the inverse eigenvalue problem of generalized reflexive matrices and its approximation national natural science foundation of china 10961010 natural science foundation of jiangxi china 2007gzs1760 jiangxi education office china gjj08432

the constrained inverse eigenvalue problem and its approximation for normal matrices abstract in this paper the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered the solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented and the expression of the solution for the optimal approximation problem is obtained

an inverse strategy for relocation of eigenfrequencies in structural design part ii second order approximate solutions this paper proposes an inverse strategy for relocation of structural natural frequencies using first order formulation and solution algorithm in the proposed method a sensitivity analysis of the systems eigenvalues with respect to material or geometrical parameters of the structure is conducted initially the required parameter variation to achieve a desired frequency shift for the structure is then computed the proposed technique incorporates the design constraints or objective functions in the system equations in such a way that a square system of equations is always preserved the formulations are general and applicable to all finite element structures because the sensitivity analysis is based on the stiffness and mass matrices regardless of the type of elements used an algorithm suitable for implementation of the technique for practical purposes is developed the accuracies of the proposed methods are tested conducting several case studies and the results are validated against exact solutions

theory and method for updating least squares finite element model of symmetric generalized centro symmetric matrices let x f be a displacement matrix and load matrix respectively c obtained by calculations or measurements is an estimate matrix of the analytical model a method is presented for correction of the model c based on the theory of inverse problem of matrices the corrected model is symmetric generalized centro symmetric with specified displacements and loads satisfying the mechanics characters of finite element model the application of the method is illustrated it is more important that a perturbation analysis is given which is not given in the earlier papers numerical results show that the method is feasible and effective

number of longest increasing subsequences we study the entropy s of longest increasing subsequences liss i e the logarithm of the number of distinct liss we consider two ensembles of sequences namely random permutations of integers and sequences drawn independent and identically distributed i i d from a limited number of distinct integers using sophisticated algorithms we are able to exactly count the number of liss for each given sequence furthermore we are not only measuring averages and variances for the considered ensembles of sequences but we sample very large parts of the probability distribution p s with very high precision especially we are able to observe the tails of extremely rare events which occur with probabilities smaller than 10 600 we show that the distribution of the entropy of the liss is approximately gaussian with deviations in the far tails which might vanish in the limit of long sequences further we propose a large deviation rate function which fits best to our observed data

large deviations of the length of the longest increasing subsequence of random permutations and random walks we study numerically the length distribution of the longest increasing subsequence lis for random permutations and one dimensional random walks using sophisticated large deviation algorithms we are able to obtain very large parts of the distribution especially also covering probabilities smaller than 10 1000 this enables us to verify for the length of the lis of random permutations the analytically known asymptotics of the rate function and even the whole tracy widom distribution we observe a rather fast convergence in the larger than typical part to this limiting distribution for the length l of lis of random walks no analytical results are known to us we test a proposed scaling law and observe convergence of the tails into a collapse for increasing system size further we obtain estimates for the leading order behavior of the rate functions in both tails

inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices the problem of generating a matrix a with specified eigenpairs where a is a tridiagonal symmetric matrix is presented a general expression of such a matrix is provided and the set of such matrices is denoted by s e moreover the corresponding least squares problem under spectral constraint is considered when the set s e is empty and the corresponding solution set is denoted by s l the best approximation problem associated with s e s l is discussed that is to find the nearest matrix a in s e s l to a given matrix the existence and uniqueness of the best approximation are proved and the expression of this nearest matrix is provided at the same time we also discuss similar problems when a is a tridiagonal bisymmetric matrix

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

the bisymmetric solutions of the matrix equation a1x1b1 a2x2b2 alxlbl c and its optimal approximation abstract a matrix a a ij r n n is said to be bisymmetric matrix if a ij a ji a n 1 j n 1 i for all 1 i j n in this paper an iterative method is constructed to find the bisymmetric solutions of matrix equation a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l c where x 1 x 2 x l is real matrices group by this iterative method the solvability of the matrix equation can be judged automatically when the matrix equation is consistent for any initial bisymmetric matrix group x 1 0 x 2 0 x l 0 a bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors and the least norm bisymmetric solution group can be obtained by choosing a special kind of initial bisymmetric matrix group in addition the optimal approximation bisymmetric solution group to a given bisymmetric matrix group x 1 x 2 x l in frobenius norm can be obtained by finding the least norm bisymmetric solution group of new matrix equation a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l c where c c a 1 x 1 b 1 a 2 x 2 b 2 a l x l b l

strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems it is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable a classical result of ky fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions which implies that the eigenvalues are semismooth functions based on a recent result of the authors it is further proved in this paper that the eigenvalues of a symmetric matrix are strongly semismooth everywhere as an application it is demonstrated how this result can be used to analyze the quadratic convergence of newton s method for solving inverse eigenvalue problems ieps and generalized ieps with multiple eigenvalues

least square solutions for inverse problems of centrosymmetric matrices let csrnxn a aij rnxn aij an 1 j i j 1 2 n in this paper we mainly discuss solving the following problems problem io given x b rnxm find a csrn n such that ax b problem i given x b rn m find a csrn n such that ax b min problem ii given a rn n find a se such that where is the frobenius norm and se is the solution set of problem i or i0 the expressions of the general solution of problems i and 10 have been derived the sufficient and necessary conditions for the solvability of problem i0 have been provided the numerical algorithm to find the optimal approximate solution and some numerical experiments have been given

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

sampling rare events statistics of local sequence alignments a method to calculate probability distributions in regions where the events are very unlikely e g p approximately 10 40 is presented the basic idea is to map the underlying model on a physical system the system is simulated at a low temperature such that preferably configurations with originally low probabilities are generated since the distribution of such a physical system is known the original unbiased distribution can be obtained as an application local alignment of protein sequences is studied the deviation of the distribution p s of optimum scores from the extreme value distribution is quantified this deviation decreases with growing sequence length

score statistics of global sequence alignment from the energy distribution of a modified directed polymer and directed percolation problem sequence alignment is one of the most important bioinformatics tools for modern molecular biology the statistical characterization of gapped alignment scores has been a long standing problem in sequence alignment research using a variant of the directed path in random media model we investigate the score statistics of global sequence alignment taking into account in particular the compositional bias of the sequences compared such statistics are used to distinguish accidental similarity due to compositional similarity from biologically significant similarity to accommodate the compositional bias we introduce an extra parameter p indicating the probability for positive matching scores to occur when p is small a high scoring alignment obviously cannot come from compositional similarity when p is large the highest scoring point within a global alignment tends to be close to the end of both sequences in which case we say the system percolates by applying finite size scaling theory on percolating probability functions of various sizes sequence lengths the critical p at infinite size is obtained for alignment of length t the fact that the score fluctuation grows as chi t 1 3 is confirmed upon investigating the scaling form of the alignment score using the kolmogorov smirnov statistics test we show that the random variable if properly scaled follows the tracy widom distributions gaussian orthogonal ensemble for p slightly larger than pc and gaussian unitary ensemble for larger p although these results deepen our understanding of the distribution of alignment scores the use of these results in practical applications remains somewhat heuristic and needs to be further developed nevertheless the possibility of characterizing score statistics for modest system size sequence lengths via proper reparametrization of alignment scores is illustrated

finite temperature local protein sequence alignment percolation and free energy distribution sequence alignment is a tool in bioinformatics that is used to find homological relationships in large molecular databases it can be mapped on the physical model of directed polymers in random media we consider the finite temperature version of local sequence alignment for proteins and study the transition between the linear phase and the biologically relevant logarithmic phase where the free energy grows linearly or logarithmically with the sequence length by means of numerical simulations and finite size scaling analysis we determine the phase diagram in the plane that is spanned by the gap costs and the temperature we use the most frequently used parameter set for protein alignment the critical exponents that describe the parameter driven transition are found to be explicitly temperature dependent furthermore we study the shape of the free energy distribution close to the transition by rare event simulations down to probabilities on the order 10 ensuremath 64 it is well known that in the logarithmic region the optimal score distribution t 0 is described by a modified gumbel distribution we confirm that this also applies for the free energy distribution tg0 however in the linear phase the distribution crosses over to a modified gaussian distribution

on large deviation properties of erdos renyi random graphs we show that large deviation properties of erdos renyi random graphs can be derived from the free energy of the q state potts model of statistical mechanics more precisely the legendre transform of the potts free energy with respect to ln q is related to the component generating function of the graph ensemble this generalizes the well known mapping between typical properties of random graphs and the q 1 limit of the potts free energy for exponentially rare graphs we explicitly calculate the number of components the size of the giant component the degree distributions inside and outside the giant component and the distribution of small component sizes we also perform numerical simulations which are in very good agreement with our analytical work finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges without recourse to the thermodynamics of the potts model

