global optimality conditions and optimization methods for constrained polynomial programming problems the general constrained polynomial programming problem gpp is considered in this paper problem gpp has a broad range of applications and is proved to be np hard necessary global optimality conditions for problem gpp are established then a new local optimization method for this problem is proposed by exploiting these necessary global optimality conditions a global optimization method is proposed for this problem by combining this local optimization method together with an auxiliary function some numerical examples are also given to illustrate that these approaches are very efficient

inclusion of the roots of a polynomial based on gerschgorin s theorem in this note a new companion matrix is presented which can be interpreted as a product of werner s companion matrices 13 gerschgorin s theorem yields an inclusion of the roots of a polynomial which is best in the sense of 4 and generalizes a result of l elsner 5 this inclusion is better than the one due to w borsch supan in 1

a high order iterative formula for simultaneous determination of zeros of a polynomial abstract we propose a hybrid method to determine all the zeros of a polynomial simultaneously by combining the single root method and the simultaneous one the present method has a high convergence order even for multiple roots by using the pade approximation

on a class of higher order methods for simultaneous rootfinding of generalized polynomials using the argument principle higher order methods for simultaneous computation of all zeros of generalized polynomials like algebraic trigonometric and exponential polynomials or exponential sums are derived the methods can also be derived following the continuation principle from 3 thereby the unified approach of 7 is enlarged to arbitrary ordern the local convergence as well as a priori and a posteriori error estimates for these methods are treated on a general level numerical examples are included

a sufficient condition for blowup solutions of nonlinear heat equations abstract the author discusses the initial boundary value problem ui t ui fi u1 um with u i 0 and ui x 0 i x i 1 m in a bounded domain r n under suitable assumptions on fi he proves that if i 1 e0 i in d i for some small e0 0 then the solutions blow up in a finite time where i is a positive solution of i fi 1 m 0 with i di 0 for i 1 m if m 1 the initial value can be negative in a subset of

an asymptotic and numerical description of self similar blow up in quasilinear parabolic equations we study the blow up behaviour of two reaction diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction we show that in each case the blow up is self similar in contrast to the linear diffusion limit of each in which the diffusion is only approximately self similar we then investigate the limit of the self similar behaviour and describe the transition from a stable manifold blow up behaviour for quasilinear diffusion to a centre manifold one for the linear diffusion

blow up for a porous medium equation with a localized source in this paper we investigate the blow up properties of the positive solutions to a localized porous medium equation v t dv m av p 1v q 1 x 0 t subject to homogeneous dirichlet condition and positive initial datum v 0 x under appropriate hypotheses we establish the local existence and obtain that in the case of p 1 q 1m the solution of the above problem blows up for large initial datum while it admits a global solution for small initial datum moreover for the special case p 1 0 q 1 m and a is large under an additional hypothesis on the initial datum we can also obtain the asymptotic behavior of the blow up solution

continuous domain dependence of the eigenvalues of the dirichlet laplacian and related operators in hilbert space in a hilbert space h is given a dense subspace w and a closed positive semidefinite quadratic form q on w w thus w is a hilbert space with the norm u 1 u 2 q u 1 2 for any closed subspace d of w 1 let a d denote the selfadjoint operator in the closure of d in h such that a d 1 2 u 2 q u for every u d for any decreasing sequence of closed subspaces di of w with intersection di d such that each a di has compact resolvent it is shown that for every n the nth eigenvalue n di of a di converges to that of a d and that a di 1 a d 1 in operator norm similar results are obtained for any order convergent sequence in the conditionally complete lattice of all closed subspaces d of w such that a d 1 has compact resolvent next these results are applied to the dirichlet laplacian and more generally to the dirichlet poly laplacian on a sequence of bounded open subsets of rn

blow up solutions for a class of nonlinear parabolic equations with dirichlet boundary conditions in the present paper the blow up of smooth local solutions for a class of nonlinear parabolic equations u t a u u f x u q t q u 2 with dirichlet boundary conditions are studied by constructing an auxiliary function and using hopf s maximum principles on it the sufficient conditions for blow up solutions are obtained and the upper bound of blow up time is given under some suitable assumptions on a f and initial date the obtained results are applied to some examples in which a and f are exponential functions

