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

hermitian hermitian r symmetric and hermitian r skew symmetric procrustes problems let r cn n be a nontrivial unitary involution i e r r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i hermitian matrices ii hermitian r symmetric matrices iii hermitian r skew symmetric matrices given z w cn m we characterize the matrices a in s that minimize az w frobenius norm and given an arbitrary e cn n we find the unique matrix among the minimizers of az w in s that minimizes a e we also obtain necessary and sufficient conditions for existence of a s such that az w and assuming that the conditions are satisfied characterize the set of all such a

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors xi i 1m in cn and a set of complex numbers i i 1m find a centrosymmetric or centroskew matrix c in rn n such that xi i 1m and i i 1m are the eigenvectors and eigenvalues of c respectively we then consider the best approximation problem for the ieps that are solvable more precisely given an arbitrary matrix b in rn n we find the matrix c which is the solution to the iep and is closest to b in the frobenius norm we show that the best approximation is unique and derive an expression for it

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

the inverse eigenproblem with a submatrix constraint and the associated approximation problem for r s symmetric matrices abstract let r r n n and s r n n be nontrivial involutions i e r r 1 i and s s 1 i a matrix a r n n is called r s symmetric if r a s a this paper presents a r s symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint the solvability condition of the constrained inverse eigenproblem is also derived the existence the uniqueness and the expression of the r s symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved respectively an algorithm is presented to compute the r s symmetric matrix solution to the best approximation problem two numerical examples are given to illustrate the effectiveness of our results

procrustes problems and associated approximation problems for matrices with k involutory symmetries abstract we say that a matrix r c n n is k involutary if its minimal polynomial is x k 1 for some k 2 so r k 1 r 1 and the eigenvalues of r are 1 2 k 1 where e 2 i k let 0 1 k 1 if r c m m a c m n s c n n and r and s are k involutory we say that a is r s symmetric if ras 1 a and a is r s symmetric if ras a let l be the class of m n r s symmetric matrices or the class of m n r s symmetric matrices given x c n t and b c m t we characterize the matrices a in l that minimize ax b frobenius norm and given an arbitrary w c m n we find the unique matrix a l that minimizes both ax b and a w we also obtain necessary and sufficient conditions for existence of a l such that ax b and assuming that the conditions are satisfied characterize the set of all such a

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

the left and right inverse eigenvalue problems of generalized reflexive and anti reflexive matrices let nxn complex matrices r and s be nontrivial generalized reflection matrices i e r r r 1 i n s s s 1 i n a complex matrix a with order n is said to be a generalized reflexive or anti reflexive matrix if ras a or ras a in this paper the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti reflexive matrices are derived and the general solutions are also given in addition the associated approximation solutions in the solution sets of the above problems are provided the results in present paper extend some recent conclusions

a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices in this paper a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions we present the general solution of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices in real number field in addition in corresponding solution set of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices the explicit expression of the nearest matrix to a given matrix in the frobenius norm is obtained

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

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

approximation of wide sense stationary stochastic processes by shannon sampling series in this paper the convergence behavior of the symmetric and the nonsymmetric shannon sampling series is analyzed for bandlimited continuous time wide sense stationary stochastic processes that have absolutely continuous spectral measure it is shown that the nonsymmetric sampling series converges in the mean square sense uniformly on compact subsets of the real axis if and only if the power spectral density of the process fulfills a certain integrability condition moreover if this condition is not fulfilled then the pointwise mean square approximation error of the nonsymmetric sampling series and the supremum of the mean square approximation error over the real axis of the symmetric sampling series both diverge this shows that there is a significant difference between the convergence behavior of the symmetric and the nonsymmetric sampling series

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

reconstruction of band limited processes from irregular samples the problem of recovering say a band limited weakly stationary process from a set of its irregularly spaced samples is studied for rather general sampling sequences some sufficient conditions ensuring mean square or pathwise reconstruction are obtained for the cases of regular samples with either finitely many missing ones and or finitely many irregular ones a necessary and sufficient condition is presented some elements of the proofs involve classical results on nonharmonic fourier series as well as more recent results on frames

