our analysis and extensive experiments show the power of the presented algorithms

blocks inequivalent to their frobenius twists abstract let k be an algebraically closed field of characteristic p we give a general method for producing examples of blocks b of finite group algebras that are not morita equivalent as k algebras to the frobenius twist b p our method produces non nilpotent blocks having one simple module and elementary abelian defect group these also provide the first known examples of blocks where there is a perfect isotypy at the level of ordinary characters with all the signs positive but no derived equivalence between the blocks we do not know of any examples of blocks b that are not morita equivalent to the second frobenius twist b p 2

the overall assurance interval for the non archimedean epsilon in dea models a partition base algorithm this study presents an algorithm for determining overall assurance interval of epsilon in dea models in the previous studies in this regard solving n linear programs are needed for this propose where n is the number of decision making units dmu involved in the evaluation this paper based on a classification of dmu s proposes a partition base algorithm that can determine the overall assurance interval of epsilon by solving a few number of linear programs the proposed algorithm is illustrated by the 44 unit data set of power plant systems of iran

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

towards donovan s conjecture for abelian defect groups abstract we define a new invariant for a p block of a finite group the strong frobenius number which we use to address the problem of reducing donovan s conjecture to normal subgroups of index p as an application we use the strong frobenius number to complete the proof of donovan s conjecture for 2 blocks with abelian defect groups of rank at most 4 and for 2 blocks with abelian defect groups of order at most 64

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

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

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

model improvement for computational difficulties of dea technique in the presence of special dmus abstract data envelopment analysis dea is a non parametric mathematical optimization technique which evaluates the relative efficiency of decision making units dmus with multiple inputs and outputs a major advantage cited in support of the use of dea in measuring efficiency is that this method does not make assumption about functional forms of production function in other words it makes a piecewise frontier efficient frontier with calculation of a maximal efficiency measure for each dmu relative to all other observed measures also it identifies a subset of efficient best practice dmus and for the remaining dmus the magnitude of their non efficiency is measured by comparing to a frontier constructed from the efficient dmus therefore in the evaluation of non efficient units by dea referenced dmus have an important role to play unfortunately dmus with extraordinary output can lead to a monopoly in reference set the fact that may call it abnormality due to outliers data in this paper we introduce a dea model for evaluating dmus under this circumstance also the model is employed to measure the bank branches efficiency of a large canadian bank the layer model can derive a ranking for dmus and obtain an improving strategy for moving toward a better layer

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

jacobi free and complex free method for finding simultaneously all zeros of polynomials having only real zeros abstract there are many methods for finding simultaneously all zeros of polynomials however a complex arithmetic or jacobian matrix is needed in these methods even if the polynomial is real and only has real zeros in this paper a new method is proposed for finding simultaneously all zeros of real polynomials having only real zeros that does not require any complex arithmetic and jacobian matrix within iteration it is based on vieta s theorem and broyden s method numerical experiences show that the new method is very effective for the test polynomials

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

a property of the nearly optimal root bound the importance of root bounds for practical and theoretical algorithms for polynomial root approximation is well known the root bound by fujiwara was shown to be near optimal by van der sluis and is the most often used in practice we show here that this bound always compares favorably with kojima s bound a question left open in the work of van der sluis

computing real roots of a polynomial in chebyshev series form through subdivision with linear testing and cubic solves abstract an arbitrary polynomial of degree n f n x can always be represented as a truncated chebyshev polynomial series chebyshev form this representation is much better conditioned than the usual power form of a polynomial we describe a new method for finding the real roots of f n x in chebyshev form the canonical interval x 1 1 is subdivided into n s subintervals each is tested for zeros using the error bound for linear interpolation on zero possible intervals f n is approximated by a cubic polynomial whose roots are then found by the usual sixteenth century formulas

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

on the infinite families of zero bounds of complex polynomials we derive several classes of zero bounds for univariate polynomials with complex coefficients the derivations are based on the iterative algorithms of graffe and jin and the approximation methods of newton and kalantari all of the proposed bounds are shown to be asymptotically sharp and the convergence speeds are compared in particular we derive an improving factor to tighten kalantari s bounds this factor and kalantari s bound can both be readily computed with a precalculated lookup table we also propose a modified version of kalantari s bound that is independent of the table and we prove that under some conditions our bound is tighter numerical examples are provided to illustrate the effectiveness of our results

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

an assurance interval for the non archimedean epsilon in dea models this paper clarifies the role of non archimedean infinitesimal e in dea models so that the associated linear programs may be infeasible for the multiplier side and unbounded for the envelopment side for certain values of e it is shown that the bound of e proposed by ali and seiford 1993 is invalid for feasibility and boundedness of the linear programs a procedure is presented for determining an assurance interval of e it is also shown that an assurance value for e can be found using a single linear program

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

on simple zeros of analytic functions of n variables we give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros

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

a fitting algorithm for real coefficient polynomial rooting a new algorithm for computing all roots of polynomials with real coefficients is introduced the principle behind the new algorithm is a fitting of the convolution of two subsequences onto a given polynomial coefficient sequence this concept is used in the initial stage of the algorithm for a recursive slicing of a given polynomial into degree 2 subpolynomials from which initial root estimates are computed in closed form this concept is further used in a post fitting stage where the initial root estimates are refined to high numerical accuracy a reduction of absolute root errors by a factor of 100 compared to the famous companion matrix eigenvalue method based on the unsymmetric qr algorithm is not uncommon detailed computer experiments validate our claims

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

overdetermined weierstrass iteration and the nearest consistent system we propose a generalization of the weierstrass iteration for overdetermined systems of equations and we prove that the proposed method is the gauss newton iteration to find the nearest system which has at least k common roots and which is obtained via a perturbation of prescribed structure in the univariate case we show the connection of our method to the optimization problem formulated by karmarkar and lakshman for the nearest gcd in the multivariate case we generalize the expressions of karmarkar and lakshman and give explicitly several iteration functions to compute the optimum the arithmetic complexity of the iterations is detailed

improved bounds for the cf algorithm we consider the problem of isolating the real roots of a square free polynomial with integer coefficients using the classic variant of the continued fraction algorithm cf introduced by akritas we compute a lower bound on the positive real roots of univariate polynomials using an exponential search this allows us to derive a worst case bound of email protected b d email protected 2 for isolating the real roots of a polynomial with integer coefficients using the classic variant of cf where d is the degree of the polynomial and t the maximum bitsize of its coefficients this improves the previous bound of sharma by a factor of d 3 and matches the bound derived by mehlhorn and ray for another variant of cf which is combined with subdivision it also matches the worst case bound of the classical subdivision based solvers sturm descartes and bernstein

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

on the morita frobenius numbers of blocks of finite reductive groups abstract we show that the morita frobenius number of the blocks of the alternating groups the finite groups of lie type in defining characteristic and the ree and suzuki groups is 1 we also show that the morita frobenius number of almost all of the unipotent blocks of the finite groups of lie type in non defining characteristic is 1 and that in the remaining cases it is at most 2

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