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

an error analysis of two related quadrature methods for computing zeros of analytic functions we present an error analysis for two related quadrature methods the delves lyness method and its modification by kravanja sakurai and van barel for computing all the zeros of an analytic function that lie inside the unit circle we consider the forward as well as the backward approximation error in case the integrals are computed via the trapezoidal rule on the unit circle contrary to the delves lyness method the quadrature error that arises from the zeros located inside the unit circle does not affect the results of the approach of kravanja et al numerical experiments illustrate our main results

polynomial zerofinding iterative matrix algorithms abstract newbery s method is completed to a method for the construction of a complex symmetric or nonsymmetric matrix with a given characteristic polynomial the methods of fiedler schmeisser and dorfler and schmeisser for similar constructions of symmetric matrices are reviewed polynomials found in the literature are solved iteratively by one of fiedler s methods with initial values supplied either by schmeisser s method or taken on a large circle or randomly in a region of the complex plane the determinental equations are solved by the qr algorithm fiedler s method used iteratively exhibits fast convergence to simple roots even in the presence of multiple roots if at some iteration step the values of the iterates which are converging to a multiple root are averaged according to the hull mathon procedure then fast convergence is also attained for multiple roots this combination appears to have nice features for polynomials of small to moderate degree

blow up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions abstract by constructing different auxiliary functions and using hopf s maximum principle the sufficient conditions for the blow up and global solutions are presented for nonlinear parabolic equation u t a u b x c t u f x u q t with different kinds of boundary conditions the upper bounds of the blow up time and the upper estimates of global solutions are provided finally some examples are presented as the application of the obtained results

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

global existence and finite time blow up for a degenerate reaction diffusion system abstract this paper investigates the blow up and global existence of solutions of the degenerate reaction diffusion system u t u m u v p v t v n u q v x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 0 and p q 0 it is proved that if m n and pq m n every nonnegative solution is global whereas if m or n or pq m n there exist both global and blow up nonnegative solutions when m n and pq m n we show that there exists 1 which depends on the parameters p q m n such that all positive solutions are global if 1 while if 1 1 all positive solutions blow up in finite time where 1 is the first dirichlet eigenvalue for the laplacian on

linking the tpr1 dpr1 and arrow head matrix structures some recent polynomial root finders rely on effective solution of the eigenproblem for special matrices such as dpr1 that is diagonal plus rank one and arrow head matrices we examine the correlation between these two classes and their links to the frobenius companion matrix and we show a gauss similarity transform of a tpr1 that is triangular plus rank one matrix into dpr1 and arrow head matrices theoretically the known unitary similarity transforms of a general matrix into a block triangular matrix with tpr1 diagonal blocks enable the extension of the cited effective eigen solvers from dpr1 and arrow head matrices to general matrices practically however the numerical stability problems with these transforms may limit their value to some special classes of input matrices

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

global existence and nonexistence for a class of degenerate parabolic systems this paper investigates the finite time blow up and global existence of the positive solution of the nonlinear degenerate parabolic system ut f1 u u a1v vt f2 v v a2u x t 0 with homogeneous dirichlet boundary data a necessary and sufficient condition for global existence of the solution is derived

analysis of blowup for a nonlinear degenerate parabolic equation in this paper the nonlinear degenerate parabolic equation ut u u u subject to dirichlet boundary conditions is studied the finite time blowup result together with an estimate of the blowup time is found the blowup set with positive measure and the asymptotic behavior of solution near the blowup time are analyzed in some detail via the similarity solutions

boundedness and blowup for nonlinear degenerate parabolic equations the author deals with the quasilinear parabolic equation ut u fi g u u bu fi 1 f u ru with dirichlet boundary conditions in a bounded domain where f and g are lower order terms he shows that under suitable conditions on f and g whether the solution is bounded or blows up in a flnite time depends only on the flrst eigenvalue of i in with dirichlet boundary condition for some special cases the result is sharp

a note on simultaneous rootfinding for algebraic exponential and trigonometric polynomials abstract a common derivation and convergence analysis is presented for two methods for the simultaneous computation of all the zeros of an algebraic exponential or trigonometric polynomial the analysis is performed considering a certain class of generalized polynomials which allow a factorization in terms of translations of a function q t such that lagrangian like interpolation is exact the methods were already known for algebraic polynomials q t t but new for exponential q t sinh t 2 and trigonometric polynomials q t sin t 2 using the properties q 0 0 q 0 0 and q 0 0 we prove the local convergence of the methods presented in this note the algorithms only require the evaluation of the generalized polynomial at certain points but no derivatives or coefficients numerical examples are included