some results on the problems of decentralized reliable stabilization concerns the problems of decentralized reliable control in linear time invariant multivariable control systems by the stable factorization approach decentralized reliable control is defined as the problem in which the closed loop system remains stable in spite of failure in the interconnection between the subplants based on the decentralized parametric characterization developed by desoer and gundes in 1988 a necessary and sufficient condition for the decentralized version of the strong stabilization problem is first proposed next it is shown that the decentralized reliable stabilization problem is solvable if and only if the augmented plant is decentralized strongly stabilizable moreover it is concluded that the problem of centralized reliable multiple control can be viewed as the problem of decentralized reliable stabilization of an augmented plant

the solvability conditions for the inverse problems of symmetric ortho symmetric matrices this paper studies the following two problems problem igiven x email protected r n x m find email protected sr n p such that ax b wheresr n p email protected sr n x n email protected sr n x n email protected or n x nsatisfyingp t p problem iigiven email protected email protected r n x n find a s e such email protected email protected a inf a s email protected email protected email protected where email protected is the frobenius norm and s e is the solution set of problem i necessary and sufficient conditions for the solvability of problem i and the general form of the solution of problem i are given for problem ii the expression of the solution is provided furthermore it is pointed that some results of math numer sin 1 2000 29 are special cases of this paper

algebraic aspects of linear control system stability after formulating the problem of linear control system stability in very general terms the problem is resolved by 1 establishing a necessary and sufficient condition for stability 2 exploring in some detail the nature of this condition and 3 developing a parametric characterization of all controllers which stabilize a given plant the problem formulation is based on a ring theoretic setting for linear systems and is distinguished by the use of a novel control system configuration any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems furthermore no other configuration has this property several benefits and extensions are accrued as a result of using this configuration first a completely general yet physically meaningful problem formulation results consequently state space and operator theoretic approaches to linear control system stability are thoroughly reconciled this reconciliation embraces continuous time as well as discrete time systems lumped as well as distributed systems and 1 d as well as n d systems second the controller parametrization affords four degrees of freedom two more than any existing parametrization the additional degrees of freedom make explicit the design opportunity associated with controller implementation and thus determine several important control system characteristics this result is expected to provide a theoretical basis for the development of comprehensive control system design methodologies wherein design considerations related to controller implementation are addressed

the solvability conditions for the inverse eigenproblems of symmetric and generalized centro symmetric matrices and their approximations let be the frobenius norm of matrices we consider i the set se of symmetric and generalized centro symmetric real n n matrices rn with some given eigenpairs j qj j 1 2 m and ii the element r in se which minimizes r r for a given real matrix r necessary and sufficient conditions for se to be nonempty are presented a general form of elements in se is given and an explicit expression of the minimizer r is derived finally a numerical example is reported

the inverse eigenvalue problem for hermitian anti reflexive matrices and its approximation in this paper we first consider the inverse eigenvalue problem as follows find a matrix a with specified eigenpairs where a is a hermitian anti reflexive matrix with respect to a given generalized reflection matrix j the sufficient and necessary conditions are obtained and a general representation of such a matrix is presented we denote the set of such matrices by s a then the best approximation problem for the inverse eigenproblem is discussed that is given an arbitrary a find a matrix email protected s a which is nearest to a in the frobenius norm we show that the best approximation is unique and provide an expression for this nearest matrix

numerical methods for inverse singular value problems3 two numerical methods one continuous and the other discrete are proposed for solving inverse singular value problems the first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function the second method generalizes an iterative process proposed originally by friedland nocedal and overton siam j numer anal 24 1987 pp 634 667 for solving inverse eigenvalue problems with the geometry understood from the first method it is shown that the second method also the method proposed by friedland nocedal and overton for inverse eigenvalue problems is a variation of the newton method while the continuous method is expected to converge globally at a slower rate in finding a stationary point of the objective function the discrete method is proved to converge locally at a quadratic rate if there is a solution some numerical examples are presented

structural optimization with frequency constraints this paper presents a design optimization algorithm for structural weight minimization with multiple frequency constraints an optimality criterion method based on uniform lagrangian density for resizing and a scaling procedure to locate the constraint boundary were used in optimization multiple frequency constraints of equality and inequality types were addressed the effectiveness of the algorithm was demonstrated by designing a number of truss structures with as many as 489 design variables no attempt was made to reduce the number of design variables by such procedures as linking and or invoking symmetry conditions the design examples include a 10 bar truss 200 bar truss a modified acoss ii and cofs control of flexible structures mast truss all the structures contain nonstructural mass besides their own mass the algorithm is extremely stable and in all cases the optimum designs were obtained in less than 20 iterations regardless of the size of the structure and the number of design variables

