on the gaussian approximation of convolutions under multidimensional analogues of s n bernstein s inequality conditions on etudie le taux d approximation de distribution de sommes de vecteurs aleatoires independants par des distributions gaussiennes correspondantes

a simple homotopy method for determining all isolated solutions to polynomial systems a new homotopy method for solving systems of polynomial equations is presented the homotopy equation is extremely simple it is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem within some limits an arbitrary starting problem can be chosen as long as its solution set is known no restrictions on the polynomial systems are made a few numerical tests are reported which show the influence of the auxiliary parameter resp the starting problem upon the computa tional cost of the method

a product decomposition bound for bezout numbers most polynomial systems that arise in practice are not completely general but have special structures a common form is that each equation must be a sum of products where each factor has an identifiable generic type a theorem is proven for such systems which offers a method for obtaining a tighter upper bound on the number of nonsingular solutions than is generally available at the same time this theorem provides an approach for solving such systems via polynomial continuation which results in less computational work to illustrate the practical usefulness of these ideas we show that a significant design of mechanisms problem can be solved with an order of magnitude less work than the published solution

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

nonlinear control of constrained linear systems via predictive reference management a method based on conceptual tools of predictive control is described for solving set point tracking problems wherein pointwise in time input and or state inequality constraints are present it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time a virtual sequence among a family of linearly parameterized command sequences by solving a convex constrained quadratic optimization problem and feeds the primal system according to a receding horizon control philosophy the overall system is proved to fulfill the constraints be asymptotically stable and exhibit an offset free tracking behavior provided that an admissibility condition on the initial state is satisfied though the cg can be tailored for the application at hand by appropriately choosing the available design knobs the required online computational load for the usual case of affine constraints is well tempered by the related relatively simple convex quadratic programming problem

the cheater s homotopy an efficient procedure for solving systems of polynomial equations a procedure is introduced for solving systems of polynomial equations that need to be solved repetitively with varying coefficients the procedure is based on the cheater s homotopy a continuation method that follows paths to all solutions all solutions are found with an amount of computational work roughly proportional to the actual number of solutions previous general methods normally require an amount of computation roughly proportional to the total degree

symmetric homotopy construction abstract homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system computations will be performed more efficiently if the symmetric structure of the system can be exploited this paper presents the construction of a symmetric homotopy using this homotopy only the paths according to the generating solutions have to be traced during continuation

on real parameter stability margins and their computation a stability margin for linear control systems subject to real parameter variations can be computed from the constraints arising from the conventional routh hurwitz criteria the proposed stability margin is similar to the real criterion but with the bound on the real parameter variations given by a differentiable function if the uncertain parameters are gaussian random variates then an upper bound on the probability of instability is constructed a computational procedure is proposed based on polynomial continuation

large deviation probabilities and dominating points for open convex sets nonlogarithmic behavior the existence of a dominating point for an open convex set and a corresponding representation formula for large deviation probabilities are established in the infinite dimensional setting under conditions which are both necessary and sufficient and follow from those used previously in mathbb r d a precise nonlogarithmic estimate of large deviation probabilities applicable to gaussian measures is also included

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

bezout number calculations for multi homogeneous polynomial systems the bezout number of a system of multi homogeneous polynomial equations is the largest number of nonsingular solutions such a system can have and it is also the number of solution paths used to compute all geometrically isolated solutions of the system using multi homogeneous polynomial continuation any polynomial system can be homogenized in a variety of ways typically yielding different bezout numbers this paper presents an efficient algorithm for computing bezout numbers and describes a procedure for finding the minimal bezout number over all possible homogenizations