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem in this paper the left and right inverse eigenpairs problem for generalized centrosymmetric matrices is considered we obtain the necessary and sufficient conditions for the solvability of the problem and we present the general expression of the solution the related optimal approximation problem to a given matrix over the solution set is solved in addition a numerical algorithm and examples to solve the problem are given

characterization and properties of r s commutative matrices let r pdiag 0im0 1im1 k 1imk 1 p 1 cm m and s qdiag 0 in0 1 in1 k 1 ink 1 q 1 cn n where m0 m1 mk 1 m n0 n1 nk 1 n 0 1 k 1 are distinct complex numbers and zktozk 0 1 k 1 we say that a cm n is r s commutative if ra as we characterize the class of r s commutative matrices and extend results obtained previously for the case where l e2 il k and l l modk 0 l k 1 with zk our results are independent of 0 1 k 1 so long as they are distinct i e if ra as for some choice of 0 1 k 1 all distinct then ra as for arbitrary of 0 1 k 1

the solutions to linear matrix equations ax b ya d with k involutory symmetries let rcmm and scnn be nontrivial k involutions if their minimal polynomials are both xk1 for some k2 i e rk1 r1i and sk1 s1i we say that acmn is r s symmetric if ras1 a and a is r s symmetric if ras a with 0 1 k1 and 0 let s be one of the subsets of all r s symmetric and r s symmetric matrices given xcnr ycsm bcmr and dcsn we characterize the matrices a in s that minimize axb2 yad2 frobenius norm under the assumption that r and s are unitary moreover among the set s x y b d s of the minimizers of axb2 yad2 min we find the optimal approximate matrix as x y b d that minimizes ag to a given unstructural matrix gcmn we also present the necessary and sufficient conditions such that ax b ya d is consistent in s if the conditions are satisfied we characterize the consistent solution set of all such a finally a numerical algorithm and some numerical examples are given to illustrate the proposed results

left and right inverse eigenvalue problem of r s symmetric matrices and its optimal approximation problem the left and right inverse eigenvalue problem which mainly arises in perturbation analysis of matrix eigenvalue and recursive matters has some practical applications in engineer and scientific computation fields in this paper we give the solvability conditions of and the general expressions to the left and right inverse eigenvalue problem for the r s symmetric and r s skew symmetric solutions the corresponding best approximation problems for the left and right inverse eigenvalue problem are also solved that is given an arbitrary complex n by n matrix a find a r s symmetric or r s skew symmetric matrix a a which is the solution to the left and right inverse eigenvalue problem such that the distance between a and a a is minimized in the frobenius norm we give an explicit solution to the best approximation problem in the r s symmetric and r s skew symmetric solution sets of the left and right inverse eigenvalue problem under the assumption that r r and s s a numerical example is given to illustrate the effectiveness of our method

matrix inverse problem and its optimal approximation problem for r symmetric matrices abstract let r cn n be a nontrivial involution i e r2 i and r i a matrix a cn n is called r skew symmetric if rar a the least squares solutions of the matrix inverse problem for r skew symmetric matrices with r r are firstly derived then the solvability conditions and the solutions of the matrix inverse problem for r skew symmetric matrices with r r are given the solutions of the corresponding optimal approximation problem with r r for r skew symmetric matrices are also derived at last an algorithm for the optimal approximation problem is given it can be seen that we extend our previous results g x huang f yin matrix inverse problem and its optimal approximation problem for r symmetric matrices appl math comput 189 2007 482 489 and the results proposed by zhou et al f z zhou l zhang x y hu least square solutions for inverse problem of centrosymmetric matrices comput math appl 45 2003 1581 1589

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

an inverse eigenproblem and an associated approximation problem for generalized reflexive and anti reflexive matrices in this paper we first give the existence of and the general expression for the solution to an inverse eigenproblem defined as follows given a set of real n vectors x i i 1 m and a set of real numbers l i i 1 m and an n by n real generalized reflexive matrix a or generalized anti reflexive matrix b such that x i i 1 m and l i i 1 m are the eigenvectors and eigenvalues of a or b respectively we solve the best approximation problem for the inverse eigenproblem that is given an arbitrary real n by n matrix email protected we find a matrix a a which is the solution to the inverse eigenproblem such that the distance between email protected and a a is minimized in the frobenius norm we give an explicit solution and a numerical algorithm for the best approximation problem over generalized reflexive or generalized anti reflexive matrices two numerical examples are also presented to show that our method is effective

