a fast numerical algorithm for the composition of power series with complex coefficients abstract the composition problem for univariate complex power series p q i e the computation of the first n 1 coefficients of the composition q p is numerically solved by interpolation methods using multitape turing machines as a model of computation the composition problem of power series with integer coefficients of modulus n is possible in time o n 2 log n log where m bounds the amount of time for the multiplication of m bit numbers e g m cm log m 1 log log m 2 for multitape turing machines this algorithm is asymptotically faster than an implementation of the brent kung algorithm on a multitape turing machine the improvement is of order n 1 2 up to logarithmic terms

multiple criterion control a convex programming approach the performance of linear dynamic systems with respect to a non linear increasing value function that aggregates convex functionals is investigated within a convex programming framework the methodology developed in this paper combines fundamental properties of convex sets in order to decompose the multicriterion control problem into a two level structure the upper level of this structure comprises only decision making aspects of the problem and involves the solution of a relaxed and static multicriterion problem the lower level of the structure comprises in turn only optimal control aspects of the problem the overall procedure is easily implemented the solution of the aggregated problem is obtained as the limit of a sequence of optimal control problems that preserve their original solution properties relationships between the approach adopted here and existing solution techniques are discussed the solutions of the classical minimax and salukvadze problems are obtained as special cases of the proposed approach

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

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

probabilistic design of lpv control systems this paper presents an alternative approach to design of linear parameter varying lpv control systems in contrast to previous methods which are focused on deterministic algorithms this paper is based on a probabilistic setting the proposed randomized algorithm provides a sequence of candidate solutions converging with probability one to a feasible solution in a finite number of steps the main features of this approach are as follows i the randomized algorithm gives a method for general lpv plants with state space matrices depending on scheduling parameters in a nonlinear manner that is the probabilistic setting does not need a gridding of the set of scheduling parameters or approximations such as a linear fractional transformation of the state space matrices ii the proposed algorithm is sequential and at each iteration it does not require heavy computational effort such as solving simultaneously a large number of linear matrix inequalities

the generalized h 2 control problem abstract in this paper we consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary but bounded energy exogenous inputs more specifically we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from l 2 0 to l 0 is below a specified level we consider both state feedback and output feedback problems in the state feedback case it is shown that if this synthesis problem is solvable then a solution can be chosen to be a constant state feedback gain necessary and sufficient conditions for the existence of solutions as well as a formula for a state feedback gain that solves this control problem are obtained in terms of a finite dimensional convex feasibility program after showing the separation properties of this synthesis problem the output feedback case is reduced to a state feedback problem it is shown that in the output feedback case generalized h 2 controllers can be chosen to be observer based controllers the theory is demonstrated with a numerical example

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

output residence time control the problem of residence time control is extended to systems with outputs necessary and sufficient conditions for output residence time controllability in linear systems with small amounts of additive noise are derived design techniques for state feedback controllers are developed and applied to a robotics control problem the approach is based on an extension of the asymptotic first passage time theory to output processes

the multiple linear quadratic gaussian problem in this paper we define and solve the multiple linear quadratic gaussian lqg problem for discrete time systems using salukvadze s ideal point method which treats dynamic but deterministic multicriteria optimization problems the solution consists in designing a scalar regulator where the weighting matrices are replaced by the sums of their counterparts in the individual criteria the resulting control is optimal in the sense that it minimizes the difference between the cost of the implemented policy from the cost that would occur if we optimized each criterion separately and added the outcomes all the nice properties of the scalar lqg regulator are enjoyed by the multicriteria regulator as well extension to the continuous time case is straightforward yielding identical results

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

short communication on multiple roots in descartes rule and their distance to roots of higher derivatives if an open interval i contains a k fold root a of a real polynomial f then after transforming i to 0 descartes rule of signs counts exactly k roots of f in i provided i is such that descartes rule counts no roots of the kth derivative of f we give a simple proof using the bernstein basis the above condition on i holds if its width does not exceed the minimum distance s from a to any complex root of the kth derivative we relate s to the minimum distance s from a to any other complex root of f using szego s composition theorem for integer polynomials log 1 s obeys the same asymptotic worst case bound as log 1 s