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

a homotopy for solving general polynomial systems that respects m homogeneous structures methods for finding all solutions to a polynomial system f z 0 of n equations in n unknowns using a generic homotopy have been presented by several researchers over the last ten years in this paper we describe a new method for defining a homotophy to find all solutions to a polynomial system a major feature of this new approach is that the homotopy need not to be generic or random the practical consequences are that the homotopy may be chosen to more closely resemble the system being solved with a consequent potential for reducing the arc length of paths and exploiting special structures in the original system in particular if the system is naturally m homogeneous then the number of homotopy paths that need to be tracked is reduced since many systems that arise in applications have such a structure this represents a significant advance over previous homotopies we also provide a proof that the homotopy parameter is strictly increasing on homotopy paths as a function of arc length a new result which has implications for the choice of numerical path tracking algorithms

on the solutions to polynomial systems obtained by homotopy methods a natural class of homotopy methods for solving polynomial systems is considered it is shown that at least one solution from each connected component of the solution set is obtained this generalizes the results of previous papers which concentrated on isolated solutions i e connected components with one single point the number of solution paths ending in a connected component is independent of the particular homotopy in use and defines in a natural way the multiplicity of the connected component a few numerical experiments illustrate the obtained results

nonlinear offset free model predictive control this paper addresses offset free reference tracking of asymptotically constant reference signals using model predictive control existing results for linear models are extended to general nonlinear models the core of the proposed method employs a disturbance model and an observer to estimate its state typical disturbance models are shown and the implications of using them are discussed conditions are given for which this setup eliminates the tracking error asymptotically basically we prove that error free output estimation and error free nominal tracking imply offset free model predictive control

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

the gbq algorithm for constructing start systems of homotopies for polynomial systems homotopy methods have become a standard tool for the computation of all solutions of a polynomial system this paper concerns the solution of deficient polynomial systems which appear to be typical in many engineering applications the gbq algorithm presented consists of two parts the computation of a generalized bezout number gb and the construction of a multi homogeneous start system q the approach generalizes m homogenization into multihomogenization it can also be regarded as a generalization towards the random product homotopy however without making assumptions on the coefficients of the polynomials in the system as is illustrated in the examples symmetric polynomial systems also can be solved more efficiently

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

algebraic methods for image processing and computer vision many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations crucial issues include the uniqueness of solution and the number of solutions if not unique and how to find numerically all the solutions the goal of this paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues three dimensional motion structure estimation is used as the context however these tools should also be helpful in other areas including surface intersection in computer aided design and inverse position problems in kinematics robotics the tools to be described are bezout numbers grobner bases homotopy methods and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example

solving polynomial systems using a branch and prune approach this paper presents tt newton a branch and prune algorithm used to find all isolated solutions of a system of polynomial constraints tt newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early the pruning in tt newton consists of enforcing at each node of the search tree a unique local consistency condition called box consistency which approximates the notion of arc consistency well known in artificial intelligence box consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the hansen sengupta narrowing operator used in interval methods as well as new operators which are more effective when the computation is far from a solution tt newton has been evaluated on a variety of benchmarks from kinematics chemistry combustion economics and mechanics on these benchmarks it outperforms the interval methods we are aware of and compares well with textit state of the art continuation methods limitations of tt newton e g a sensitivity to the size of the initial intervals on some problems are also discussed of particular interest is the mathematical and programming simplicity of the method

numerical decomposition of the solution sets of polynomial systems into irreducible components in engineering and applied mathematics polynomial systems arise whose solution sets contain components of different dimensions and multiplicities in this article we present algorithms based on homotopy continuation that compute much of the geometric information contained in the primary decomposition of the solution set in particular ignoring multiplicities our algorithms lay out the decomposition of the set of solutions into irreducible components by finding at each dimension generic points on each component as by products the computation also determines the degree of each component and an upper bound on its multiplicity the bound is sharp i e equal to one for reduced components the algorithms make essential use of generic projection and interpolation and can if desired describe each irreducible component precisely as the common zeroes of a finite number of polynomials

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

predictive inverse model allocation for constrained over actuated linear systems the paper presents a model predictive allocation scheme for constrained over actuated linear systems for which input redundancy entails the existence of multiple trajectories in the state space yielding a given reference output the method relies upon the concept of inverse model allocation where dynamic allocation of reference state and input trajectories is accomplished within the framework of output regulation while maintaining invariance of the error zeroing subspace the study focuses on the design of model predictive allocator to achieve constraint satisfaction and asymptotic evolution of the trajectories to a pre computed steady state target in particular the objective of this study is the analysis of the stability and feasibility properties of the proposed schemes and the characterization of suitable sufficient conditions for stability in geometric terms in support of the theoretical findings this study presents an example where the proposed methodology is applied to solve a constrained tracking control problem for the linearized model of the air path system of a turbocharged diesel engine