multi parameter perturbation methods for the eigensolution sensitivity analysis of nearly resonant non defective multi degree of freedom systems the dynamic behavior of structural systems may be strongly characterized by the occurrence of multiple internal resonances for particular combinations of the mechanical parameters the linear models governing these resonant or nearly resonant systems tend to exhibit high sensitivity of the eigenvalues and eigenvectors to small parameter modifications this pathological condition is recognized as a source of relevant phenomena such as frequency veering and mode localization or hybridization the paper presents the generalization of uniformly valid perturbation methods to perform eigensolution sensitivity analyses in multi degree of freedom hamiltonian systems with a generic number of close eigenvalues the leading idea is to treat systematically nearly resonant systems as multi parameter perturbations of a perfectly resonant non defective though a priori unknown reference system given a single nearly resonant system a multi parameter perturbation method is presented to achieve a two fold objective first identify a close resonant system suited to serve as a starting point for sensitivity analyses inverse problem second to approximate asymptotically the eigensolution of all the nearly resonant systems which may arise from its generic perturbation direct problem the conditions of existence and uniqueness of the inverse problem solution are discussed the direct problem solution is analyzed with a focus on the eigensolution sensitivity to parameter perturbations with different physical meanings such as a slight geometric disorder or weak elastic coupling in periodic structures finally the procedure is verified on a prototypical structural system describing the section dynamics of a suspended bridge

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

statistical mechanics of an np complete problem subset sum we study the statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics the problem is a generalization of the number partitioning problem which is also an npcomplete problem and has been studied in the physics literature the asymptotic expressions for the number of solutions are obtained these results applied to the number partitioning problem as a special case are compared with those which were previously obtained by a different method we discuss the limit of applicability of the techniques of statistical mechanics to the present problem

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

whittaker type derivative sampling reconstruction of stochastic lα ω processes abstract mean square and almost sure whittaker type derivative sampling theorems are obtained for the class l f p 0 2 of stochastic processes having spectral representation with the aid of the weierstras function functions of this class are represented by interpolatory series the results are valid for harmonizable and stationary processes 2 as well the formulae are interpreted in the mean sense and also in the almost sure p sense when the initial signal function and its derivatives up to some fixed order are sampled at the points of the integer lattice z 2 the circular truncation error is introduced and used in the truncation error analysis finally sampling sum convergence rate is provided

inverse eigenproblem for r symmetric matrices and their approximation let email protected c n x n be a nontrivial involution i e r r 1 i n we say that email protected c n x n is r symmetric if rgr g the set of all nxnr symmetric matrices is denoted by gsc n x n in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors x i i 1 m in c n and a set of complex numbers l i i 1 m find a matrix email protected gsc n x n such that x i i 1 m and l i i 1 m are respectively the eigenvalues and eigenvectors of a we then consider the following approximation problem given an nxn matrix email protected find email protected email protected s e such that email protected email protected email protected min a s email protected email protected email protected where s e is the solution set of iep and email protected email protected is the frobenius norm we provide an explicit formula for the best approximation solution email protected by means of the canonical correlation decomposition

constrained inverse eigenproblem and associated approximation problem for anti hermitian r symmetric matrices abstract let r c n n be a nontrivial involution i e r 2 i and r i a c n n is called anti hermitian r symmetric if a a and rar a the presentation and some properties for an arbitrary anti hermitian r symmetric matrix with r r and the relations between the eigenproblem for a and the corresponding eigenproblems for anti hermitian matrices are given then the solutions of constrained inverse eigenproblem and approximation problem are essentially decomposed into the same kind subproblems for anti hermitian matrices in complex field with smaller dimensions the explicit solutions for the later subproblems are arrived the corresponding problems which are the formulations of constrained inverse eigenproblem and approximation problem in complex field was first given then the solutions of constrained inverse eigenproblem and approximation problem with r r are derived

a generalized inverse eigenvalue problem in structural dynamic model updating this paper is concerned with the problem of the best approximation for a given matrix pencil under a given spectral constraint and a submatrix pencil constraint such a problem arises in structural dynamic model updating by using the moore penrose generalized inverse and the singular value decomposition svd matrices the solvability condition and the expression for the solution of the problem are presented a numerical algorithm for solving the problem is developed

the p q skew symmetric extremal rank solutions to a system of quaternion matrix equations abstract let h m n denote the set of all m n matrices over the quaternion algebra h and p h m m q h n n be involutions we say that a h m n is p q symmetric or p q skewsymmetric if a paq or a paq we in this paper mainly investigate the p q skew symmetric maximal and minimal rank solutions to the system of quaternion matrix equations ax b xc d we present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with p q symmetry and p q skewsymmetry to the system the expressions of such solutions to this system are also given when the solvability conditions are satisfied a numerical example is presented to illustrate our results the findings of this paper extend some known results in this literature