global existence and blow up problems for reaction diffusion model with multiple nonlinearities abstract in this paper we study the following reaction diffusion model u t a u b x c t u g x t f u in d 0 t u n h x t r u on d 0 t u x 0 u 0 x 0 in d where d is a bounded domain in r n with smooth boundary d n 2 this paper deals with interactions among three kinds of nonlinear mechanisms nonlinear reaction nonlinear convection and nonlinear boundary flux the existence theorems of blow up positive solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global positive solutions and upper estimates of global positive solutions are given under suitable assumptions on a b c f g h r and initial data u 0 x

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

iterative methods for multiple zeros of a polynomial by clustering abstract for the numerical determination of zeros of a complex polynomial by simultaneous iterative methods we show theorems on the fact that the approximations for a multiple zero gather around the exact zero when they become close approximations application of our theorems makes it possible to estimate the multiplicity of zeros dynamically as well as to improve the convergence to multiple zeros for the known methods e g durand kerner s or tanabe s methods

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

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 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

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

coefficient free adaptations of polynomial root finders we adapt two celebrated polynomial root finders performing one of them we involve only the scaled values of the input polynomial c l at the points approximating the roots and recursively updated performing another root finder we also compute the values of the derivative c l at these points in neither case do we use the coefficients of c l we also relate our algorithms to approximating the eigenvalues of a matrix

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

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

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

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

improvement of convergence of an iterative method for finding polynomial factors of analytic functions in this paper we consider an iterative method for evaluating the coefficients of a monic factor of an analytic function using complex circular arithmetic in a previous paper the authors presented a factoring method that finds a cluster of zeros as a polynomial factor we analyze the convergence behavior of this method and discuss a technique for improving convergence numerical examples illustrate the aspects of the improved method

some degenerate and quasilinear parabolic systems not in divergence form abstract this paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form u t u p u av v t v q v bu with null dirichlet boundary conditions and positive initial conditions where p q a and b are all positive constants the local existence and uniqueness of classical solution are proved moreover it will be proved that all solutions exist globally if and only if ab 1 2 where 1 is the first eigenvalue of in with homogeneous dirichlet boundary condition

global existence and nonexistence for some degenerate and quasilinear parabolic systems abstract the author discusses the degenerate and quasilinear parabolic system u t u v u a u p v q and v t u v v b u r v s with dirichlet boundary conditions in a bounded domain and shows that the global existence depends crucially on the sign of the difference q r 1 p 1 s and the domain

polynomial roots from companion matrix eigenvalues in classical linear algebra the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial an algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition we derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm our error analysis expands on work by van dooren and dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully regarding the stronger concept of a small componentwise backward error our analysis predicts a small such error on a test suite of eight random polynomials suggested by toh and trefethen however we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice we extend our results to polynomial matrices where the result is essentially the same but the geometry becomes more complicated

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

boundary value problem for a class of degenerate quasilinear parabolic equations with singularity this paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy the existence and uniqueness of generalized solutions to a kind of boundary value problem is established

linear construction of companion matrices this note is concerned with the following problem for a given matrix a e c ix and a vector a e c does there exist a mapping 2 assigning to each manic polynomial f of degree n a vector x f e c such that the matrix b a a x f is a companion matrix off i e the characteristic polynomial of b is t l ftf the classes of suitable matrices a and vectors a are characterized and some properties of b are described the corresponding unique mapping is determined by a system of linear equations the cases of a triangular bidiagonal or diagonal matrix a are discussed explicitly and many known companion matrices are obtained as particular cases then gershgorin s theorem is applied yielding error estimates for polynomial roots finally the extension to block companion matrices and an example of nonlinear construction are discussed

global existence and blowup solutions for quasilinear parabolic equations the authors discuss the quasilinear parabolic equation ut g u u h u u f u with u 0 u x 0 x i ff g and h are polynomials with proper degrees and proper coefficients they show that the blowup property only depends on the first eigenvalue of in with dirichlet boundary condition for a special case they obtain a sharp result