homotopies exploiting newton polytopes for solving sparse polynomial systems this paper is concerned with the problem of finding all isolated solutions of a polynomial system the bkk bound defined as the mixed volume of the newton polytopes of the polynomials in the system is a sharp upper bound for the number of isolated solutions in mathbb c 0 n mathbb c 0 mathbb c backslash 0 of a polynomial system with a sparse monomial structure first an algorithm is described for computing the bkk bound following the lines of bernshtei n s proof the algorithmic construction of the cheater s homotopy or the coefficient homotopy is obtained the mixed homotopy methods can be combined with the random product start systems based on a generalized bezout number applications illustrate the effectiveness of the new approach

moderate deviation probabilities for open convex sets nonlogarithmic behavior precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite and infinite dimensional settings our results are based on the existence of dominating points for these sets a related representation formula and asymptotics for the integral term in this formula

brief paper mpc for tracking piecewise constant references for constrained linear systems in this paper a novel model predictive control mpc for constrained non square linear systems to track piecewise constant references is presented this controller ensures constraint satisfaction and asymptotic evolution of the system to any target which is an admissible steady state therefore any sequence of piecewise admissible setpoints can be tracked without error if the target steady state is not admissible the controller steers the system to the closest admissible steady state these objectives are achieved by i adding an artificial steady state and input as decision variables ii using a modified cost function to penalize the distance from the artificial to the target steady state iii considering an extended terminal constraint based on the notion of invariant set for tracking the control law is derived from the solution of a single quadratic programming problem which is feasible for any target furthermore the proposed controller provides a larger domain of attraction for a given control horizon than the standard mpc and can be explicitly computed by means of multiparametric programming tools on the other hand the extra degrees of freedom added to the mpc may cause a loss of optimality that can be arbitrarily reduced by an appropriate weighting of the offset cost term

a homotopy method for eigenvalue assignment using decentralized state feedback this paper addresses the following problem given an interconnected system m composed of n subsystems of the form a i b i k i i 1 n a i b i a controllable pair and where the off diagonal blocks of m lie in the image of the appropriate b i then is it possible to arbitrarily assign the characteristic polynomial of m by a suitable selection of the characteristic polynomials of a i b i k i moreover is it possible to compute the appropriate characteristic polynomials of the a i b i k i or equivalently construct the k i needed to do so the first question is answered by constructing a mapping f r n rightarrow r n which maps a prescribed set of n of the feedback gains elements of k i i 1 n to the n coefficients of the characteristic polynomial of m the question then becomes given a p in r n does f x p have a solution the answer is found by constructing a homotopy h r n x o 1 rightarrow r n where h x 1 f x and h x 0 is some trivial function degree theory is then applied to guarantee that there exists an x t such that h x t t p for all t in 0 1 the parameterized sard s theorem is then utilized to prove that with probability 1 x t is a smooth curve and hence can be followed numerically from x 0 to x 1 by the solution of a differential equation davidenko s method

a new start system for solving deficient polynomial systems using continuation we are interested in all finite solutions of polynomial systems the start system normally used in the homotopy causes divergent continuation paths when applied to a deficient system our aim is to avoid this wasted computation by constructing a new start system the start system presented in this paper respects the m homogeneous structure of the system we want to solve

an efficient start system for multi homogeneous polynomial continuation when solving a system of polynomial equations using homotopy continuation the number of solution paths can often be significantly reduced by casting the equations in multi homogeneous form this requires a multi homogeneous start system having a full set of nonsingular solutions that can be easily computed we give a general form of such a start system that works for any multi homogeneous structure and that can be used efficiently in continuation

membership tests for images of algebraic sets by linear projections given a witness set for an irreducible variety v and a linear map p we describe membership tests for both the constructible algebraic set p v and the algebraic set p v we also provide applications and examples of these new tests including computing the codimension one components of p v p v additionally we also describe computing the geometric genus of a curve section of an irreducible component of the solution set of a polynomial system and a test for deciding whether a plane quartic curve is a luroth quartic