p skew symmetric common solutions to a pair of quaternion matrix equations abstract an n n quaternion matrix a is termed p symmetric or p skewsymmetric if a pap or a pap where p is an n n nontrivial quaternion involution in this paper we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations a 1 x 1 c 1 a 2 x 2 c 2 a 3 x 1 b 1 a 4 x 2 b 2 c b then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of p symmetric and p skewsymmetric solutions to the system of quaternion matrix equations a a x c a and a b xb b c b furthermore we establish representations of p symmetric and p skewsymmetric quaternion matrices a numerical example is presented to illustrate the results of this paper

an inverse eigenvalue problem and an associated approximation problem for generalized k centrohermitian matrices a partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized k centrohermitian matrices are considered it is shown under which conditions the inverse eigenproblem has a solution an expression of its general solution is given in case a solution of the inverse eigenproblem exists the optimal approximation problem can be solved the formula of its unique solution is given

linear parameterized inverse eigenvalue problem of bisymmetric matrices abstract in this paper we consider the linear parameterized inverse eigenvalue problem of bisymmetric matrices which is described as follows problem i given real bisymmetric matrices b 0 b 1 b 2 b n and real numbers 1 2 n find values of c c 1 c 2 c n t such that the eigenvalues of the matrix b c b 0 c 1 b 1 c 2 b 2 c n b n are precisely 1 2 n we first establish the structure of bisymmetric matrices based on the newton type iterative method we then turn problem i into the inverse eigenvalue problems of symmetric matrices with small sizes and solve these problems finally we present numerical examples to illustrate our results

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

linear restriction problem of hermitian reflexive matrices and its approximation in this paper we consider a linear restriction problem of hermitian reflexive matrices and its approximation by using the properties and structure of hermitian reflexive matrices and the special properties of reflexive vectors and anti reflexive vectors we convert the linear restriction problem to an equivalence problem trickily which is a special feature of this paper and is a different method from other articles then we solve this problem completely and also obtain its optimal approximate solution moreover an algorithm provided for it and the numerical examples show that the algorithm is feasible

approaches to robust pole assignment abstract robust pole assignment is a non linear optimization problem in many variables we describe numerical methods for determining robust or well conditioned so lutions to the problem of pole assignment by state feedback the solutions are chosen to minimize various objective functions based on the condition number of the eigenvector matrix careful choice of parametrization and objective function avoids singularities and artificial variable constraints explicit formulae for the gradient and hessian permit rigorous stopping criteria and rapid local convergence several computational examples are included

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

iterative method to solve the generalized coupled sylvester transpose linear matrix equations over reflexive or anti reflexive matrix abstract the iterative method of generalized coupled sylvester transpose linear matrix equations a x b c y t d s 1 e x t f g y h s 2 over reflexive or anti reflexive matrix pair x y is presented on the condition that the coupled matrix equations are consistent we show that the solution pair x y proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff error for any initial value given a reflexive or anti reflexive matrix moreover the optimal approximation reflexive or anti reflexive matrix solution pair to an arbitrary given reflexive or anti reflexive matrix pair can be derived by searching the least frobenius norm solution pair of the new generalized coupled sylvester transpose linear matrix equations finally some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient

a matrix inverse eigenvalue problem and its application the problem of generating a matrix a with specified eigenvalues which maps a given set of vectors of another given set is presented an existence theorem is given and proved a stable algorithm for producing the matrix a is discussed the relation between this problem and the pole assignment problem in control theory is investigated the application of this problem in the design of neural networks is discussed

on the eigenvectors of symmetric toeplitz matrices this paper presents a number of results concerning the eigenvectors of a symmetric toeplitz matrix and the location of the zeros of the filters eigenfilters whose coefficients are the elements of the eigenvectors one of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues if distinct have their zeros on the unit circle while the zeros of the other eigenfilters may or may not have their zeros on the unit circle even if the zeros of the eigenfilters of a matrix are all on the unit circle the matrix need not be toeplitz examples are given to illustrate the different properties

