linear programming system identification the general nonnegative parameters case abstract by linear programming system identification we mean the problem of estimating the objective function coefficient vector and the technological coefficient matrix a for a linear programming system that best explains a set of input output vectors input vectors are regarded as available resources output vectors are compared to imputed optimal ones by a decisional efficiency measure and a likelihood function is constructed in an earlier paper we obtained results for a simplified version of the problem in this paper we propose a genetic algorithm approach for the general case in which and a are of arbitrary finite dimensions and have nonnegative components a method based on householder transformations and monte carlo integration is used as an alternative to combinatorial algorithms for the extreme points and volumes of certain required convex polyhedral sets the method exhibits excellent face validity for a published test data set in data envelopment analysis

high probability analysis of the condition number of sparse polynomial systems let f f1 fn be a random polynomial system with fixed n tuple of supports our main result is an upper bound on the probability that the condition number of f in a region u is larger than 1 e the bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the newton polytopes and underlying variances are all identical we also consider polynomials with real coefficients and give bounds for the expected number of real roots and restricted condition number using a kahler geometric framework throughout we also express the expected number of roots of f inside a region u as the integral over u of a certain mixed volume form thus recovering the classical mixed volume when u c n

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

lattice polytopes cut out by root systems and the koszul property we show that lattice polytopes cut out by root systems of classical type are normal and koszul generalizing a well known result of bruns gubeladze and trung in type a we prove similar results for cayley sums of collections of polytopes whose minkowski sums are cut out by root systems the proofs are based on a combinatorial characterization of diagonally split toric varieties

regenerative cascade homotopies for solving polynomial systems abstract a key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set in many problems involving a solution set of a polynomial system the witness superset contains all the needed information sommese and wampler gave the first numerical method to compute witness supersets based on dimension by dimension slicing of the solution set by generic linear spaces followed later by the cascade homotopy of sommese and verschelde recently the authors of this article introduced a new method regeneration to compute solution sets of polynomial systems tests showed that combining regeneration with the dimension by dimension algorithm was significantly faster than naively combining it with the cascade homotopy however in this article we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets computational tests give strong evidence that regenerative cascade is superior in practice to previous methods

parametric estimation of affine deformations of planar shapes we consider the estimation of affine transformations aligning a known 2d shape and its distorted observation the classical way to solve this registration problem is to find correspondences between the shapes and then compute the transformation parameters from these landmarks here we propose a novel approach where the exact transformation is obtained as the solution of a polynomial system of equations the method has been tested on synthetic as well as on real images and its robustness in the presence of segmentation errors and additive geometric noise has also been demonstrated we have successfully applied the method for the registration of hip prosthesis x ray images the advantage of the proposed solution is that it is fast easy to implement has linear time complexity works without established correspondences and provides an exact solution regardless of the magnitude of transformation

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

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

regularity results for solving systems of polynomials by homotopy method the homotopy method for solving systems of polynomial equations proposed by li also morgan is believed to be one of the simplest for computer implementation it is shown here that the parameters of the homotopy can be chosen from an open and dense set such that all isolated solutions of polynomial systems can be obtained

minimizing multi homogeneous bezout numbers by a local search method consider the multi homogeneous homotopy continuation method for solving a system of polynomial equations for any partition of variables the multi homogeneous bezout number bounds the number of isolated solution curves one has to follow in the method this paper presents a local search method for finding a partition of variables with minimal multi homogeneous bezout number as with any other local search method it may give a local minimum rather than the minimum over all possible homogenizations numerical examples show the efficiency of this local search method

central limit theorems for permuted regenerative estimators we prove strong laws of large numbers and central limit theorems for some permuted estimators from regenerative simulations these limit theorems provide the basis for constructing asymptotically valid confidence intervals for the permuted estimators

