a verified method for bounding clusters of zeros of analytic functions in this paper we propose a verified method for bounding clusters of zeros of analytic functions our method gives a disk that contains a cluster of m zeros of an analytic function f z complex circular arithmetic is used to perform a validated computation of n degree taylor polynomial p z of f z some well known formulae for bounding zeros of a polynomial are used to compute a disk containing a cluster of zeros of p z a validated computation of an upper bound for taylor remainder series of f z and a lower bound of p z on a circle are performed based on these results rouche s theorem is used to verify that the disk contains the cluster of zeros of f z this method is efficient in computation of the initial disk of a method for finding validated polynomial factor of an analytic function numerical examples are presented to illustrate the efficiency of the proposed method

pseudozeros of multivariate polynomials the pseudozero set of a system f of polynomials in n complex variables is the subset of cn which is the union of the zero sets of all polynomial systems g that are near to f in a suitable sense this concept is made precise and general properties of pseudozero sets are established in particular it is shown that in many cases of natural interest the pseudozero set is a semialgebraic set also estimates are given for the size of the projections of pseudozero sets in coordinate directions several examples are presented illustrating some of the general theory developed here finally algorithmic ideas are proposed for solving multivariate polynomials

high order terms for condition estimation of univariate polynomials the established expressions for the condition numbers of a root of a polynomial assume that the perturbations in its coefficients are small such that only the lowest order terms in a taylor series need be considered this restriction to the lowest order terms is considered in this paper and it is shown that there exists a well defined class of polynomial for which it is not valid in particular if a root x 0 of a polynomial p x is close to but distinct from its neighboring roots x i i 1 dots r it is shown that it is necessary to consider higher order terms in the taylor series these additional terms imply that a closed form expression for the condition number of x 0 cannot be obtained and numerical methods must therefore be employed in this paper a high order taylor series is used to compute the region in the complex plane in which the perturbed root x 0 delta x 0 lies and this enables the variation of the condition number of x 0 with the distances to its neighboring roots to be established it is shown that this extra information which cannot be obtained if only the lowest order terms are used is significant

on condition numbers of polynomial eigenvalue problems in this paper we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients generalizing classical results of matrix perturbation theory we provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate we obtain that if a simple eigenvalue of a matrix polynomial is ill conditioned in some respects then it is close to be multiple and we construct an upper bound for this distance measured in the euclidean norm we also derive a new expression for the condition number of a simple eigenvalue which does not involve eigenvectors moreover an elsner like perturbation bound for matrix polynomials is presented

a unified approach to computing the nearest complex polynomial with a given zero suppose we have a complex polynomial f z whose coefficients are inaccurate and a prescribed complex number such that f 0 we study the problem of computing a complex polynomial f z such that f 0 and the distance between f and f i e f f is minimal considering that previous works usually took the usual l p norm weighted l p norm and block wise norm as distance measures we first introduce a new defined synthetic norm that integrates all these norms then we propose a unified approach to study the proposed problem and succeed in giving explicit expressions of the nearest polynomial the effectiveness of our approach is illustrated by two examples one of which shows an extension of finding the nearest complex polynomial with a zero in a given domain finally as an application of the new defined norm we discuss a matrix valued optimization problem that is very common in machine learning this paper introduces a new synthetic block wise norm and proves its dual norm which is very important here this paper for the first time proposes a unified approach to compute the nearest complex polynomial with a given zero this paper discusses a matrix valued optimization problem that is very common in machine learning as an application of the introduced norm

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

a bound for the degree of nonholonomy in the plane let v1 vs be a system made with vector fields v1 vs in rn whose coordinates are polynomials of degree d to such a system is associated the control system x u1v1 it is proven that in the case n 2 the degree of nonholonomy of such a system is bounded by a function 2 d 6d2 2d 2