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

economic fluctuations and statistical physics quantifying extremely rare and less rare events in finance one challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time we present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations one of these puzzles is how to describe outliers phenomena that lie outside of patterns of statistical regularity we review evidence consistent with the possibility that such outliers may not exist this possibility is supported by recent analysis of databases containing information about each trade of every stock r 2007 elsevier b v all rights reserved

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

brief paper guaranteed cost regulator design a probabilistic solution and a randomized algorithm this paper presents a gradient based randomized algorithm to design a guaranteed cost regulator for a plant with general parametric uncertainties the algorithm either provides with high confidence a probabilistic solution that satisfies the design specification with high probability for a randomly sampled uncertainty or claims that the feasible set of the design parameters is too small to contain a ball with a given radius in both cases the number of iterations executed in the algorithm is of polynomial order of the problem size and is independent of the dimension of the uncertainty

direct variance design a multiobjective control theory a theory for designing linear state feedback controllers to directly accommodate variance constraints imposed on the system states is offered such controllers guarantee an average closed loop response to gaussian input disturbances the method provides a mechanism for directly assigning all variance values the diagonal elements of the covariance matrix to the closed loop system the theory may be viewed as offering direct access to a new set of design variables the state variances

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

on general multiple linear quadratic control problems general multiple linear quadratic control problems are studied where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem the auxiliary lagrangian problem formulated at the lower level is of a parametric linear quadratic structure in which the weighting vector is adjusted by the upper level until a derived optimal condition is met

brief paper variance constrained self tuning control a self tuning regulator for a variance constrained optimal control problem is given the criterion for control is to minimize the stationary variance of the output in the cases when the regulator which gives minimum variance requires too large control signals an inequality constraint on the input variance is introduced in practice it is easier to select a constraint on the variance of the input than to choose the relative weights in a quadratic loss function the self tuning regulator applies the robbins monro scheme to adjust the lagrange multiplier of the variance constrained control problem the behaviour of the algorithm is illustrated by a simulated example the asymptotic behaviour of the regulator is studied using a set of associated ordinary differential equations

matrix criterion robust linear quadratic control problem this paper studies a robust linear quadratic lq control problem with multiple quadratic performance indices specifically a linear model is assumed to be available for each of a prespecified set of operating points of a system a vector valued quadratic control performance index is given at each operating point it is then desired to find the constant feedback gains of a linear controller to obtain satisfactory control performance over all the operating points this is a realistic case involving the minimization of a vector criterion of vector criteria i e a matrix valued criterion the matrix valued criterion gives insight into standard robust lq design and leads to several related vector minimization problems and to natural procedures for computing a satisfactory non inferior solution to the matrix valued minimization problem

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

on multiple criteria stationary linear quadratic control multiple control criteria and design specifications are in general present in control systems design in this study all pareto optimal solutions are characterized for a large class of multiple criteria stationary linear quadratic control problems in terms of associated scalar criterion linear quadratic control problem solutions based on a supporting hyperplane argument and thus the results are fully constructive this argument which was used to establish these results is believed to be new

a bicriteria stationary lq control problem abstract this note gives a simple derivation of the minimal loss vectors and the pareto optimal solutions to a certain bicriteria stationary lq control problem

percolation model for growth rates of aggregates and its application for business firm growth motivated by recent empirical studies of business firm growth we develop a dynamic percolation model which captures some of the features of the economical system i e merging and splitting of business firms represented as aggregates on a d dimensional lattice we find the steady state distribution of the aggregate size and explore how this distribution depends on the model parameters we find that at the critical threshold the standard deviation of the aggregate growth rates increases with aggregate size s as s where can be explained in terms of the connectedness length exponent and the fractal dimension df with 1 2df 0 20 for d 2 and 0 125 for d the distributions of aggregate growth rates have a sharp peak at the center and pronounced wings extending over many standard deviations giving the distribution a tent shape form the laplace distribution the distributions for different aggregate sizes scaled by their standard deviations collapse onto the same curve

mixed l2 and l problems by weight selection in quadratic optimal control in an attempt to achieve more realistic control objectives the weighting matrices in the standard lq1 problem are usually chosen by the designer in an ad hoc manner this paper shows several optimal control design problems that minimize a quadratic function of the control vector subject to multiple inequality constraints on the output l 2 norms l norms covariance matrix and the maximum singular value of the output covariance matrix the solutions of all four of these problems reduce to standard lqi control problems with different choices of weights this paper shows how to construct these different weights the practical significance of these results is that many robustness properties relate directly to these four entities hence the given control design algorithm delivers a specified degree of robustness to both parameter errors and disturbances the results are presented in the deterministic terms of the linear quadratic impulse lqi for continuous and discrete systems problem rather than the stoc