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

a numerical algorithm to find all feedback nash equilibria in scalar affine quadratic differential games this note deals with solving scalar coupled algebraic riccati equations these equations arise in finding linear feedback nash equilibria of the scalar n player affine quadratic differential game a numerical procedure is provided to compute all the stabilizing solutions the main idea is to reformulate the riccati equations into an extended eigenvalue eigenvector problem for a specific parametrized matrix u in i r 2 n times 2 n since the size of u increases exponentially on n the algorithm only applies for games where the number of players is not too large

on the nullspace the rangespace and the characteristic polynomial of euclidean distance matrices we present a characterization of the nullspace and the rangespace of a euclidean distance matrix edm d in terms of the vector of all ones and in terms of the gale subspace g d and the realization matrix p corresponding to d this characterization is then used to compute the characteristic polynomial of d we also present some results concerning edms generated by regular figures and edms generated by centrally symmetric points

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

on positivity of principal minors of bivariate bezier collocation matrix it is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct thus the corresponding interpolation matrix m is nonsingular schumaker stated the conjecture that all principal submatrices of m are nonsingular too furthermore all of the corresponding determinants the principal minors are conjectured to be positive this result would solve the constrained interpolation problem in this paper the conjecture on minors for polynomial degree 17 and conjecture for some particular configurations of domain points are confirmed

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 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

heuristic methods for computing the minimal multi homogeneous bezout number the multi homogeneous bezout number of a polynomial system is the number of paths that one has to follow in computing all its isolated solutions with continuation method each partition of variables corresponds to a multi homogeneous bezout number it is a challenging problem to find a partition with minimal multi homogeneous bezout number two heuristic numerical methods for computing the minimal multi homogeneous bezout number are presented in this paper some analysis of computational complexity are given numerical examples show the efficiency of these two methods

a subdivision based algorithm for the sparse resultant multivariate resultants generalize the sylvester resultant of two polynomials and characterize the solvability of a polynomial system they also reduce the computation of all common roots to a problem in linear algebra we propose a determinantal formula for the sparse resultant of an arbitrary system of n 1 polynomials in n variables this resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their bezout number our algorithm uses a mixed polyhedral subdivision of the minkowski sum of the newton polytopes in order to construct a newton matrix its determinant is a nonzero multiple of the sparse resultant and the latter equals the gcd of at most n 1 such determinants this construction implies a restricted version of an effective sparse nullstellensatz for an arbitrary specialization of the coefficients there are two methods that use one extra variable and yield the sparse resultant this is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n we conjecture its extension to producing an exact rational expression for the sparse resultant

mixed multiplicities of ideals versus mixed volumes of polytopes the main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as samuel s multiplicities in particular we can give a purely algebraic proof of bernstein s theorem which asserts that the number of common zeros of a system of laurent polynomial equations in the torus is bounded above by the mixed volume of their newton polytopes

macaulay style formulas for sparse resultants we present formulas for computing the resultant of sparse polynomials as a quotient of two determinants the denominator being a minor of the numerator these formulas extend the original formulation given by macaulay for homogeneous polynomials

the local structure of a chaotic attractor in four dimensions abstract a simple system of four ordinary differential equations exhibiting chaotic behavior is introduced the lyapunov exponents are approximately 3 34 0 00 1 79 and 5 55 points on the attractor where two variables simultaneously assume prechosen values are located by a numerical integration procedure in which suitable antecedent conditions are obtained by successive approximations the points are organized into curves which appear more numerous under higher resolution a backward integration procedure indicates that a line transverse to the curves intersects each curve rather than passing through gaps while the set of intersections forms a cantor set thus the attractor appears to be locally the product of three continua and one cantor set the continua are found to extend in the expanding neutral and less rapidly contracting local directions while the cantor set extends in the more rapidly contracting direction

