a framework of verified eigenvalue bounds for self adjoint differential operators for eigenvalue problems of self adjoint differential operators a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues in the case of the laplacian operator by applying crouzeix raviart finite elements an efficient algorithm is developed to bound the eigenvalues for the laplacian defined in 1d 2d and 3d spaces moreover for nonconvex domains for which case there may exist singularities of eigenfunctions around re entrant corners the proposed algorithm can easily provide eigenvalue bounds by further adopting the interval arithmetic the explicit eigenvalue bounds from numerical computations can be mathematically correct

the heun equation and the calogero moser sutherland system i the bethe ansatz method we propose and develop the bethe ansatz method for the heun equation as an application holomorphy of the perturbation for the bc1 inozemtsev model from the trigonometric model is proved

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

numerical verification of solutions for obstacle problems this article is an extension of the previous paper numer math 81 1998 305 by the same authors we propose a method to prove the existence of solutions for more general obstacle problems by numerical computations the main task in this paper consists of the numerical determination of some constants which appear in a priori error estimations for the finite element approximation of a simple variational inequality we present a numerical example for verification

numerical verification of solutions for obstacle problems using a newton like method abstract this paper is a continuation of the preceding study 1 in which we described a method which automatically proves the existence of solutions for variational inequalities by computer we newly formulate a verification method using a newton like method this approach enables us to remove the restriction in the previous paper to the retraction property of the operator in a neighborhood of the solution we show some numerical examples which confirm that the method is really applicable to problems which have no retraction property

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

pohlmeyer reduced form of string theory in ads5 s5 semiclassical expansion we consider the pohlmeyer reduced formulation of the ads5 s5 superstring it is constructed by introducing new variables which are algebraically related to supercoset current components so that the virasoro conditions are automatically solved the reduced theory is a gauged wzw model supplemented with an integrable potential and fermionic terms that ensure its uv finiteness the original superstring theory and its reduced counterpart are closely related at the classical level and we conjecture that they remain related at the quantum level as well in the sense that their quantum partition functions evaluated on respective classical solutions are equal we provide evidence for the validity of this conjecture at the one loop level i e at the first non trivial order of the semiclassical expansion near several classes of classical solutions

numerical verification of solutions for variational inequalities in this paper we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities this technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates for obstacle problems we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example for an obstacle problem is presented

using confidence limits for the global optimum in combinatorial optimization let z be the unknown optimal value of a combinatorial optimization problem in minimization form and let z be a constant satisfying probz z 1 for a fixed 0 1 then z is called a statistical or probabilistic lower bound at significance level 1 we discuss how statistical lower bounds can be used to measure the effectiveness of an approximate solution and to fathom subproblems in a branch and bound approach after reviewing some methods for obtaining statistical lower bounds we report on extensive computational experience for two notoriously difficult combinatorial optimization problems the traveling salesman problem and the quadratic assignment problem from our experience we conclude that statistical bounds are highly useful for quadratic assignment problems in this problem context the quality of simple analytic formulas is comparable to the quality of the maximum likelihood formula in general using statistics to aid in the solution of combinatorial optimization problems should be most useful when such problems are large

interval slopes for rational functions and associated centered forms for an arithmetic expression f x involving n rational operations an o n algorithm is given which computes an interval enclosure for the set of slopes f x z where x ranges over an interval x applications to real and complex centered forms are given resulting in improvements over previous results by ratschek siam j numer anal 17 1980 pp 656 662 and petkovic freiburger intervall berichte 83 2 1983 pp 33 50

a stochastic method for global optimization a stochastic method for global optimization is described and evaluated the method involves a combination of sampling clustering and local search and terminates with a range of confidence intervals on the value of the global optimum computational results on standard test functions are included as well