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

solving inverse eigenvalue problems via householder and rank one matrices abstract a brief and practical algorithm is introduced to solve symmetric inverse eigenvalue problems which we call hrou algorithm the algorithm is based on householder transformations and rank one updating we give some basic properties and the computational amount and develop sensitivity analysis of hrou algorithm furthermore we develop hrou algorithm into a multi level and adaptive one named mlahrou to solve symmetric nonnegative inverse eigenvalue problems new sufficient conditions to ensure symmetric nonnegative matrices and symmetric m matrices are given many numerical examples are given to verify our theory compare with existing results and show the efficiency of our algorithms

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

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

on the generalized reflexive and anti reflexive solutions to a system of matrix equations let p and q be two generalized reflection matrices i e p ph p2 i and q qh q2 i an n n matrix a is said to be generalized reflexive generalized anti reflexive with respect to the matrix pair p q if a paq a paq it is obvious that any n m matrix is also a generalized reflexive with respect to the matrix pair in im by extending the conjugate gradient least square cgls approach the present paper treats two iterative algorithms to solve the system of matrix equations f1 x a1 f2 x a2 fm x am including the sylvester and lyapunov matrix equations as special cases over the generalized reflexive and anti reflexive matrices where f1 f2 fm are the linear operators from cn n onto cri si and ai cri si for i 1 2 m when this system is consistent over the generalized reflexive generalized anti reflexive matrix it is proved that the first second iterative algorithm converges to a generalized reflexive generalized anti reflexive solution for any initial generalized reflexive generalized anti reflexive matrix also the first second iterative algorithm can obtain the the least frobenius norm generalized reflexive generalized anti reflexive solution for special initial generalized reflexive generalized anti reflexive matrix furthermore the optimal approximation generalized reflexive generalized anti reflexive solution to a given generalized reflexive generalized anti reflexive matrix can be derived by finding the least frobenius norm generalized reflexive generalized anti reflexive solution of a new system of matrix equations finally we test the proposed iterative algorithms and show their effectiveness using numerical examples

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

phase transition in the number partitioning problem universit at magdeburg institut fu r theoretische physik universit atsplatz 2 d 39106 magdeburg germany february 1 2008 number partitioning is an np complete problem of com binatorial optimization a statistical mechanics analysis re veals the existence of a phase transition that separates theeasy from the hard to solve instances and that re ects thepseudo polynomiality of number partitioning the phase di agram and the value of the typical ground state energy arecalculated 64 60 cn 02 60 pn 02 70 lq 89 80 h

typical solution time for a vertex covering algorithm on finite connectivity random graphs we analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex cover problem on finite connectivity random graphs we find two different transitions the first one is algorithm dependent and marks the dynamical transition from linear to exponential solution times the second one gives the maximum computational complexity and is found exactly at the threshold where the system undergoes an algorithm independent phase transition in its solvability analytical results are corroborated by numerical simulations

phase transitions of subset sum and shannon s limit in source coding abstract we consider statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics after introducing the statistical mechanical treatment of the problem the phase transition behaviors are discussed we also introduce a source coding scheme which is based on the idea of the subset sum it is shown that the phase transitions in the subset sum is related to shannon s limit for source coding

fundamental study a concise functional neural network computing the largest modulus eigenvalues and their corresponding eigenvectors of a real skew matrix quick extraction of the largest modulus eigenvalues of a real antisymmetric matrix is important for some engineering applications as neural network runs in concurrent and asynchronous manner in essence using it to complete this calculation can achieve high speed this paper introduces a concise functional neural network fnn which can be equivalently transformed into a complex differential equation to do this work after obtaining the analytic solution of the equation the convergence behaviors of this fnn are discussed simulation result indicates that with general initial complex values the network will converge to the complex eigenvector which corresponds to the eigenvalue whose imaginary part is positive and modulus is the largest of all eigenvalues comparing with other neural networks designed for the like aim this network is applicable to real skew matrices

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

further comments on on the eigenvectors of symmetric toeplitz matrices attention is drawn to some previous work on the matrices discussed in the comments of datta and morgera on the above paper

probabilistic analysis of the number partitioning problem given a sequence of n positive real numbers the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized in the case in which the s are statistically independent random variables uniformly distributed in the unit interval this np complete problem is equivalent to the problem of finding the ground state of an infinite range random antiferromagnetic ising model we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large n limit furthermore we calculate analytically the fraction of metastable states i e states that are stable against all single spin flips and found that it vanishes like