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

homotopy method for generalized eigenvalue problems ax λbx abstract generalized eigenvalue problems can be considered as a system of polynomials the homotopy continuation method is used to find all the isolated zeros of the polynomial system which corresponds to the eigenpairs of the generalized eigenvalue problem a special homotopy is constructed in such a way that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs since the curves followed by general homotopy curve following scheme are computed independently of one another the algorithm is a likely candidate for exploiting the advantages of parallel processing to the generalized eigenvalue problems

random polynomials with prescribed newton polytope the newton polytope pf of a polynomial f is well known to have a strong impact on its zeros as in the kouchnirenko bernstein theorem on the number of simultaneous zeros of m polynomials with given newton polytopes in this article we show that pf also has a strong impact on the distribution of zeros of one or several polynomials we equip the space of holomorphic polynomials of degreen in m complex variables with its usual su m 1 invariant gaussian measure and then consider the conditional measures np induced on the subspace of polynomials whose newton polytope pfnp when p the unit simplex then it is obvious and well known that the expected distribution of zeros zf1 fk regarded as a current of k polynomials f1 fk of degree n is uniform relative to the fubini study form our main results concern the conditional expectation e np zf1 fk of zeros of k polynomials with newton polytope npnp where p deg p these results are asymptotic as the scaling factor n 1 we show that e np zf1 fk is asymptotically uniform on the inverse image ap of the open scaled polytope 1 p via the moment map cp m for projective space however the zeros have an exotic distribution outside of ap and when k m the case of the kouchnirenko bernstein theorem the percentage of zeros outside ap approaches 0 as n 1

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

homotopies for solving polynomial systems within a bounded domain abstract the problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain proving rouche s theorem by homotopy continuation concepts yields a new class of homotopy methods the so called regional homotopy methods these methods rely on isolating a part of the system to be solved which dominates the rest of the system on the border of the domain as the dominant part has a sparser structure it is easier to solve it will be used as start system in the regional homotopy the paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain applications illustrate the usefulness of the approach

balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methods polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems the numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of the powers of the continuation parameter to reduce this height we propose a procedure that operates as an intermediate stage between the mixed volume computation and the tracing of solution curves this procedure computes new lifting values of the support of a polynomial system these values preserve the structure of the mixed cell configuration obtained from the mixed volume computation and produce better balanced powers of the continuation parameter in the polyhedral homotopies

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

on steplength algorithms for a class of continuation methods the continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form fx b f d subset r n 1 to r n for given b in r n while these methods are similar in structure to those used for ode solvers their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step this suggests the use of a posteriors estimates of the convergence radii of the corrector it is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about f however it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors this is used for the design of various step algorithms for continuation processes two of them are based on a newton corrector while the third o

permuted derivative and importance sampling estimators for regenerative simulations abstract in a previous paper we introduced a new variance reduction technique for regenerative simulations based on permuting regeneration cycles in this paper we apply this idea to new classes of estimators in particular we derive permuted versions of likelihood ratio derivative estimators for steady state performance measures importance sampling estimators of the mean cumulative reward until hitting a set of states and tin estimators for steady state ratio formulas empirical results are presented showing that modest to significant variance reductions can be obtained

imperialist competitive algorithm for solving systems of nonlinear equations solving systems of nonlinear equations is a relatively complicated problem in which arise a diverse range of sciences there are a number of different approaches that have been proposed in this paper we employ the imperialist competitive algorithm ica for solving systems of nonlinear equations some well known problems are presented to demonstrate the efficiency of this new robust optimization method in comparison to other known methods

circum euclidean distance matrices and faces abstract we study the structure of circum euclidean distance matrices those euclidean distance matrices generated by points lying on a hypersphere we show for example that such euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all euclidean distance matrices also we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk d r 1 finally we obtain a geometric characterization of the faces of this cone given a configuration of points and its euclidean distance matrix d any matrix in the minimal face containing d comes from a configuration that is a linear perturbation of the points that generate d