condition numbers and their condition numbers abstract various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized new results are derived for the cases where two common noninduced matrix norms are used and where different vector norms are used for the domain and range of the matrix condition numbers that respect the structure of symmetric problems are also analyzed the sensitivity of the condition number itself is then investigated and we obtain sharp examples of demmel s general result that for certain problems in numerical analysis the condition number of the condition number is the condition number finally upper bounds are derived for the sensitivity of componentwise condition numbers

bayesian stopping rules for multistart global optimization methods by far the most efficient methods for global optimization are based on starting a local optimization routine from an appropriate subset of uniformly distributed starting points as the number of local optima is frequently unknown in advance it is a crucial problem when to stop the sequence of sampling and searching by viewing a set of observed minima as a sample from a generalized multinomial distribution whose cells correspond to the local optima of the objective function we obtain the posterior distribution of the number of local optima and of the relative size of their regions of attraction this information is used to construct sequential bayesian stopping rules which find the optimal trade off between reliability and computational effort

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

accurate solution of dense linear systems part ii algorithms using directed rounding in part i and this part ii of our paper we investigate how extra precise evaluation of dot products can be used to solve ill conditioned linear systems rigorously and accurately in part i only rounding to nearest is used in this part ii we improve the results significantly by permitting directed rounding linear systems with tolerances in the data are treated and a comfortable way is described to compute error bounds for extremely ill conditioned linear systems with condition numbers up to about u 2 n where u denotes the relative rounding error unit in a given working precision we improve a method by hansen bliek rohn ning kearfott neumaier of the known methods by krawczyk rump hansen et al ogita and nguyen we show that our presented algorithm lsserrbnd seems the best compromise between accuracy and speed moreover for input data with tolerances a new method to compute componentwise inner bounds is presented for not too wide input data they demonstrate that the computed inclusions are often almost optimal all algorithms are given in executable matlab code and are available from my homepage

bounding the search space for global optimization of neural networks learning error an interval analysis approach training a multilayer perceptron mlp with algorithms employing global search strategies has been an important research direction in the field of neural networks despite a number of signifcant results an important matter concerning the bounds of the search region typically defined as a box where a global optimization method has to search for a potential global minimizer seems to be unresolved the approach presented in this paper builds on interval analysis and attempts to define guaranteed bounds in the search space prior to applying a global search algorithm for training an mlp these bounds depend on the machine precision and the term guaranteed denotes that the region defined surely encloses weight sets that are global minimizers of the neural network s error function although the solution set to the bounding problem for an mlp is in general non convex the paper presents the theoretical results that help deriving a box which is a convex set this box is an outer approximation of the algebraic solutions to the interval equations resulting from the function implemented by the network nodes an experimental study using well known benchmarks is presented in accordance with the theoretical results

componentwise perturbation theory for linear systems with multiple right hand sides existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right hand sides and to a general class of componentwise measure of perturbations involving holder p norms it is shown that for a system of order n with r right hand sides the componentwise backward error can be computed by finding the minimum p norm solutions to n underdetermined linear systems and an explicit expression is obtained in the case r 1 a perturbation bound is derived and from this the componentwise condition number is obtained to within a multiplicative constant applications of the results are discussed to invariant subspace computations quasi newton methods based on multiple secant equations and an inverse ode problem

single point seshadri constants on rational surfaces abstract motivated by a similar result of dumnicki kuronya maclean and szemberg under a slightly stronger hypothesis we exhibit irrational single point seshadri constants on a rational surface x obtained by blowing up very general points of p c 2 assuming only that all prime divisors on x of negative self intersection are smooth rational curves c with c 2 1 this assumption is a consequence of the shgh conjecture but it is weaker than assuming the full conjecture

inner and outer bounds for the solution set of parametric linear systems consider linear systems involving affine linear dependencies on interval parameters presented is a free c xsc software implementing a generalized parametric fixed point iteration method for verified enclosure of the parametric solution set some specific features of the corresponding algorithm concerning sharp enclosure of the contracting matrix and inner approximation of the solution enclosure are discussed

an enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems abstract we consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation this problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems we formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation we also consider the indices of eigenvalues especially the first eigenvalue of such a problem in these enclosing procedures the finite dimensional verified computations for linear and nonlinear system of equations play an essential role a numerical example is presented

towards ideal multistart a stochastic approach for locating the minima of a continuous function inside a bounded domain a stochastic global optimization method based on multistart is presented in this the local search is conditionally applied with a probability that takes in account the topology of the objective function at the detail offered by the current status of exploration as a result the number of unnecessary local searches is drastically limited yielding an efficient method results of its application on a set of common test functions are reported along with a performance comparison against other established methods of similar nature

rigorous sensitivity analysis for parameter dependent systems of equations abstract in this paper we construct rigorous and realistic enclosures for the solutions of parameter dependent systems of equations f x t 0 the enclosure of the solution x x t is given in the form x 0 s t t 0 with interval vectors x0 t0 and an interval matrix s this allows a detailed analysis of the sensitivity of the solution x t to small changes of t and the influence of specific changes in selected parameters on the solution the linear case is treated separately

rigorous sensitivity analysis for systems of linear and nonlinear equations methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent noncorrelated data for general systems of linear and nonlinear equations the methods may serve for the following two purposes first to bound the dependency of the solution on changes in the input data in contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed second to estimate the true solution set for problems the input data of which are afflicted with tolerances the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear systems demonstrating that the computed bounds are in general very sharp interesting comparisons to traditional condition numbers are given

identification of region of attraction for global optimization problem using interval symmetric operator in this paper some of the experience of the interval arithmetic methods for obtaining and locating the region of attraction of a global optimizer of a real continuous function f defined on a compact set d of r n is described

topological structure preserving numerical simulations of dynamical models this paper brings together two methods producing numerical solutions with a statement of their quality the nonstandard finite difference method and the method of validated computing it deals with the construction and the analysis of reliable numerical discretizations of dynamical systems by employing these two techniques an epidemiological model is used as a model example for their combined application

quantum mass correction for the twisted kink we present an analytic result for the 1 loop quantum mass correction in semiclassical quantization for the twisted 4 kink on s1 without explicit knowledge of the fluctuation spectrum for this purpose we use the contour integral representation of the spectral zeta function by solving the bethe ansatz equations for the n 2 lam equation we obtain an analytic expression for the corresponding spectral discriminant we discuss the renormalization issues of this model an energetically preferred size for the compact space is finally obtained

on nagata s conjecture abstract this paper gives an improved lower bound on the degrees d such that for general points p 1 p n p 2 and m 0 there is a plane curve of degree d vanishing at each p i with multiplicity at least m

an efficient numerical verification method for the kolmogorov problem of incompressible viscous fluid some computer assisted proofs of nontrivial steady state solutions for the kolmogorov flows are presented the method is based on the infinite dimensional fixed point theorem using a newton like operator with a numerical verification algorithm that automatically generates a set that includes the exact nontrivial solution when discussing the numerical results we consider the effects of rounding errors in the floating point computations this is a continuation of our study that was presented in watanabe 2009

numerical verifications of solutions for elliptic equations in nonconvex polygonal domains in this paper methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied in order to verify solutions using computer it is necessary to determine some constants which appear in a priori error estimations we propose some methods for determination of these constants in numerical examples calculating these constants for anl shaped domain we verify the solution of a nonlinear elliptic equation

rationality of seshadri constants and the segre harbourne gimigliano hirschowitz conjecture abstract in this paper we relate the shgh conjecture to the rationality of one point seshadri constants on blow ups of the projective plane

on the solution of hill s equation using milne s method milne s method is applied to the hill equation i e the schrodinger equation for a periodic one dimensional potential the method allows a compact discussion of the organization and parameter dependence of the stability bands as well as an efficient numerical computation of the band edges as an example a class of potentials which show a transition from two to one minimum per period is studied numerically furthermore periodic solutions of the milne equation are discussed and constructed

numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element the verifications of solutions to weakly nonlinear elliptic equations by the method described e g by nakao 1988 1989 etc are sometimes hardly accomplished when the right hand sides of the equations are very large to overcome such difficulties a residual iteration technique with approximate solution was introduced by nakao 1993 in the present paper we propose an a posteriori method for the residual iteration and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element

on very accurate enclosure of the optimal constant in the a priori error estimates for h02 projection we present constructive a priori error estimates for h 0 2 projection into a space of polynomials on a one dimensional interval here constructive indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form using the properties of legendre polynomials we consider a method by which to determine these constants to be as small as possible using the proposed technique the optimal constant could be enclosed in a very narrow interval with result verification furthermore constructive error estimates for finite element h 0 2 projection in one dimension are presented these types of estimates will play an important role in the numerical verification of solutions for nonlinear fourth order elliptic problems as well as in the guaranteed a posteriori error analysis for the finite element method or the spectral method e g hashimoto et al 2006 2 nakao et al 2008 3 watanabe et al 2009 11

case study on statistically estimating minimum makespan for flow line scheduling problems abstract lower bounds are typically used to evaluate the performance of heuristics for solving combinatorial minimization problems in the absence of tight analytical lower bounds optimal objective function values may be estimated statistically in this paper extreme value theory is used to construct confidence interval estimates of the minimum makespan achievable when scheduling nonsimilar groups of jobs on a two stage flow line experimental results based on randomly sampled solutions to each of 180 randomly generated test problems revealed that i least squares parameter estimators outperformed standard analytical estimators for the weibull approximation to the distribution of the sample minimum makespan ii to evaluate each weibull fit reliably both the anderson darling and kolmogorov smirnov goodness of fit tests should be used and iii applying a local improvement procedure to a large sample of randomly generated initial solutions improved the probability that the resulting weibull fit yielded a confidence interval covering the minimum makespan

the impact of accelerating tools on the interval subdivision algorithm for global optimization abstract the interval subdivision method of moore and skelboe is considered for global optimization the new implementation allows the use of a long list in the procedure which is limited only by the available computer memory standard global optimization test problems are used to measure the efficiency of different versions with or without the monotonicity and the cut off tests of the algorithm the new implementation and the inclusion of the monotonicity test made it possible for the first time for all standard test problems to be solved completely by an interval method moreover in some problems the efficiency of the algorithm was better than that of the best traditional techniques which do not give guaranteed reliability results

pure adaptive search in monte carlo optimization pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space at any stage the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence we show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem this compares to exponential growth in iterations required for pure random search

some classes of nonsingular matrices with applications to localize the real eigenvalues of real matrices abstract we first present some classes of nonsingular matrices which are then applied to give some simple nonsingular criteria inclusion and exclusion regions of real eigenvalues for real matrices in particular the real eigenvalues different from 1 of any stochastic matrix are localized more precisely

overestimation in linear interval equations bounds are given for the overestimation of the solution set of a system of linear interval equations by various inclusion intervals for this set in particular a theorem of gay this journal 19 1982 pp 858 870 on the quadratic approximation property of the preconditioned fixpoint inverse is strengthened

convergence qualification of adaptive partition algorithms in global optimization following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems necessary and sufficient convergence conditions are formulated in terms of respectively implied or postulated properties of the partition operator the convergence results obtained are pertinent to a number of deterministic algorithms in global optimization permitting their diverse modifications and generalizations

verified eigenvalue evaluation for the laplacian over polygonal domains of arbitrary shape the finite element method fem is applied to bound leading eigenvalues of the laplace operator over polygonal domains compared with classical numerical methods most of which can only give concrete eigenvalue bounds over special domains of symmetry our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape even when the eigenfunction has a singularity the problem of eigenvalue estimation is solved in two steps first we construct a computable a priori error estimation for the fem solution of poisson s problem which holds even for nonconvex domains with reentrant corners second new computable lower bounds are developed for the eigenvalues because the interval arithmetic is implemented throughout the computation the desired eigenvalue bounds are expected to be mathematically correct we illustrate several computation examples such as the cases of an l shaped domain and a crack domain to demonstrate the efficiency and flexibility of the proposed method