failure of convergence of the newton weierstrass iterative method for simultaneous rootfinding of generalized polynomials abstract many problems in mathematics and other natural sciences and techniques reduce themselves to determining all roots of generalized polynomial equations we consider in this paper the situations in which the critical initial approximations z i 0 i 1 n fail when the iterative methods of newton weierstrass type are applied

global existence and blowup for quasilinear parabolic equations not in divergence form abstract this paper deals with global and blowup solutions of the quasilinear parabolic equation u t x u u u f x u u with homogeneous dirichlet boundary conditions we will give sufficient conditions such that the solutions either exist globally or blow up in a finite time for any smooth initial values in special cases a necessary and sufficient condition for global existence is given

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

simultaneous factorization of a polynomial by rational approximation in this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously our approach gives methods of arbitrary order grau s method carstensen 1992 grau 1971 is obtained as the second order method which is durand kerner s method when we have linear factors for linear factors our approach yields the simultaneous methods introduced in sakurai et al 1991 we prove local convergence and estimate the r order of the total step version as well as the single step version of the methods we derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted

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

design variables for bias distribution in transfer function estimation estimation of transfer functions of linear systems is one of the most common system identification problems several different design variables chosen by the user for the identification procedure affect the properties of the resulting estimate in this paper it is investigated how the choices of prefilters noise models sampling interval and prediction horizon i e the use of k step ahead prediction methods influence the estimate an important aspect is thai the true system is not assumed to be exactly represented within the chosen model set the estimate will thus be biased it is shown how the distribution of bias in the frequency domain is governed by a weighting function which emphasizes different frequency bands the weighting function in turn is a result of the previously listed design variables it is shown e g thai the common least squares method has a tendency to emphasize high frequencies and that this can be counteracted by prefiltering it is also shown that asymptotically it is only the prediction horizon itself and not how it is split up into sampling interval times number of predicted sampling instants that affects this weighting function

comparison of the closed loop identfication methods in terms of the bias distribution several schemes for plant model identification in closed loop operation including classical direct method two step identification and closed loop output error algorithms are considered these methods are analyzed and compared in terms of the bias distribution of the estimates for the case that the noise model is estimated as well as the case that a fixed model of noise is considered output error structure the problems concerning the filtered direct method which is often used in the iterative identification and control scheme are mentioned it is shown that these problems may be solved by the closed loop output error identification method

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

recursive algorithms for identification in closed loop a unified approach and evaluation a unified presentation of recursive algorithms for plant model identification in closed loop is given from the basic formulation of the problem of finding the plant model that gives the best predictor for the closed loop system two families of algorithms using either a reparameterized predictor for the closed loop or a plant predictor operating on filtered data are presented and their asymptotic properties are examined validation tests for the models identified in closed loop are proposed a comparative evaluation of the various algorithms in simulation and on real time experiments concludes the paper

regularity in sobolev spaces for the fast diffusion and the porous medium equation abstract the degenerate parabolic equation u t u m 1 u m 0 is considered in a cylinder 0 t under homogeneous dirichlet boundary values the regularity of weak solutions for the fast diffusion equation m 1 and the porous medium equation m 1 are investigated regularity of u and u m 1 u in weighted sobolev spaces and in fractional order nikolskii spaces are proved

a composite polynomial zerofinding matrix algorithm abstract a globally convergent matrix algorithm is presented for finding the real and complex zeros of a complex polynomial p x a combination of schmeisser s and fiedler s matrices is used in the algorithm first the greatest common divisor g x of p x and its derivative p x is obtained and used to reduce p x to a polynomial q x p x q x with simple zeros second the zeros of q x are computed either by finding the eigenvalues of the first block of schmeisser s matrix or by applying fiedler s algorithm recursively in either case the eigenvalues are obtained by the qr algorithm third the multiplicity of each zero of p x is calculated by means of lagouanelle s modified limiting formula sample numerical results are given to demonstrate the effectiveness of the algorithm

focusing solutions of porous medium equations with reaction in this paper we study the existence of focusing solution to a class of porous medium equations taking the form ut um f x u u where m 1 focusing solution has the property that its initial distribution is in the exterior of a finite domain that is there is a hole in the support of initial value and in finite time t the hole disappears we show there exists a focusing solution for a number of important models in physics and biology such solution is an example of a self similar solution of the second kind that is the similarity variables cannot be determined a priori from dimensional consideration furthermore it serves the purpose of supplying concrete bounds for the optimal regularity of general solutions of the equation the p laplacian counterpart of this equation is also studied