generic properties of dispersion relations for discrete periodic operators an old problem in mathematical physics deals with the structure of the dispersion relation of the schrodinger operator v x in rn with periodic potential near the edges of the spectrum i e near extrema of the dispersion relation a well known and widely believed conjecture says that generically with respect to perturbations of the periodic potential the extrema are attained by a single branch of the dispersion relation are isolated and have nondegenerate hessian i e dispersion relations are graphs of morse functions the important notion of effective masses in solid state physics as well as the liouville property green s function asymptotics and so on hinges upon this property the progress in proving this conjecture has been slow it is natural to try to look at discrete problems where the dispersion relation is in appropriate coordinates an algebraic rather than analytic variety moreover such models are often used for computation in solid state physics the tight binding model alas counterexamples exist even for schrodinger operators on simple 2d periodic two atomic structures showing that the genericity fails in some discrete situations we start with establishing in a very general situation the following natural dichotomy the non degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters thus a random choice of a point in the parameter space gives the correct answer with probability one noticing that the known counterexample has only two free parameters one can suspect that this might be too tight for the genericity to hold we thus consider the maximal z2 periodic two atomic nearest cell interaction graph which has nine edges per unit cell and the discrete laplace beltrami operator on it which has nine free parameters we then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph since the proof is non trivial and would be much harder for more general structures we show three different approaches to the genericity which might be suitable in various situations it is also proven in this case that adding more parameters indeed cannot destroy the genericity result this allows us to list all bad periodic subgraphs of the one we consider and discover that in all these cases the genericity fails for trivial reasons only

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

fixed points of belief propagation an analysis via polynomial homotopy continuation belief propagation bp is an iterative method to perform approximate inference on arbitrary graphical models whether bp converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model to understand this dependence it is interesting to find all fixed points in this work we formulate a set of polynomial equations the solutions of which correspond to bp fixed points to solve such a nonlinear system we present the numerical polynomial homotopy continuation nphc method experiments on binary ising models and on error correcting codes show how our method is capable of obtaining all bp fixed points on ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties we further asses the accuracy of the marginals and weighted combinations thereof weighting marginals with their respective partition function increases the accuracy in all experiments contrary to the conjecture that uniqueness of bp fixed points implies convergence we find graphs for which bp fails to converge even though a unique fixed point exists moreover we show that this fixed point gives a good approximation and the nphc method is able to obtain this fixed point

numerical solution of bivariate and polyanalytic polynomial systems finding the real solutions of a bivariate polynomial system is a central problem in robotics computer modeling and graphics computational geometry and numerical optimization we propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition in contrast to existing eigen based solvers the proposed algorithm does not depend on grobner bases or normal sets nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution the method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and newton raphson refinement numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the sa

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

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

chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations abstract this study proposes a novel chaotic quantum behaved particle swarm optimization algorithm for solving nonlinear system of equations different chaotic maps are introduced to enhance the effectiveness and robustness of the algorithm several benchmark studies are carried out logistic map gives the best results and is utilized in solving nonlinear equation sets nine well known problems are solved with our algorithm and results are compared with quantum behaved particle swarm optimization intelligent tuned harmony search gravitational search algorithm and literature studies comparison results reveal that the proposed algorithm can cope with the highly non linear problems and outperforms many algorithms which exist in the literature

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

polynomial equation solving by lifting procedures for ramified fibers let be given a parametric polynomial equation system which represents a generically unramified family of zero dimensional algebraic varieties we exhibit an efficient algorithm which computes a complete description of the solution set of an arbitrary parameter instance from a complete description of the infinitesimal structure of a particular ramified parameter instance of our family this generalizes in the case of space curves previous methods of heintz et al and schost which require the given parameter instance to be unramified we illustrate our method solving particular polynomial equation systems by deformation techniques

on cell matrices a class of euclidean distance matrices abstract in this paper we study the set of cell matrices and its relationship with the cone of positive semidefinite diagonal matrices the set forms a convex polyhedral cone in the linear space of symmetric matrices we describe the faces of the cone and its polar we also provide a new linear inequality associated with cell matrices

on properties of cell matrices in this paper properties of cell matrices are studied a determinant of such a matrix is given in a closed form in the proof a general method for determining a determinant of a symbolic matrix with polynomial entries based on multivariate polynomial lagrange interpolation is outlined it is shown that a cell matrix of size n 1 has exactly one positive eigenvalue using this result it is proven that cell matrices are circum euclidean distance matrices c edm and their generalization k cell matrices are cedm under certain natural restrictions a characterization of k cell matrices is outlined