newtonian clustering an approach based on molecular dynamics and global optimization given a data set a dynamical procedure is applied to the data points in order to shrink and separate possibly overlapping clusters namely newton s equations of motion are employed to concentrate the data points around their cluster centers using an attractive potential constructed specially for this purpose during this process important information is gathered concerning the spread of each cluster in succession this information is used to create an objective function that maps each cluster to a local maximum global optimization is then used to retrieve the positions of the maxima that correspond to the locations of the cluster centers further refinement is achieved by applying the em algorithm to a gaussian mixture model whose construction and initialization is based on the acquired information to assess the effectiveness of our method we have conducted experiments on a plethora of benchmark data sets in addition we have compared its performance against four clustering techniques that are well established in the literature

a sharp version of bauer fike s theorem in this paper we present a sharp version of bauer fike s theorem we replace the matrix norm with its spectral radius or sign complex spectral radius for diagonalizable matrices 1 norm and norm for non diagonalizable matrices we also give the applications to the pole placement problem and the singular system

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

ellifit an unconstrained non iterative least squares based geometric ellipse fitting method a novel ellipse fitting method which is selective for digital and noisy elliptic curves is proposed in this paper the method aims at fitting an ellipse only when the data points are highly likely belong to an ellipse this is achieved using the geometric distances of the ellipse from the data points the proposed method models the non linear problem of ellipse fitting as a combination of two operators with one being linear numerically stable and easily invertible while the other being non linear but unique and easily invertible operator as a consequence the proposed ellipse fitting method has several salient properties like unconstrained stable non iterative and computationally inexpensive the efficacy of the method is compared against six contemporary and recent algorithms based on the least squares formulation using five experiments of diverse practical challenges like digitization incomplete ellipses and gaussian noise up to 30 three of the experiments comprise of a total of 44 400 ellipses positive test data while the other two are tested on 320 000 non elliptic conics negative test data the results show that the proposed method is quite selective to elliptic shapes only and provides accurate fitting results indicating potential application in medical robotics object detection and other image processing industrial applications

numerical verification of solutions for variational inequalities of the second kind abstract the purpose of this paper is to present an approach to the numerical proof of existence of solutions for the problem of the flow of a viscous plastic fluid in a pipe using the finite element approximations and the explicit a priori error estimates for the problem of the flow of a viscous plastic fluid in a pipe we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example is presented

verified solution of large systems and global optimization problems recent results on the solution of large banded or sparse systems and on global unconstrained optimization problems including verification of the correctness of the result will be discussed the computing time for the verification for large linear systems is the same as for the decomposition the total computational effort for verifying the global optimum value is for well known test examples competitive to those of pure floating point algorithms computational examples will be demonstrated

periodic functional determinants examines and evaluates the functional determinants associated with a class of periodic operators which arise in the quantum theory of solitons the functional determinants are expressed in terms of a finite subset of the spectrum of the operator a subset which corresponds to the edges of the valency or conduction bands of the schrodinger equation

microphone array optimization by stochastic region contraction the authors deal with optimal microphone placement and gain for a linear one dimensional array often in a confined environment a power spectral dispersion function psd is used as a core element for a min max objective function psdx derivation of the optimal spacings and gains of the microphones is a hard computational problem since the min max objective function exhibits multiple local minima hundreds or thousands the authors address the computational problem of finding the global optimal solution of the psdx function a new method stochastic region contraction src is proposed it achieves a computational speedup of 30 50 when compared to the commonly used simulated annealing method src is ideally suited for coarse gain parallel processing