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

multiple objective optimization approach to adaptive and learning control this paper formulates a new approach to the classical learning adaptive control problem our approach is based on two key observations 1 the inherent conflict between control and identification as they compete for the only available resource namely the input to the plant 2 when designing and optimizing the performance of a control system the current task as well as the repertoire of other typical future tasks which the system may encounter during its life time should be considered our approach is formulated for a general nonlinear time varying plant thus unlike existing adaptive control theory the theory for a linear time invariant system evolves as a special case of the general case the design for the full lifetime of the system creates a methodology that specifies what current actions should be taken in addition to the tracking of the current reference trajectory at the expense of some performance degradation in the current task so as to improve the performance of future tasks this is

efficient isolation of polynomial s real roots this paper revisits an algorithm isolating the real roots of a univariate polynomial using descartes rule of signs it follows work of vincent uspensky collins and akritas johnson krandick our first contribution is a generic algorithm which enables one to describe all the known algorithms based on descartes rule of sign and the bisection strategy in a unified framework using that framework a new algorithm is presented which is optimal in terms of memory usage and as fast as both collins and akritas algorithm and krandick s variant independently of the input polynomial from this new algorithm we derive an adaptive semi numerical version using multi precision interval arithmetic we finally show that these critical optimizations have important consequences since our new algorithm still works with huge polynomials including orthogonal polynomials of degree 1000 and more which were out of reach of previous methods

multiple models multiplicative noise and linear quadratic control algorithmic aspects numerical solution of certain linear quadratic lq control problems for robust design is addressed an iterative method is suggested and analysed for solving a minimax multiple model lq control problem a convergent iterative method is studied for finding the constant feedback gains of a given linear controller so as to solve a spectral radius functional minimization problem for multiple plants furthermore a convergent iterative method is proposed for solving the maximum entropy parametric lq control problem with multiplicative noise

a convergent algorithm for the output covariance constraint control problem this paper considers the optimal control problem of minimizing control effort subject to multiple performance constraints on output covariance matrices y i of the form y i leq overline y i where overline y i is given the contributions of this paper are a set of conditions that characterize global optimality and an iterative algorithm for finding a solution to the optimality conditions this iterative algorithm is completely described up to a user specified parameter we show that under suitable assumptions on problem data the iterative algorithm converges to a solution of the optimality conditions provided that this parameter is properly chosen both discrete and continuous time problems are considered

distributionally robust control of constrained stochastic systems we investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process i e when only the first two moments of the disturbance distribution are known we consider two types of distributionally robust constraints in the first case we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments these constraints are commonly referred to as distributionally robust chance constraints in the second case we impose conditional value at risk cvar constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information such constraints are referred to as distributionally robust cvar constraints with second order moment specifications we propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems we prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust cvar and chance constrained problems the proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives

universal features in the growth dynamics of complex organizations we analyze the fluctuations in the gross domestic product gdp of 152 countries for the period 1950 1992 we find that i the distribution of annual growth rates for countries of a given gdp decays with fatter tails than for a gaussian and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with results on firm growth these results are consistent with the hypothesis that the evolution of organizations with complex structure is governed by similar growth mechanisms s0031 9007 98 07339 6 pacs numbers 02 50 ey 01 75 m 05 40 j 05 45 b in the study of physical systems the analysis of the scaling properties of the fluctuations has been shown to give important information regarding the underlying processes responsible for the observed macroscopic behavior in contrast most studies on the time evolution of economic time series have concentrated on average growth rates 1 18 here we investigate the possibility that the study of fluctuations in economics may also lead to a better understanding of the mechanisms responsible for the observed dynamics 19 23 we therefore analyze the fluctuations in the growth rate of the gross domestic product gdp of 152 countries during the period 1950 1992 24 we will show that i the distribution of annual growth rates for countries of a given gdp is consistent for a certain range with an exponential decay and ii the width of the distribution scales as a power law of gdp with a scaling exponent b 0 15 both findings are in surprising agreement with