determining articulated motion from perspective views a decomposition approach abstract we consider the problem of estimating the 3d motion of an articulated object such as a robot arm or a human body from a monocular sequence of 2d perspective views we advocatte an approach of decomposition the object under analysis is decomposed into simpler parts each containing a small number of links we first estimate the motion of the simplest part s and then propagate the analysis to the remaining parts of the object human gait is used as an example however the approach is general to use this decomposition approach we need a repertoire of results for the motion of simple articulated objects motion estimation algorithms uniqueness and number of solutions especially how many views are needed for uniqueness with the help of techniques in algebraic geometry we have results for a number of cases which are particularly useful for human gait analysis

the random product homotopy and deficient polynomial systems most systems ofn polynomial equations inn unknowns arising in applications aredeficient in the sense that they have fewer solutions than that predicted by the total degree of the system we introduce the random product homotopy an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems in many cases the amount of computation required to find all solutions can be made roughly proportional to the number of solutions

coefficient parameter polynomial continuation we establish the theory for a very general approach to constructing homotopies for computing all geometrically isolated solutions to polynomial systems we assume that the coefficients for a polynomial system are given by parameters in engineering applications these would be derived from physical parameters since the coefficients themselves can be the parameters this is completely general we show that by considering the space of parameters instead of the space of coefficients we can exploit any special structure the coefficient parameter formulas impose on the solution set in particular we prove that we can ignore generic positive dimensional solution sets we also show that any side conditions additional equations can be used to reduce the number of paths these results unify and generalize all current approaches to polynimial continuation the effect of these new results is to greatly increase the efficiency of polynomial continuation in several engineering areas for example we establish a 36 path homotopy for a electrocardiographic application that needed 193 paths under the previous most appropriate theory also we show that a 16 path homotopy suffices for the inverse kinematics problem for 6 revolute joint manipulators a problem previously requiring 32 paths

stability results and strong invariance principles for partial sums of banach space valued random variables a general stability theorem for b valued random variables is obtained which refines a result of kuelbs and zinn our proof is based on two exponential inequalities for sums of independent b valued r v s essentially due to yurinskii and appears particularly simple we then use our theorem to prove strong invariance principles lil results and other related stability results for sums of i i d b valued r v s in the domain of attraction of a gaussian law most of these results seem to be still unknown for real valued r v s

nonlinear homotopies for solving deficient polynomial systems with parameters by a deficient polynomial system of n polynomial equation in n unknowns a system is meant that has fewer solutions than that predicted by the total degree or the bezout number of the system in practice a deficient polynomial system is often associated with a set of parameters q q 1 cdots q s in this paper nonlinear homotopies for numerically determining all isolated solutions of deficient m homogeneous systems are introduced the procedure is that when a system p q x 0 with particular parameter q 0 is solved it can be used to solve p q x 0 for any parameter q it is assumed that for a generic q p q x shares the same kind of deficiency at infinity thus far fewer paths need to be followed as an application it is shown how to compute the 32 solutions of the tsai morgan manipulator problem with 26 parameters by following only 32 solution paths

regeneration homotopies for solving systems of polynomials we present a new technique based on polynomial continuation for solving systems of n polynomials in n complex variables the method allows equations to be introduced one by one or in groups obtaining at each stage a representation of the solution set that can be extended to the next stage until finally obtaining the solution set for the entire system at any stage where positive dimensional solution components must be found they are sliced down to isolated points by the introduction of hyperplanes by moving these hyperplanes one may build up the solution set to an intermediate system in which a union of hyperplanes regenerates the intersection of the component with the variety of the polynomial or system of polynomials brought in at the next stage the theory underlying the approach guarantees that homotopy paths lead to all isolated solutions and this capability can be used to generate witness supersets for solution components at any dimension the first step in computing an irreducible decomposition of the solution set of a system of polynomial equations the method is illustrated on several challenging problems where it proves advantageous over both the polyhedral homotopy method and the diagonal equation by equation method formerly the two leading approaches to solving sparse polynomial systems by numerical continuation

dominating points and large deviations for random vectors we establish a representation formula useful for obtaining precise large deviation probabilities for convex open subsets of a banach space these estimates are based on the existence of dominating points in this setting

the continuation method for rational and deficient systems deficient polynomial systems those with fewer than the full number of solutions cannot be solved efficiently with the usual continuation methods this paper describes a continuation technique which allows some deficient and rational systems of equations to be solved with little wasted computation several examples from the literature are discussed