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

characterization of controllers in simultaneous stabilization this paper gives a convenient parametrization for the class of all stabilizing controllers for two or more plants the result represent a generalization of the youla parametrization of the class of all stabilizing controllers in terms of an arbitrary stable proper transfer function although as expected the additional constraints for simultaneous stabilization are not readily incorporated into h h2 optimization procedures as in the standard case there is immediate application of the theory for reduction of controllers which simultaneously stabilize two or more plants

algebraic theory of linear time invariant feedback systems with two input two output plant and compensator this paper present an algebraic theory for linear time invariant multi input multi output systems with two input two output plant and compensator using the factorization approach this system configuration considered by doyle and nett is most general in that any interconnection of two systems can be represented in terms of this scheme other system configurations encountered in compensator design problems are special cases of this system configuration the analysis and synthesis applies to continuous time as well as discrete time systems and to lumped as well as distributed systems since the algebraic setting is completely general the compensator parametrization has four degrees of freedom the paper is self contained and is tutorial in the sense that it develops the main results of nett without introducing the concept of containment

the solvability conditions for the inverse eigenvalue problems of centro symmetric matrices this paper involves related inverse eigenvalue problems of centro symmetric matrices and their optimal approximation the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue and the general provided form of the solution furthermore the algorithm to compute the optimal approximate solution and some numerical experiments are given

solutions to an inverse monic quadratic eigenvalue problem abstract given n 1 pairs of complex numbers and vectors closed under complex conjugation the inverse quadratic eigenvalue problem is to construct real symmetric or anti symmetric matrix c and real symmetric matrix k of size n n so that the quadratic pencil q 2 i n c k has the given n 1 pairs as eigenpairs necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are obtained numerical algorithms for solving the problem are developed numerical examples illustrating these solutions are presented

distribution of diameters for erdős renyi random graphs we study the distribution of diameters d of erd s renyi random graphs with average connectivity c the diameter d is the maximum among all the shortest distances between pairs of nodes in a graph and an important quantity for all dynamic processes taking place on graphs here we study the distribution p d numerically for various values of c in the nonpercolating and percolating regimes using large deviation techniques we are able to reach small probabilities like 10 100 which allow us to obtain the distribution over basically the full range of the support for graphs up to n 1000 nodes for values c 1 the distribution is more complex and no complete analytical results are available for this parameter range p d exhibits an inflection point which we found to be related to a structural change of the graphs for all values of c we determined the finite size rate function d n and were able to extrapolate numerically to n indicating that the large deviation principle holds

frequency modification using newton s method and inverse iteration eigenvector updating an iterative procedure is proposed for determining structural changes necessary for modifying natural frequencies of a structure in a prescribed manner the procedure uses a newton s method approach to determine a structural change that satisfies the equation of motion of the modified structure the newton equation is the same as that derived from sensitivity analysis and from linear perturbation methods iterative application of this equation can be used to predict accurately the structural change for either large or small frequency modifications inverse iteration is used to update the structural eigenvector between newton iterations the application of this procedure to a finite element analysis is described natural frequency modification of a finite element structural model is shown to converge rapidly to accurate results

inverse problems for r s symmetric matrices in structural dynamic model updating let r s c n n be nontrivial involutions i e r r 1 i n and s s 1 i n a matrix a c n n is referred to as r s symmetric if and only if r a s a the set of all r s symmetric matrices of order n is denoted by c s n n r s given a full column rank matrix x c n m a matrix b c m m and a matrix a c n n in structural dynamic model updating we usually consider the sets s 1 a a c s n n r s x h a x b and s 2 a a c s n n r s x h a x b min in the frobenius norm sense where the superscript h denotes conjugate transpose then we characterize the unique matrices a arg min a s 1 a a and a arg min a s 2 a a when r r h and s s h by using the generalized singular value decomposition gsvd the necessary and sufficient conditions for the non emptiness of s 1 and the general representations of the elements in s 1 and a are derived respectively the analytical expressions of a s 2 and a are also obtained by using the gsvd the canonical correlation decomposition ccd and the projection theorem finally a corresponding numerical algorithm and some illustrated examples are presented