a unified approach to methods for the simultaneous computation of all zeros of generalized polynomials applying newton s method to a particular system of nonlinear equations we derive methods for the simultaneous computation of all zeros of generalized polynomials these generalized polynomials are from a function space satisfying a condition similar to haar s condition by this approach we bring together recent methods for trigonometric and exponential polynomials and a well known method for ordinary polynomials the quadratic convergence of these methods is an immediate consequence of our approach and needs not to be proved explicitly moreover our approach yields new interesting methods for ordinary trigonometric and exponential polynomials and methods for other functions occuring in approximation theory

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

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real

inverse power and durand kerner iterations for univariate polynomial root finding abstract univariate polynomial root finding is the oldest classical problem of mathematics and computational mathematics and is still an important research topic due to its impact on computational algebra and geometry the weierstrass durand kerner approach and its variations as well as matrix methods based on the qr algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial we propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root finding demonstrate its effectiveness and relate its study to unifying the derivation of the weierstrass durand kerner algorithm having quadratic convergence and its extensions having convergence rates 4 6 8 our experiments show substantial improvement versus the latter algorithm even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots

global existence of solutions of a strongly coupled quasilinear parabolic system with applications to electrochemistry this paper consists of two parts in the first part we proved the global existence of weak solutions of a strongly coupled quasilinear parabolic system in rn using weak compactness method in the second part we considered the electrochemistry model studied in choi and lui j differential equations 116 1995 306 where the poisson equation governing the electric potential is replaced by a local electro neutrality condition in one space dimension the equations for the model is of the form considered in the first part of this paper except that the coefficient matrix is discontinuous at places where all the charged ions vanish we approximate the equations by nicer operators and pass to the limit to obtain global existence of weak solutions the non negativity of weak solutions and l2 stability of the steady state solutions are also shown under additional hypotheses

on grau s method for simultaneous factorization of polynomials this note is concerned with a a grau s method siam j numer anal 8 1971 pp 425 438 for simultaneous factorization of polynomials as a generalisation of the famous durand kerner method grau s method is discussed on a general level such that certain invariance properties are obtained this yields a matrix interpretation that facilitates error estimates by gershgorin s theorem a convergence result is proved that increases the usual quadratic convergence finally a single step version is stated which has r order of convergence greater than 2

a supplementary bibliography on roots of polynomials in 1 this author presented a bibliography on roots of polynomials covering most published works on this topic from the beginnings until approximately mid 1990 see 1 for some very brief remarks on the history and applications of the subject now we present a supplementary bibliography this contains references to works not included in the original bibliography either because they were not yet published or because they had not yet been discovered by the author formerly the bibliography was presented on a diskette included with the journal issue together with a c program which facilitated access to the material e g certain categories could be selected now however the material is being made available via the world wide web instructions for access are as follows 1 the bibliography can be accessed from the homepage of the journal of computational and applied mathematics

blow up solutions and global solutions for a class of quasilinear parabolic equations with robin boundary conditions the type of problem under consideration isu email protected email protected a u b x u g x t f u indx 0 t email protected email protected x t u 0 email protected dx 0 t u x 0 u 0 x 0 ind where d is a smooth bounded domain of r n by constructing an auxiliary function and using hopf s maximum principles on it existence theorems of blow up solutions upper bound of blow up time upper estimates of blow up rate existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a b f g s and initial data u 0 x the obtained results are applied to some examples in which a b f g and s are power functions or exponential functions

some remarks on the bias distribution analysis of discrete time identification algorithms based on pseudo linear regressions abstract in 1998 a karimi and i d landau published in this journal an article entitled comparison of the closed loop identication methods in terms of bias distribution one of its main purposes was to provide a bias distribution analysis in the frequency domain of closed loop output error identication algorithms that had been recently developed the expressions provided in that paper are only valid for prediction error identification methods pem not for pseudo linear regression plr ones for which we give the correct frequency domain bias analysis both in open and closed loop although plr was initially and is still considered as an approximation of pem we show that it gives better results at high frequencies