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

methods for constructing distance matrices and the inverse eigenvalue problem abstract let d 1 r k k and d 2 r l l be two distance matrices we provide necessary conditions on z r k l in order that d d 1 z z t d 2 r n n be a distance matrix we then show that it is always possible to border an n n distance matrix with certain scalar multiples of its perron eigenvector to construct an n 1 n 1 distance matrix we also give necessary and sufficient conditions for two principal distance matrix blocks d 1 and d 2 be used to form a distance matrix as above where z is a scalar multiple of a rank one matrix formed from their perron eigenvectors finally we solve the inverse eigenvalue problem for distance matrices in certain special cases including n 3 4 5 6 any n for which there exists a hadamard matrix and some other cases

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

euclidean graph distance matrices of generalizations of the star graph abstract in this paper a relation between graph distance matrices of the star graph and its generalizations and euclidean distance matrices is considered it is proven that distance matrices of certain families of graphs are circum euclidean their spectrum and generating points are given in a closed form

the cell matrix closest to a given euclidean distance matrix abstract cell matrix introduced by jaklic and modic 6 is a special euclidean distance matrix edm that has a quite attractive and simple structure it is of interest to approximate an edm by a cell matrix in this paper we consider the problem of finding the cell matrix that is closest to a given edm with respect to the frobenius norm the majorization ordering of the eigenvalues of a cell matrix is also discussed

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 note on methods for constructing distance matrices and the inverse eigenvalue problem abstract in this paper a symmetric nonnegative matrix with zero diagonal and given spectrum where exactly one of the eigenvalues is positive is constructed this solves the symmetric nonnegative eigenvalue problem sniep for such a spectrum the construction is based on the idea from the paper hayden reams wells methods for constructing distance matrices and the inverse eigenvalue problem some results of this paper are enhanced the construction is applied for the solution of the inverse eigenvalue problem for euclidean distance matrices under some assumptions on the eigenvalues

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

unification and extension of intersection algorithms in numerical algebraic geometry the solution set of a system of polynomial equations called an algebraic set can be decomposed into finitely many irreducible components in numerical algebraic geometry irreducible algebraic sets are represented by witness sets whereas general algebraic sets allow a numerical irreducible decomposition comprising a collection of witness sets one for each irreducible component we denote the solution set of any system of polynomials f c n c n as v f c n given a witness set for some algebraic set z c n and a system of polynomials f c n c n the algorithms of this paper compute a numerical irreducible decomposition of the set z v f while extending the types of intersection problems that can be solved via numerical algebraic geometry this approach is also a unification of two existing algorithms the diagonal intersection algorithm and the homotopy membership test the new approach includes as a special case the extension problem where one wishes to intersect an irreducible component a of v g x with v f x y where f introduces new variables y for example this problem arises in computing the singularities of a when the singularity conditions are expressed in terms of new variables associated to the tangent space of a several examples are included to demonstrate the effectiveness of our approach applied in a variety of scenarios

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

deformation techniques to solve generalised pham systems in heintz et al electron j sadio 1 1 1998 37 castro et al found comput math 2003 to appear and pardo proceedings eaca 2000 2000 pp 25 51 the authors have shown that universal solving procedures require exponential running time roughly speaking a universal solving procedure takes as input a system of multivariate polynomial equations and outputs complete symbolic information on the solution variety here we introduce a nonuniversal solving procedure adapted to generalised pham systems the aim is to compute partial information of the variety defined by the input system the algorithm is based on an homotopic deformation and on a non archimedean lifting procedure from a non singular zero of the homotopy curve the complexity of the procedure is also stated and it depends on some intrinsic quantity called the deformation degree of the given input system