inverse structural modification using constraints in a structural modification problem the mass and stiffness matrices are modified to obtain a desired spectrum in this paper this is done by imposing constraints on the structure the undamped natural vibrations of a constrained linear structure are calculated by solving a generalized eigenvalue problem derived from the equations of motion for the constrained system involving lagrangian multipliers the coefficients of the constraint matrix are taken as design variables and a set of equations defining the inverse structural modification problem is formulated this modification problem requires an iterative method for its solution an algorithm based on newton s method is employed each iteration step involves the calculation of a rectangular jacobian and the solving of an associated underdetermined system of linear equations the system can be solved by using the moore penrose inverse the method is demonstrated in some numerical examples

the general solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti symmetric matrices this paper gives the precise solutions on the minimum residual problem and the matrix nearness problem for symmetric matrices or anti symmetric matrices firstly an equivalent linear equation is given for the minimum residual problem for symmetric matrices then by generalized singular value decomposition gsvd the expression of a general solution on the minimum residual problem for symmetric matrices is arrived furthermore for the matrix nearness problem for symmetric matrices an unique optimal approximate solution is given by first find the general symmetric solution of a corresponding equivalent matrix equation of the matrix nearness problem in addition explicit solutions on the minimum residual problem and the matrix nearness problem for anti symmetric matrices are derived

on the inverse eigenvalue problem for matrices atomic corrections the inverse eigenvalue problem for matrices is studied with the objective of obtaining an efficient method for correcting energy levels in atomic systems though the results are applicable to any eigenvalue problem the approach is a development of earlier work by s friedland 1977 the diagonal elements of a real symmetric matrix with given off diagonal elements are adjusted to yield a given spectrum the authors discuss cases where there are real solutions and no real solutions with particular emphasis on the latter problems of slow convergence arise they demonstrate the cause of this slow convergence give a geometrical interpretation of the problem and show how it can be avoided also the matrices encountered arise from complex calculations and are subject to error they develop an error analysis that permits us among other things to judge whether corrections in any particular case are justified in view of anticipated errors in the given computed off diagonal matrix elements finally the method is demonstrated in an application to certain sets of levels in 12 times ionized neon like titanium

sensitivity of repeated eigenvalues to perturbations two methods for calculating the derivatives of a repeated eigenvalue of viscously damped vibrating systems with respect to a parameter are given the first method implements the subspace spanned by the eigenvectors corresponding to the repeated eigenvalue the second method is based on an explicit formula that uses the characteristic equation directly without explicitly employing the eigenvector data examples demonstrate the various results

the solvability conditions for the inverse eigenvalue problems of reflexive matrices this paper involves related inverse eigenvalue problems of reflexive matrices and their optimal approximation the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue and the general provided form of the solution furthermore the algorithm to compute the optimal approximate solution and some numerical experiments are given

on the construction of band matrices from spectral data abstract we consider 2m 1 banded hermitian skew hermitian or complex symmetric matrices a we derive a set of eigenvalues of a and certain of its submatrices whose knowledge enables us to reconstruct a uniquely the dependence of a on the spectral data is shown to be continous

construction of band matrices from spectral data abstract a stable numerical algorithm is presented to generate a symmetric p band matrix from the given eigenvalues of the p greatest leading submatrices the algorithm consists of two parts first a matrix with the given spectral data is constructed then this matrix is transformed into a p band matrix leaving invariant the eigenvalues of the p greatest leading submatrices

a newton iteration process for inverse eigenvalue problems we suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases for the numerical solution of this problem we propose a newton iteration process and compare it with a known method finally we apply it to a numerical example

the hermitian reflexive solutions to the matrix inverse problem ax b in this paper the necessary and sufficient conditions for the solvability of the matrix inverse problem ax b in hermitian reflexive matrix set hj n x n are given and the solution expression of this matrix inverse problem is also presented moreover an optimal approximation to any matrix in the above solution set in frobenius norm is provided

an algorithm for symmetric generalized inverse eigenvalue problems abstract using qr like decomposition with column pivoting and least squares techniques we propose a new and efficient algorithm for solving symmetric generalized inverse eigenvalue problems and give its locally quadratic convergence analysis we also present some numerical experiments which illustrate the behaviour of our algorithm