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

on regular and singular interval systems we give a survey on interval linear systems discussing problems for regular systems as well as for singular ones we consider several solution sets and direct methods to enclose them moreover we study iterative methods particularly the total step method as the basis for other ones we also use this method for enclosing solutions of singular linear systems

a computer assisted proof for the kolmogorov flows of incompressible viscous fluid a computer assisted proof of non trivial steady state solutions for the kolmogorov flows is described the method is based on the infinite dimensional fixed point theorem using newton like operator this paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact non trivial solution with local uniqueness all discussed numerical results take into account the effects of rounding errors in the floating point computations

mass corrections to periodic solitons the quantum mass corrections to static periodic solitons are considered and expressed in terms of a functional quantity the discriminant of the hill s operator appearing several general expansions for this and hence the mass corrections are given in particular the connection with integrable systems and inverse scattering is clearly seen this enables one to obtain simple expressions for these corrections

on computational proofs of the existence of solutions to nonlinear parabolic problems abstract this paper is an extension of the preceding study nakao this journal 1991 in which we described a numerical verification method of the solution for one space dimensional parabolic problems to the several space dimensional case here numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer we reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed point problem of a compact operator on certain function spaces as in the preceding study based upon a simple c 0 finite element approximation and its constructive a priori error estimates a numerical verification procedure is presented with some numerical examples

stochastic global optimization methods part 1 clustering methods in this stochastic approach to global optimization clustering techniques are applied to identify local minima of a real valued objective function that are potentially global three different methods of this type are described their accuracy and efficiency are analyzed in detail

componentwise pseudospectrum of a matrix abstract given a nonnegative weight matrix e defining componentwise structured perturbations e e we introduce a componentwise pseudospectrum of a square matrix a as contour sets of the function c a i 1 e where m is the spectral radius of a square matrix m and discuss how to compute it

on an alternative to gerschgorin circles and ovals of cassini an alternative to gerschgorin circles for the localization of the real eigenvalues of a real matrix was studied in 8 in this paper we present a similar alternative to the brauer s theorem on ovals of cassini

the application of a unified bayesian stopping criterion in competing parallel algorithms for global optimization abstract the unconstrained global programming problem is addressed using a multistart multialgorithminfrastructure in which different algorithms compete in parallel for a contribution towards a single global stopping criterion denoted the unified bayesian global stopping criterion the use of different algorithms is motivated by the observation that no single global optimization algorithm consistently outperforms all other algorithms when applied to large sets of problems from different classes the bayesian stopping criterion is based on the single assumption that the probability of each algorithm converging to the global optimum is at least as large as the probability of convergence to any other local minimum this assumption is often valid in the case of practical problems of physical origin e g determining physical configurations corresponding to minimum potential energy results for parallel clusters of up to 128 machines are presented

conservatism of the circle criterion solution of a problem posed by a megretski in the collection of open problems in mathematical systems and control theory blondel et al 1999 megretski poses a problem related to the degree of conservativism of the well known circle criterion follows we solve this problem

algodesk an experimental comparison of eight evolutionary heuristics applied to the quadratic assignment problem abstract this work compares the effectiveness of eight evolutionary heuristic algorithms applied to the quadratic assignment problem qap reputedly one of the most difficult combinatorial optimization problems qap is merely used as a carrier for the comparison we do not attempt to compare any heuristics with solving algorithms specific for it results are given both with respect to the best result achieved by each algorithm in a limited time span and to its speed of convergence to that result

numerical verification of solutions for signorini problems using newton like method we proposed a numerical method to verify the existence of solutions for a simplified signorini problem comput math appl 2000 40 1003 1013 using sequential iteration method we numerically constructed a set containing solutions that satisfies the hypothesis of schauder s fixed point theorem in a certain sobolev space it is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution in this paper in order to overcome such a difficulty we describe an alternative approach to this problem numerical examples are presented copyright 2007 john wiley sons ltd

worst case bounds on the point wise discretization error in boundary element method for the elasticity problem in this work point wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation the formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem from these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods

structured perturbations and symmetric matrices abstract for a given n n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal bauer skeel condition number cannot be larger than 3 2 2 n in this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation for relative perturbations of the individual matrix components it is shown that a symmetric linear system can be arbitrarily ill conditioned while any symmetric and entrywise relative perturbation of the matrix of less than 100 does not produce a singular matrix that means that the inverse of the condition number and the distance to the nearest ill posed problem can be arbitrarily far apart finally we prove that restricting structured perturbations to symmetric entrywise perturbations cannot change the condition number by more than a factor 3 2 2 n

theorems of perron frobenius type for matrices without sign restrictions abstract the paper attempts to solve a problem which was called a challenge for the future in linear algebra appl we define and investigate a new quantity for real matrices the sign real spectral radius and derive various characterizations bounds and properties of it in certain aspects our quantity shows similar behavior to the perron root of a nonnegative matrix it is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix relations to the perron root of the entrywise absolute value of the matrix and to the number are given as well

connections between kelvin functions and zeta functions with applications a very simple method for resuming infinite series of the type sup approximately l 1 or 1 l 1 l n 1 2 cosh la k n 1 2 lb is presented as a result a series expansion in powers of a and b is obtained relevant to many physical applications

numerical verification of solutions for a simplified signorini problem abstract we proposed some numerical methods for the automatic proof of existence of solutions for variational inequalities i e obstacle problems and elastoplastic torsion problems it is based on nakao s method by using finite element approximation and its explicit error estimates for the problem in this paper we present a numerical verification algorithm of exact solutions for a simplified signorini problem in a computer a numerical example for the signorini problem is presented

numerical verification of solutions for some unilateral boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some unilateral boundary value problems using the finite element approximations and explicit a priori error estimates we construct in a computer a set of solutions which satisfies the hypothesis of schauder s fixed point theorem for a compact map on a certain sobolev space further the conditions of verifiability by this method are considered and some numerical examples are presented

estimation of the sensitivity of linear and nonlinear algebraic problems abstract methods are presented for performing a rigorous sensitivity analysis for general systems of linear and nonlinear equations w r t weighted perturbations in the input data the weights offer the advantage that all or part of the input data may be perturbed relatively or absolutely system zeros may depending on the application stay zero or not the main purpose of the paper is to give methods for computing rigorous bounds on the sensitivity of each individual component of the solution on the computer the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear and nonlinear systems demonstrating that the computed bounds are in general very sharp interesting comparisons with traditional condition numbers are given for linear systems the solution set for finite perturbations in the coefficients is estimated moreover some theoretical results for eigenvectors eigenvalues and singular values are given

probabilistic modeling and recognition of 3 d objects this paper introduces a uniform statistical framework for both 3 d and 2 d object recognition using intensity images as input data the theoretical part provides a mathematical tool for stochastic modeling the algorithmic part introduces methods for automatic model generation localization and recognition of objects 2 d images are used for learning the statistical appearance of 3 d objectss both the depth information and the matching between image and model features are missing for model generation the implied incomplete data estimation problem is solved by the expectation maximization algorithm this leads to a novel class of algorithms for automatic model generation from projections the estimation of pose parameters corresponds to a non linear maximum likelihood estimation problem which is solved by a global optimization procedure classification is done by the bayesian decision rule this work includes the experimental evaluation of the various facets of the presented approach an empirical evaluation of learning algorithms and the comparison of different pose estimation algorithms show the feasibility of the proposed probabilistic framework

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

local discretization error bounds using interval boundary element method in this paper a method to account for the point wise discretization error in the solution for boundary element method is developed interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented the developed method does not assume any special properties besides the laplace equation being a linear elliptic partial differential equation whose green s function for an isotropic media is known numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error copyright 2008 john wiley sons ltd