perturbed homotopies for finding all isolated solutions of polynomial systems abstract given a polynomial system f c n c n the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of f z 0 as well as points on any positive dimensional components of the solution set v f some of these methods are guaranteed to find all isolated solutions nonsingular and singular alike while others may not find singular solutions one of the most recent advances in this field is regeneration an equation by equation solver that is often more efficient than other methods however the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation the aim of this article is two fold first more generally we consider the use of perturbed homotopies for solving polynomial systems in particular we propose first solving a perturbed version of the polynomial system followed by a parameter homotopy to remove the perturbation such perturbed homotopies are sometimes more efficient than regular homotopies though they can also be less efficient second a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions including all singular isolated solutions this version of regeneration perturbed regeneration can decrease the efficiency of regeneration but increases its applicability

adaptive multiprecision path tracking this article treats numerical methods for tracking an implicitly defined path the numerical precision required to successfully track such a path is difficult to predict a priori and indeed it may change dramatically through the course of the path in current practice one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved to avoid unnecessary computational cost it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path we present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates we then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations

nonlinear reduction for solving deficient polynomial systems by continuation methods neither continuation methods nor symbolic elimination methods can be directly applied to compute all finite solutions to polynomial systems because the amount of computational time is mostly not proportional to the dimension of the system and to the number of finite solutions the notion of s polynomials is used to developed a reduction algorithm to lower the total degree of the deficient polynomial system so that computing the solutions at infinity can be avoided applying the reduction algorithm before solving the system with continuation methods yields a reliable solution method

finding all solutions to a system of polynomial equations given a polynomial equation of degree d over the complex domain the fundamen tal theorem of algebra tells us that there are d solutions assuming that the solutions are counted by multiplicity these solutions can be approximated by deforming a standard n th degree equation into the given equation and following the solutions through the deformation this is called the homotopy method the fundamental theorem of algebra can be proved by the same technique in this paper we extend these results and methods to a system of n polynomial equations in n complex variables we show that the number of solutions to such a system is the product of the degrees of the equations assuming that infinite solutions are included and solutions are counted by multiplicity the proof is based on a homotopy or deformation from a standard system of equations with the same degrees and known solutions this homotopy provides a computational method of approximating all solutions computational results demonstrating the feasibility of this method are also presented

moderate deviations and associated laplace approximations for sums of independent random vectors let xj be an i i d sequence of banach space valued r v s and let sn z 1 xj for certain positive sequences bn oo we determine the exact asymptotic behavior of e exp b2 n d sn bn where d is a smooth function we also prove a large deviation principle for 2y sn bn

algebraic c actions and the inverse kinematics of a general 6r manipulator abstract let x be a smooth quadric of dimension 2 m in p c 2 m 1 and let y z x be subvarieties both of dimension m which intersect transversely in this paper we give an algorithm for computing the intersection points of y z based on a homotopy method the homotopy is constructed using a c action on x whose fixed points are isolated which induces bialynicki birula decompositions of x into locally closed invariant subsets as an application we present a new solution to the inverse kinematics problem of a general six revolute serial link manipulator

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

complete stagnation of gmres we study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations our tool for analysis is a nonlinear system of equations the stagnation system that characterizes this behavior we focus on complete stagnation for which there is no progress until the last iteration we give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues finally we show that there are real matrices for which complete stagnation occurs for certain complex right hand sides but not for real ones 2003 elsevier science inc all rights reserved

on input allocation based regulation for linear over actuated systems results concerning the output regulation problem for over actuated linear systems are presented in this paper the focus is on the characterization of the solution of the full information regulator problem for systems which are right invertible but not left invertible and the input operator is injective the intrinsic redundancy in the plant model is exploited by parameterizing all solutions of the regulator equations and performing a static or dynamic optimization on the space of solutions this approach effectively shapes the non unique steady state of the system so that the long term behavior optimizes a given performance index in particular nonlinear cost functions that account for constraints on the inputs are considered within the general form of a hybrid system assumed for the allocation mechanism an example is given to illustrate the proposed methodology