large deviations of convex hulls of self avoiding random walks a global picture of a random particle movement is given by the convex hull of the visited points we obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self avoiding random walks namely the classical self avoiding walk the smart kinetic self avoiding walk and the loop erased random walk to obtain a comprehensive description of the measured random quantities we applied sophisticated large deviation techniques which allowed us to obtain the distributions over a large range of support down to probabilities far smaller than p 10 ensuremath 100 we give an approximate closed form of the so called large deviation rate function mathrm ensuremath phi which generalizes above the upper critical dimension to the previously studied case of the standard random walk further we show correlations between the two observables also in the limits of atypical large or small values

approximate cayley transform methods for inverse eigenvalue problems and convergence analysis abstract based on an approximation to cayley transform we propose an approximate cayley transform method and its inexact version for solving inverse eigenvalue problems which has the advantage over other known methods in the sense that it avoids solving systems in obtaining the approximate eigenvectors under the nonsingular condition used in d sun and j sun 29 we show that the proposed methods converge at least superlinearly moreover numerical experiments are given which illustrate that comparing with the cayley transform methods our methods need much less inner iterations and cpu time

efficient computation of mode shape derivatives for large dynamic systems this paper is concerned with the simplified numerical computation of eigenvector derivatives for very large problems with symmetric banded matrices such as those ordinarily encountered in structural dynamic systems starting with a brief review of existing alternate computation schemes the paper focuses on nelson s work which is at present generally accepted as the most computationally efficient method two competitive and more traditional alternatives to nelson s approach one direct and one iterative are offered no claim is made that either of these methods are superior to nelson s but they do offer viable and more conventional alternatives also discussed is the common case of repeated or closely spaced eigenvalues which was not treated by nelson a procedure for extending nelson s ideas for such cases is proposed

a numerical method of structure preserving model updating problem and its perturbation theory a method is presented to update a special finite element fe analytical model based on matrix approximation theory with spectral constraint at first the model updating problem is treated as a matrix approximation problem dependent on the spectrum data from vibration test and modal parameter identification the optimal approximation is the first modified solution of fe model an algorithm is given to preserve the sparsity of the model by multiple correction the convergence of the algorithm is investigated and perturbation of the modified solution is analyzed finally a numerical example is provided to confirm the convergence of the algorithm and perturbation theory

an inverse eigenvalue formulation for optimizing the dynamic behaviour of pin jointed structures abstract an efficient relationship between geometric and material properties of pin jointed truss structures and their eigenvalues is established the problem is formulated as an inverse eigenvalue problem this formulation allows the determination of the required modifications on the structural members to achieve specified eigenfrequencies in addition to the modification of the existing structural elements the formulation also allows addition of new structural elements to obtain the desired frequencies using the proposed inverse method two cases of plane as well as space truss structures are studied and the results are compared with those obtained using the conventional optimization techniques adopted by commercial finite element codes

distribution of shortest path lengths in subcritical erdős renyi networks networks that are fragmented into small disconnected components are prevalent in a large variety of systems these include the secure communication networks of commercial enterprises government agencies and illicit organizations as well as networks that suffered multiple failures attacks or epidemics the structural and statistical properties of such networks resemble those of subcritical random networks which consist of finite components whose sizes are nonextensive surprisingly such networks do not exhibit the small world property that is typical in supercritical random networks where the mean distance between pairs of nodes scales logarithmically with the network size unlike supercritical networks whose structure has been studied extensively subcritical networks have attracted relatively little attention a special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies the overall statistics of the network can be obtained by a summation over all the components with suitable weights we use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths dspl on components of given sizes and topologies in subcritical erd ifmmode mbox h o else h o fi s r enyi er networks from this expansion we obtain an exact analytical expression for the dspl of the entire subcritical network in the asymptotic limit the dspl which accounts for all the pairs of nodes that reside on the same finite component fc is found to follow a geometric distribution of the form p mathrm fc l ensuremath ell ll ensuremath infty 1 ensuremath c c ensuremath ell ensuremath 1 where cl1 is the mean degree using computer simulations we calculate the dspl in subcritical er networks of increasing sizes and confirm the convergence to this asymptotic result we also obtain exact asymptotic results for the mean distance ensuremath langle l ensuremath rangle mathrm fc and for the standard deviation of the dspl ensuremath sigma l mathrm fc and show that the simulation results converge to these asymptotic results using the duality relations between subcritical and supercritical er networks we obtain the dspl on the nongiant components of er networks above the percolation transition