a posteriori error bounds for two point boundary value problems consider a general two point boundary value problem tpbvp begin gathered y t f t y hfill b 1 y a b 2 y b w quad hfill end gathered a leqq t leqq b where f r n 1 to r n f in c 2 b 1 and b 2 are n times n matrices and w in r n it is shown how one can bound a posteriori the error made in the numerical solution of the tpbvp the error bounds obtained are rigorous and include the truncation and the roundoff error in addition the computations establish the existence of solutions to the tpbvp numerical schemes are developed for the case where f t y is a polynomial in t and y examples are given of computational existence proofs for problems where analytical existence proofs are not known

a numerical approach to the proof of existence of solutions for some generalized obstacle problems in 8 we proposed some numerical verification methods for automatic proof of the existence of solution for obstacle problems in this paper we propose a new iterative algorithm to automatically prove the existence of solutions for some generalized obstacle problems

systems of fuzzy equations in structural mechanics systems of linear and nonlinear equations with fuzzy parameters are relevant to many practical problems arising in structure mechanics electrical engineering finance economics and physics in this paper three methods for solving such equations are discussed method for outer interval solution of systems of linear equations depending linearly on interval parameters fuzzy finite element method proposed by rama rao and sensitivity analysis method the performance and advantages of presented methods are described with illustrative examples extended version of the present paper can be downloaded from the web page of the utep i skalna m v rama rao a pownuk systems of fuzzy equations in structural mechanics the university of texas at el paso department of mathematical sciences research reports series texas research report no 2007 01 2007

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

an optimization approach to robust nonlinear control design abstract a new procedure of robust control design is presented for nonlinear systems with parametric uncertainty and disturbance this nonlinear optimization based approach can design robust nonlinear controllers of widely different structures and arbitrary complexity by taking the worst case design approach a minimax optimization problem is formulated a number of state of the art optimization procedures are explored as possible candidates for minimaximization it is determined that the tools of non smooth analysis and optimization are the most useful for the efficient solution of practical size worst case design problems included in the paper are the significant details of a special two tier structured minimax programming algorithm implemented in this work the overall algorithm is used in solving two robust nonlinear control design problems the controller designs produced by the minimax optimization algorithm are discussed and the performance of the controllers is studied by simulations

the globally convexized filled functions for global optimization a new filled function the globally convexized filled function is proposed the value of this function tends to positive infinity as the norm of the variable vector tends to infinity and it has no stationary point in the region in which the objective function value is greater than or equal to the least value found so far but it does have a minimizer not only along the radius direction from the minimizer with the least objective function value found so far like filled function found earlier by ge in the region in which the objective function value is less than the least value found so far but with a prefixed minimizer how to adjust parameters in the filled function is considered such a filled function can be appropriately combined with stochastic stopping rules proposed by boender rinnooy kan and timmer the computational results show that this algorithm is quite efficient and reliable

numerical verification of solutions for nonlinear elliptic problems using anl residual method abstract we consider a numerical enclosure method with guaranteedl error bounds for the solution of nonlinear elliptic problems of second order by using an a posteriori error estimate for the approximate solution of the problem with a higher orderc0 finite element it is shown that we can obtain the guaranteedl error bounds with high accuracy a particular emphasis is that our method needs no assumption of the existence of the solution of the original nonlinear equation but it follows as the result of computation itself a numerical example that confirms the effectiveness of the method is presented

ill conditioned matrices are componentwise near to singularity for a square matrix normed to 1 the normwise distance to singularity is well known to be equal to the reciprocal of the condition number in this paper we give an elementary and self contained proof for the fact that an ill conditioned matrix is also not far from a singular matrix in a componentwise sense this is shown to be true for any weighting of the componentwise distance in other words for matrix inversion ill conditioned means nearly ill posed in the normwise and also in the componentwise sense

perron frobenius theory for complex matrices abstract the purpose of this paper is to present a unified perron frobenius theory for nonnegative for real not necessarily nonnegative and for general complex matrices the sign real spectral radius was introduced for general real matrices this quantity was shown to share certain properties with the perron root of nonnegative matrices in this paper we introduce the sign complex spectral radius again this quantity extends many properties of the perron root of nonnegative matrices to general complex matrices various characterizations will be given and many open problems remain