solving secular and polynomial equations a multiprecision algorithm we present an algorithm for the solution of polynomial equations and secular equations of the form s x 0 for s x i 1 na ix b i 1 0 which provides guaranteed approximation of the roots with any desired number of digits it relies on the combination of two different strategies for dealing with the precision of the floating point computation the strategy used in the package mpsolve of d bini and g fiorentino d a bini g fiorentino design analysis and implementation of a multi precision polynomial rootfinder numer algorithms 23 2000 127 173 and the strategy used in the package eigensolve of s fortune s fortune an iterated eigenvalue algorithm for approximating the roots of univariate polynomials j symbolic comput 33 5 2002 627 646 the algorithm is based on the ehrlich aberth ea iteration and on several results introduced in the paper in particular we extend the concept and the properties of root neighborhoods from polynomials to secular functions provide perturbation results of the roots obtain an effective stop condition for the ea iteration and guaranteed a posteriori error bounds we provide an implementation released in the package mpsolve 3 0 based on the gmp library from the many numerical experiments it turns out that our code is generally much faster than mpsolve 2 0 and of the package eigensolve for certain polynomials like the mandelbrot or the partition polynomials the acceleration is dramatic the algorithm exploits the parallel architecture of the computing platform

finite determination of accessibility and singular points of nonlinear systems an algebraic approach abstract exploiting tools from algebraic geometry the problem of determination of accessibility strong accessibility is investigated for polynomial systems and also for analytic systems that are immersible into polynomial systems the results are constructive and algorithms are given to find the maximum depth of lie brackets necessary for deciding accessibility strong accessibility of the system at any point called here accessibility strong accessibility index of the system and known as the degree of non holonomy in the literature alternatively upper bounds on the accessibility strong accessibility index are obtained which can be computed easier in each approach the entire set of accessibility strong accessibility singular points are obtained as a limiting algebraic set of a strictly increasing chain of ideals that stabilizes in finite time several examples demonstrate the applicability of the results using computer algebra tools

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

algebro geometric solution of the 2 1 dimensional burgers equation with a discrete variable the quasiperiodic solution of the 2 1 dimensional burgers equation with a discrete variable is obtained through three steps a decomposition into a symplectic map plus two finite dimensional hamiltonian systems b straightening out of both the discrete and the continuous flows in the jacobian variety c inversion into the original variables inner relation with the modified kadomtsev petviashvili equation is presented the explicit theta function solutions for these two 2 1 integrable models are given

the deduction of the lax representation for constrained flows from the adjoint representation for x and tn finite dimensional hamiltonian systems obtained from the decompositions of zero curvature equations it is shown that their lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems as a consequence the zero curvature representation for soliton hierarchy with source is presented

from integrable to superintegrable it is shown that the presence of multiple eigenvalue parameters may result in the superintegrability of the restricted soliton flows the restricted akns flow the neumann system and the restricted mkdv flow with an eigenvalue parameter whose multiplicity is greater than two are demonstrated to be superintegrable

the nearest polynomial with a zero in a given domain for a real univariate polynomial f and a closed domain email protected c whose boundary c is represented by a piecewise rational function we provide a rigorous method for finding a real univariate polynomial email protected such that email protected has a zero in d and email protected email protected is minimal first we prove that if a nearest polynomial exists there is a nearest polynomial email protected such that the absolute value of every coefficient of email protected is email protected email protected with at most one exception using this property and the representation of c we reduce the problem to solving systems of algebraic equations each of which consists of two equations with two variables

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

more on pseudospectra for polynomial eigenvalue problems and applications in control theory definitions and characterizations of pseudospectra are given for rectangular matrix poly nomials expressed in homogeneous form p da d d 1 a d 1 da 0 it is shown that problems with infinite pseudo eigenvalues are elegantly treated in this framework for such problems stereographic projection onto the riemann sphere is shown to provide a convenient way to visualize pseudospectra lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system with equality for standard state space systems are obtained in terms of pseudospectra showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory how and why to incorporate linear structure into pseudospectra is also discussed by example

some algebro geometric solutions for the coupled modified kadomtsev petviashvili equations arising from the neumann type systems a treatment is described for getting some algebro geometric solutions of the coupled modified kadomtsev petviashvili cmkp equations and a hierarchy of 1 1 dimensional integrable nonlinear evolution equations inlees by using the neumann type systems through three steps i according to the nonlinearization of lax pair the cmkp equations and the 1 1 dimensional inlees are reduced to a family of compatible neumann type systems on a symplectic submanifold whose involutive solutions give rise to the finite parametric solutions of the inlees in both 2 1 and 1 1 dimensions ii from the holomorphic differentials and the abel map on a hyperelliptic curve of riemman surface the abel jacobi variables are introduced to straighten out the neumann type flows the 1 1 and 2 1 dimensional flows giving the abel jacobi solutions iii based on the riemann theorem and the trace formulas the jacobi inversion is applied to the straightened flows for getting some new algebro geometric solutions of the cmkp equations and the 1 1 dimensional integrable hierarchy