on the construction of a jacobi matrix from its mixed type eigenpairs in this paper an inverse eigenvalue problem of constructing a jacobi matrix from its mixedtype eigenpairs is considered the necessary and sufficient conditions for the existence and uniqueness of the solutions are derived one numerical algorithm and two numerical examples are given

a robust reliable stabilization scheme for single input single output systems using transcendental methods abstract a stabilization scheme reliable against a single controller failure originally proposed by vidyasagar and viswanadham is analysed a connection between the problem of robust reliable stabilization and the problem of stabilization by a stable minimum phase compensator is established via interpolation theory

large deviation properties of the extended moran model the distributions of the times to the first common ancestor t mrca is numerically studied for an ecological population model the extended moran model this model has a fixed population size n the number of descendants is drawn from a beta distribution beta alpha 2 alpha for various choices of alpha this includes also the classical moran model alpha 0 as well as the uniform distribution alpha 1 using a statistical mechanics based large deviation approach the distributions can be studied over an extended range of the support down to probabilities like 10 70 which allowed us to study the change of the tails of the distribution when varying the value of alpha in 0 2 we find exponential distributions p t mrca delta t mrca in all cases with systematically varying values for the base delta only for the cases alpha 0 and alpha 1 analytical results are known i e delta exp 2 n 2 and delta 2 3 respectively we recover these values confirming the validity of our approach finally we also study the correlations between t mrca and the number of descendants

on the real symmetric inverse eigenvalue problem abstract the inverse eigenvalue problem of a real symmetric matrix dependent on several parameters is studied a new solution based on obtaining perturbation expansions of the eigensystem of such a matrix is presented the proposed solution is a modification of the well known newton method based on investigating the analyticity of the eigenvalues and the eigenvectors of the matrix

on the least squares solution of inverse eigenvalue problems an inverse eigenvalue problem where a matrix is to be constructed from some or all of its eigenvalues may not have a real valued solution at all an approximate solution in the sense of least squares is sometimes desirable two types of least squares problems are formulated and explored in this paper in spite of their different appearance the two problems are shown to be equivalent thus one new numerical method modified from the conventional alternating projection method is proposed the method converges linearly and globally and can be used to generate good starting values for other faster but more expensive and locally convergent methods the idea can be applied to multiplicative inverse eigenvalue problems for the purpose of preconditioning numerical examples are presented

objective method for estimating asymptotic parameters with an application to sequence alignment sequence alignment is an indispensable computational tool in modern molecular biology the model underlying biological sequence alignment is of interest to physicists because it approximates the statistical mechanics of dna and protein annealing while bearing an intimate relationship to models of directed polymers in random media recent methods for determining the statistics of random sequence alignments have reduced the computation time to less than 1 s opening up some interesting possibilities for online computation with biological search engines before implementation however the methods required an objective technique for computing regression coefficients pertinent to an asymptotic regime typically physicists estimate parameters pertinent to an asymptotic regime subjectively they eyeball their data estimate the asymptotic regime where the regression model holds with reasonable accuracy and then regress data only within the estimated asymptotic regime our publicly available computer program arrp replaces the subjective assessment of the asymptotic regime with an objective change point detection method increasing confidence in the scientific objectivity of the parameter estimates asymptotic regression has potential applications across most of physics

inverse eigenvalue problem in structural design a method is presented for the design of a structure with specified low order natural frequencies the method can also be used for design problems in which certain bands devoid of any frequency constraints are specified this can further be used to generate initial feasible designs for optimum design problems with frequency constraints

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

least squares solution for inverse eigenpair problem of nonnegative definite matrices suppose we know some eigenvalues i and eigenvectors xi associated with i i 1 2 m for a positive semidefinite may be unsymmetric matrix let x x1 x2 xm diag 1 2 m in this paper we mainly discuss solving the following two problems problem i given x rn m diag 1 m find matrices a such that ax x min where a is a positive semidefinite may be unsymmetric matrix problem ii given a rn n find a se such that a a mina se a a where is frobenius norm and se denotes the solution set of problem i an existence theorem of solution for problems i and ii has been given and proved and the general solutions of problem i have been derived sufficient conditions that prove an explicit solution have been provided