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

computing all nonsingular solutions of cyclic n polynomial using polyhedral homotopy continuation methods all isolated solutions of the cyclic n polynomial equations are not known for larger dimensions than 11 we exploit two types of symmetric structures in the cyclic n polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solutions numerical results on the cyclic 8 to the cyclic 12 polynomial equations including their solution information are given

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

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

applying genetic algorithms for solving nonlinear algebraic equations solving of nonlinear algebraic equations is a prominent problem in science and engineering in this study the systems of these equations were solved using genetic algorithms three distinctive groups of the equations were studied a single nonlinear equation a simple set of nonlinear equations and a complex set of nonlinear equations in the two first cases the problem was converted into the unconstrained optimization problem with some simple mathematical relations then the genetic algorithm was used to solve the problem in the complex nonlinear equations the problem was changed into the constrained optimization problem and genetic algorithm in conjunction with augmented lagrangian function has been used to find the solution the proposed scheme has been evaluated with five benchmarks with different applications moreover the sensitivity analysis of the method to initial interval and size of population has been studied the results have a high precision and show good agreement in comparison with conventional methods

solving the scalar feedback nash algebraic riccati equations an eigenvector approach in this note we present an algorithm to compute all solutions of the scalar algebraic riccati equations that play an important role in finding feedback nash equilibria of the scalar n player linear quadratic differential game we show that all appropriate solutions can be obtained by analyzing the eigenstructure of a related matrix

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

exact algorithms for linear matrix inequalities let a x a 0 x 1a 1 cdots x na n be a linear matrix or pencil generated by given symmetric matrices a 0 a 1 ldots a n of size m with rational entries the set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron described by a linear matrix inequality we design an exact algorithm that up to genericity assumptions on the input matrices computes an exact algebraic representation of at least one point in the spectrahedron or decides that it is empty the algorithm does not assume the existence of an interior point and the computed point minimizes the rank of the pencil on the spectrahedron the degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil we provide explicit bounds for the complexity of our algorithm proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a mu

a numerical algorithm to find soft constrained nash equilibria in scalar lq games this paper provides a numerical algorithm to calculate all soft constrained nash equilibria in a regular scalar indefinite linear quadratic game the algorithm is based on the calculation of the eigenstructure of a certain matrix the analysis follows the lines of the approach taken by engwerda 2003 to calculate the solutions of a set of scalar coupled feedback nash algebraic riccati equations

linear programming system identification we define a version of the inverse linear programming problem that we call linear programming system identification this version of the problem seeks to identify both the objective function coefficient vector and the constraint matrix of a linear programming problem that best fits a set of observed vector pairs one vector is that of actual decisions that we call outputs these are regarded as approximations of optimal decision vectors the other vector consists of the inputs or resources actually used to produce the corresponding outputs we propose an algorithm for approximating the maximum likelihood solution the major limitation of the method is the computation of exact volumes of convex polytopes a numerical illustration is given for simulated data

the expected number of nash equilibria of a normal form game fix finite pure strategy sets s1 s and let s s x x s n in our model of a random game the agents payoffs are statistically independent with each agent s payoff uniformly distributed on the unit sphere in r a for given nonempty t 1 subset of s 1 t n subset of s n we give a computationally implementable formula for the mean number of nash equilibria in which each agent i s mixed strategy has support t i the formula is the product of two expressions the first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets ti the second may be construed as the probability that such an equilibrium remains an equilibrium when the strategies in the sets s i t i become available

sensitivity analysis for simulations via likelihood ratios we present a simple method of estimating the sensitivity of quantities obtained from simulation with respect to a class of parameters here sensitivity means the derivative of an expectation with respect to a parameter the class of parameters includes for example poisson rates discrete probabilities and the mean and variance of a normal distribution the method is extremely well suited to regenerative simulation and can be implemented on extant simulations with little effort increase in running time or memory requirements it is based on some change of measure formulas derived from likelihood ratios in addition to the theorems that underly the technique we present some numerical examples