root neighborhoods of a polynomial the root neighborhoods of p z a polynomial over the complex field are the sets of complex numbers that are the roots of polynomials which are near to p z the term near means that the coefficients of the polynomials are within some fixed magnitude of the coefficients of p z a necessary and sufficient condition for a complex number to be in the root neighborhoods is given and it is proved that each root neighborhood contains at least one root of p z and the same number of roots of each near polynomial finally a necessary and sufficient condition is given for a root neighborhood to contain more than one root of p z and consequently more than one root of any of the near polynomials

the finite band solution of the jaulent miodek equation a method to construct the finite band solution to the soliton equation is presented we take the jaulent miodek equation as an example with the use of the nonlinearization of lax pair the jaulent miodek equation is decomposed into two finite dimensional integrable systems whose properties are studied from the view of an r matrix then through solving these finite dimensional integrable systems the finite band solution of the jaulent miodek equation is obtained

an iterative method for computing robustness of polynomial stability we propose a method for computing the distance of a stable polynomial to the set of unstable ones both in the hurwitz and in the schur case the method is based on the reformulation of the problem as the structured distance to instability of a companion matrix associated to a polynomial we first introduce the structured e pseudospectrum of a companion matrix and write a system of ordinary differential equations which maximize the real part or the absolute value of elements of the structured e pseudospectrum and then exploit the knowledge of the derivative of the maximizers with respect to e to devise a quadratically convergent iteration furthermore we use a variant of the same odes to compute the boundary of structured pseudospectra and compare them to unstructured ones an extension to constrained perturbations is also considered

on pseudospectra of matrix polynomials and their boundaries in the first part of this paper sections 2 4 the main concern is with the boundary of the pseudospectrum of a matrix polynomial and particularly with smoothness properties of the boundary in the second part sections 5 6 results are obtained concerning the number of connected components of pseudospectra as well as results concerning matrix polynomials with multiple eigenvalues or the proximity to such polynomials

quasi periodic solutions of the modified kadomtsev petviashvili equation the 2 1 dimensional modified kadomtsev petviashvili equation is decomposed into systems of integrable ordinary differential equations resorting to the nonlinearization of lax pairs abel jacobi coordinates are introduced to straighten the flows from which quasi periodic solutions of the modified kadomtsev petviashvili equation are obtained in terms of riemann theta functions

quasi periodic wave solution and asymptotic behavior for the 2 1 dimensional toda lattice equation abstract one and two periodic wave solutions for the 2 1 dimensional toda lattice equation are presented based on the hirota bilinear method and the riemann theta function the asymptotic behaviors of these two solutions are considered and the rigorous proof is given that the periodic wave solutions tend to the soliton solutions in an appropriate limiting procedure

algebro geometric solutions for the gerdjikov ivanov hierarchy this paper is dedicated to provide theta function representations of algebro geometric solutions and related crucial quantities for the gerdjikov ivanov gi hierarchy our main tools include the polynomial recursive formalism the hyperelliptic curve with finite number of genus the baker akhiezer functions the meromorphic function the dubrovin type equations for auxiliary divisors and the associated trace formulas with the help of these tools the explicit representations of the baker akhiezer functions the meromorphic function and the algebro geometric solutions are obtained for the entire gi hierarchy

the nearest complex polynomial with a zero in a given complex domain given a univariate complex polynomial f and a closed complex domain d whose boundary c is a curve parameterized by a piecewise rational function we propose two computational algorithms for finding a univariate complex polynomial email protected such that email protected has a zero in d and the distance between f and email protected is minimal our approach is composed of two steps first in the case of d consisting of one point a we give explicit formulas of email protected and the minimal distance in terms of a next the case of a general closed domain d is considered by using the property that a nearest polynomial email protected has a zero on the boundary c the curve c is parameterized piecewisely and on each piece we search for the minimum of the distance between f and email protected at this step we exploit the explicit formula of the minimal distance as a function of a point a then the global minimum and the nearest polynomial are obtained by comparing the piecewise minima some examples are presented one of them confirms that the distance between a nearest complex polynomial and a given polynomial is less than that between a nearest real polynomial and the given polynomial