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

the bkk root count in c n the root count developed by bernshtein kushnirenko and khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus c n in this paper we modify this bound slightly so that it counts the number of isolated zeros in c n our bound is apparently significantly sharper than the recent root counts found by rojas and in many cases easier to compute as a consequence of our result the huber sturmfels homotopy for finding all the isolated zeros of a polynomial system in c n can be slightly modified to obtain all the isolated zeros in c n

a convex geometric approach to counting the roots of a polynomial system consider a polynomial system f f1 fn in n variables with complex coefficients a standard way to bound the number of isolated roots of f in cn is to multiply and add the degrees of the fi in a straightfoward computation described explicitly by some variant of bezout s theorem unfortunately even if one knows a priori which monomial terms will appear this upper bound can be far from exact in practice one frequently knows which monomial terms will appear in a polynomial system before one actually decides to count or approximate its roots so a fast accurate root count which takes this additional information into account is vital we propose a much tighter upper bound on the number of isolated roots of f in cn and give an explicit combinatorial geometric criterion for when it is exact as a corollary we obtain a geometric characterization of the average case computational complexity of root counting our root count is a formula derived from the theory of toric varieties and its computation involves finding the mixed volume of n shadowed polytopes in rn our formula generalizes the bkk bound bernshtein 1975 kushnirenko 1976 khovanskii 1978 canny and rojas 1991 in four ways 1 for any r 0 n our formula can be used to count the roots whose first r coordinates are nonzero the bkk bound only handles the r n case as a corollary we obtain that for fixed n all of these root counts for generic f can be done in time polynomial in the numbers of monomial terms of the fi 2 we refine bernshtein s 1975 algebraic criterion for exactness of the bkk bound into a complete classification of the subsets of the coefficients of f whose genericity guarantees exactness in our generalized bound 3 our results are stated in terms of arbitrary algebraically closed fields of any characteristic the bkk bound was originally stated only over the complex numbers 4 we generalize all of the above to results on the number of n k dimensional components of the zero set of f1 fk when k n

a homotopy for solving polynomial systems this paper presents a continuation method for finding all solutions to polynomial systems it features a simpler homotopy than has been previously published

numerical solution of a class of deficient polynomial systems most systems of polynomials which arise in applications have fewer than the expected number of solutions the amount of computation required to find all solutions of such a deficient system using current homotopy continuation methods is proportional to the expected number of solutions and roughly to the size of the system much time is wasted following paths which do not lead to solutions we suggest methods for solving some deficient polynomial systems for which the amount of computational effort is instead proportional to the number of solutions

a methodology for solving chemical equilibrium systems this paper describes a methodology for solving chemical equilibrium systems the methodology is especially appropriate when many systems with the same structure must be solved quickly as in finite difference models of fluid flow and combustion the approach features identifying a canonical form for such systems and exploiting this canonical form to effect a preliminary algebraic reduction then numerical scaling and iterative solution techniques are evoked both homotopy continuation and a variant of newton s method are effective solution methods this methodology is a significant advance over previous approaches to solving chemical equilibrium systems the problem of solving such systems has heretofore been regarded as challenging solution techniques have tended to emphasize case by case analyses and hit or miss numerical methods we show that the problem can be solved by a unified approach quickly and reliably

computing isolated roots of sparse polynomial systems in affine space we present a symbolic probabilistic algorithm to compute the isolated roots in c n of sparse polynomial equation systems as some already known numerical algorithms solving this task our procedure is based on polyhedral deformations and homotopies but it amounts to solving a smaller number of square systems of equations and in fewer variables the output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system the complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre processing yielding the mixed cells in a subdivision of the family of these supports

uniqueness of solutions to three perspective views of four points we show that there is in general a unique solution for the relative orientation of three cameras simultaneously photographing four feature points on a fixed object however multiple solutions are possible in rare cases even when the four feature points are not coplanar

incomplete grobner basis as a preconditioner for polynomial systems precondition plays a critical role in the numerical methods for large and sparse linear systems it is also true for nonlinear algebraic systems in this paper incomplete grobner basis igb is proposed as a preconditioner of homotopy methods for polynomial systems of equations which transforms a deficient system into a system with the same finite solutions but smaller degree the reduced system can thus be solved faster numerical results show the efficiency of the preconditioner

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable