solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

a relaxation procedure for domain decomposition methods using finite elements we present the convergence analysis of a new domain decomposition technique for finite element approximations this technique was introduced in 11 and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent no global preconditioner is needed in the practice but simply single domain finite element solvers are required an optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated applications are given to both elliptic equations and incompressible stokes equations

explicit polar decomposition and a near characteristic polynomial the 2 2 case abstract explicit algebraic formulas for the polar decomposition of a nonsingular real 2 2 matrix a are given as well as a classification of all integer 2 2 matrices that admit a rational polar decomposition these formulas lead to a functional identity which is satisfied by all nonsingular real 2 2 matrices a as well as by exactly one type of exceptional matrix a n for each n 2

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

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

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

minimal norm static feedbacks using dissipative hamiltonian matrices abstract in this paper we characterize the set of static state feedbacks that stabilize a given continuous linear time invariant system pair using dissipative hamiltonian matrices this characterization results in a parametrization of feedbacks in terms of skew symmetric and symmetric positive semidefinite matrices and leads to a semidefinite program that computes a static state stabilizing feedback this characterization also allows us to propose an algorithm that computes minimal norm static feedbacks the theoretical results extend to the static output feedback sof problem and we also propose an algorithm to compute the minimal norm sof we illustrate the effectiveness of our algorithm compared to state of the art methods for the sof problem on numerous numerical examples from the compleib library

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

on the kahan parlett jiang theorem a globally optimal backward perturbation error for two sided invariant subspaces abstract a crucial question in large non hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations for this problem the famous kahan parlett jiang theorem kahan et al 1982 15 seems to be the best compromise between generality and sharpness in essence it presents an a posteriori error bound on approximate solutions to eigenproblems which provides a powerful tool to evaluate quality of the computed two sided invariant subspaces of large scale non hermitian matrices however the perturbation error determined by this theorem is only locally optimal rather than globally optimal in this work we revisit this problem and derive a globally optimal backward perturbation error for given two sided approximate invariant subspaces our theorem enhances the kahan parlett jiang theorem and numerical experiments demonstrate the effectiveness of our theoretical results

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

parallel algorithms for algebraic riccati equations the matrix sign function is the basis of a parallel algorithm for solving the generalized algebraic riccati equation three forms of the algorithm were implemented and tested on a distributed memory hypercube multiprocessor performance results indicate that the method is an excellent means of solving large scale problems on a parallel computer

effective bounds for the number of transcendental points on subvarieties of semi abelian varieties let a be a semi abelian variety and x a subvariety of a both denned over a number field assume that x does not contain x x2 for any positive dimensional subvarieties x x2 of a let r be a subgroup of a c of finite rational rank we give doubly exponential bounds for the size of x n 0 x q among the ingredients is a uniform bound doubly exponential in the data on finite sets which are quantifier free definable in differentially closed fields we also give uniform bounds on x n v in the case where x contains no translate of any semi abelian subvariety of a and r is a subgroup of a c of finite rational rank which has trivial intersection with a q here a is assumed to be defined over a number field but x need not be algebraic torus c 2 and both a and x are defined over a number field the question asked by bombieri was to find uniform bounds for the number of generic points of x fl t namely those points which are not rational over the al gebraic closure of q buium obtained uniform bounds of an iterated exponential nature in the data deg x 9 rank t buium used differential algebraic methods the aim in the present paper is to improve the bounds to being doubly exponen tial in the relevant data as well as to treat the more general case of semi abelian varieties we are also able to slightly weaken the hypothesis on x the methods are also differential algebraic closely following buium but there are additional model theoretic ingredients coming from the theory of commutative groups of finite morley rank and there is also one more key observation regarding the differential algebraic set up which yields the doubly exponential bounds before stating the results let us discuss the nature of the data which for us will be quite naive we work for now over the complex numbers let a be a semi abelian variety of dimension n9 and x a subvariety of dimension ra both defined over q denote q by k assume a to be given as a locally closed

symmetric matrix pencils abstract a significant number of matrix eigenvalue problems quadratic or linear are best reformulated as pencils a m in which both a and m are real and symmetric some examples are given and then the canonical forms are re examined to explain the role of the sign characteristic attached to real eigenvalues in addition we examine the limitations on the use of the rayleigh quotient functional x ax x mx in describing the eigenvalues this sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations the reduction of indefinite pencils to useful sparse forms is mentioned

effective uniform bounding in partial differential fields abstract motivated by the effective bounds found in 12 for ordinary differential equations we prove an effective version of uniform bounding for fields with several commuting derivations more precisely we provide an upper bound for the size of finite solution sets of partial differential polynomial equations in terms of data explicitly given in the equations and independent of parameters our methods also produce an upper bound for the degree of the zariski closure of solution sets whether they are finite or not

factorizations into normal matrices in indefinite inner product spaces abstract we show that any nonsingular real or complex square matrix can be factorized into a product of at most three normal matrices one of which is unitary another is selfadjoint with eigenvalues in the open right half plane and the third one is normal involutory with a neutral negative eigenspace we call the last matrix normal neutral involutory here the words normal unitary selfadjoint and neutral are understood with respect to an indefinite inner product

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

numerically stable improved chebyshev halley type schemes for matrix sign function a general family of iterative methods including a free parameter is derived and proved to be convergent for computing matrix sign function under some restrictions on the parameter several special cases including global convergence behavior are dealt with it is analytically shown that they are asymptotically stable a variety of numerical experiments for matrices with different sizes is considered to show the effectiveness of the proposed members of the family

constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming it is known that the karush kuhn tucker kkt conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices we show in this paper that the primal and dual constraint nondegeneracies the strong regularity the nonsingularity of the b subdifferential of this nonsmooth system and the nonsingularity of the corresponding clarke s generalized jacobian at a kkt point are all equivalent moreover we prove the equivalence between each of these conditions and the nonsingularity of clarke s generalized jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing newton methods in particular we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies but without the strict complementarity

an iterative procedure with interface relaxation for domain decomposition methods a domain decomposition method for second order elliptic problems is considered an iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed at each iteration a relaxation is accomplished at the subdomain interfaces in several circumstances a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found moreover when such a value is not available an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed this iterative method is then applied to the spectral collocation approximation of the differential problem the same kind of convergence results are proven many numerical experiments show the effectiveness of the method proposed here

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

convergence of the domain decomposition finite element boundary element coupling methods in this work we analyze three available domain decomposition methods we also establish the convergence conditions the theoretical analysis provides an interval in which a relaxation parameter has to be chosen in order to achieve convergence moreover it allows the selection of the relaxation parameter so that convergence is achieved with the minimum number of iterations several example problems are given for elaboration

a domain decomposition approach for coupled modelling of nonlinear soil structure interaction abstract this paper proposes a new approach for modelling coupled nonlinear soil structure interaction problems by domain decomposition it is assumed that the soil structure coupled system is physically partitioned into independently modelled soil and structure sub domains a coupling procedure based on the sequential iterative dirichlet neumann coupling algorithm is presented which utilizes the condensed tangent stiffness matrices at the soil structure interface to ensure and accelerate convergence to compatibility in successive update of the boundary conditions as a special case of the proposed approach the condensed tangent stiffness matrix is approximated via reduced order models of the partitioned sub domains throughout the coupling iterations a simulation environment has been developed utilizing discipline oriented solvers for nonlinear structural and geotechnical analysis and is discussed this tool is used to demonstrate the applicability of the presented coupling algorithm in modelling nonlinear soil structure interaction problems highlighting its relative merits compared to interface relaxation algorithms furthermore the developed tool is employed for a case study involving nonlinear soil structure interaction analysis between a plane frame and soil subjected to ground excavation where the computational and modelling benefits of the proposed approach and the developed tool are clearly established

geometric axioms for differentially closed fields with several commuting derivations abstract a geometric first order axiomatization of differentially closed fields of characteristic zero with several commuting derivations in the spirit of pierce and pillay 1998 13 is formulated in terms of a relative notion of prolongation for kolchin closed sets

on iterative algorithms for the polar decomposition of a matrix and the matrix sign function in this paper we consider relations between the principal iterations from the pade family of kenney and laub for computing the matrix sign function and the principal iterations from the reciprocal pade family of greco iannazzo and poloni and the dual pade family of zi tak we show global convergence of the principal reciprocal pade iterations and the principal dual pade iterations we adopt the dual pade family of iterations to which the newton method belongs for computing the unitary polar factor of a nonsingular matrix we present numerical experiments with the scaled newton method of higham for computing the unitary polar factor which show how the quality of the computed inverses of matrices in the scaled newton method affects the accuracy of the computed polar factorization of a nonsingular matrix it indicates that assumptions under which the backward stability of the scaled newton method has been proved in the literature cannot be weaker

a note on the approximation of two sided rayleigh quotient to eigenvalues abstract we consider the two sided rayleigh quotient introduced by ostrowski as convergence acceleration tool for eigenvalue solvers an interesting property of the two sided rayleigh quotient is established its approximation to eigenvalues depends on the eigenvalue condition number this fact is valid not only for matrix but also for matrix polynomial multiparameter system and periodic matrix pairs

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

eigenvalues of rayleigh quotient matrices support thata is ann n hermitian matrix andx 1 y 1 aren 1 matrices satisfyingx 1 h x 1 y 1 h y 1 i 1 l n such that the column space x 1 is an eigenspace ofa and y 1 is an approximation of x 1 let a 1 x 1 h ax 1 h 1 y 1 h ay 1 r ay 1 y 1 h 1 and theta arccos x 1 h y 1 y 1 h x 1 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 geqslant 0 the matricesa 1 andh 1 are called the rayleigh quotient matrices ofa with respect tox 1 andy 1 respectively this paper derives new upper bounds of the difference between h 1 and a 1 the set of the eigenvaluesh 1 anda 1 in terms of the quantities sin and or r in a simple way here denotes a unitarily invariant norm

backward errors for eigenvalue and singular value decompositions we present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two norm and in the frobenius norm through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition when the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured analogous results hold for the singular vectors we indicate the effect of our error bounds on implementations for eigenvector and singular vector computation in a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal qr factor is not much larger than its distance to the closest orthogonal matrix

on the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

the hermitian p k 1 anti reflexive solutions of a linear matrix equation in this paper we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the hermitian p k 1 anti reflexive solutions of the matrix equation a x b and meanwhile the best approximation solution is considered then if the solvability conditions are not satisfied the least squares hermitian p k 1 anti reflexive solutions and the least squares hermitian p k 1 anti reflexive solutions with the minimum norm of the above matrix equation are respectively derived in addition two algorithms are shown to compute the least squares hermitian p k 1 anti reflexive solutions and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

an inequality for hilbert schmidt norm for the absolute value c c c 1 2 and the hilbert schmidt norm c hs trc c 1 2 of an operatorc the following inequality is proved for any bounded linear operatorsa andb on a hilbert space a b hs leqq 2 1 2 a b hs the corresponding inequality for two normal state and of a von neumann algebram is also proved in the following form d varphi psi leqq xi varphi xi psi leqq 2 1 2 d varphi psi here denotes the unique vector representative of a state in a natural positive conep form andd denotes the bures distance defined as the infimum which is also the minimum of the distance of vector representatives of and in particular xi varphi 1 xi varphi 2 leqq 2 1 2 xi 1 xi 2 for any vector representatives j of j j 1 2

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

continuity properties of the 1 inverse and perturbation bounds for the drazin inverse there are known results about the continuity of the important pseudoinverses we extend some well known results to the 1 inverse as for the drazin inverse which is not a 1 inverse we present new perturbation bounds

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction recently matrix based methods have gained wide attentions in pattern recognition and machine learning communities the generalized low rank approximations of matrices glram and the bilinear lanczos components blc algorithm are two popular algorithms that treat data as the native two dimensional matrix patterns however these two algorithms often require heavy computation time and memory space in practice especially for large scale problems in this paper we propose inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction we first introduce the thick restarting strategy to the blc algorithm and present a thick restarted lanczos components trblc algorithm in this algorithm we use the ritz vectors as approximations to dominant eigenvectors instead of the lanczos vectors in our implementation the iterative matrices are neither formed nor stored explicitly thanks to the characteristic of the lanczos procedure then we explore the relationship between the reconstruction error and the accuracy of the ritz vectors so that the computational complexities of eigenpairs can be reduced significantly as a result we propose an inexact thick restarted lanczos components inex trblc algorithm moreover we investigate the problem of incremental generalized low rank approximations of matrices and propose an incremental and inexact trblc incr trblc algorithm numerical experiments illustrate the superiority of the new algorithms over the glram algorithm and its variations as well as the blc algorithm for some real world image reconstruction and face recognition problems highlightswe proposed an inexact algorithm for image reconstructionthis algorithm is efficient both in storage and running time we analyze its convergence and backward errors an incremental version of the algorithm is given to handle dynamic increased data set

computing fundamental matrix decompositions accurately via the matrix sign function in two iterations the power of zolotarev s functions the symmetric eigenvalue decomposition and the singular value decomposition svd are fundamental matrix decompositions with many applications conventional algorithms for computing these decomposi

nonnegative inverse eigenvalue problems with partial eigendata in this paper we consider the inverse problem of constructing an n n real nonnegative matrix a from the prescribed partial eigendata we first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem to find a nonnegative solution we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth newton type method for solving its equivalent nonsmooth equation under some mild assumptions the global and quadratic convergence of our method is established we also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings

primal dual interior point methods for semidefinite programming in finite precision recently a number of primal dual interior point methods for semidefinite programming have been developed to reduce the number of floating point operations each iteration of these methods typically performs block gaussian elimination with block pivots that are close to singular near the optimal solution as a result these methods often exhibit complex numerical properties in practice we consider numerical issues related to some of these methods our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well conditioned but are unstable otherwise with this result we explain why one particular method the one introduced by alizadeh haeberly and overton is in general more stable than others we also explain why the so called least squares variation introduced for some of these methods does not yield more numerical accuracy in general finally we present results from our numerical experiments to support our analysis

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

the mosaic of high performance domain decomposition methods for structural mechanics formulation interrelation and numerical efficiency of primal and dual methods a multitude of domain decomposition methods ddm for structural mechanics is available in the literature today a unified framework for formulating primal and dual ddm is thus presented in this paper aiming at providing a mathematical platform for a uniform treatment of high performance ddm in structural mechanics a novel approach for developing new ddm from existing methods is also proposed and is applied to dual and primal methods in the field of the feti methods this approach leads to a new category of methods derived from existing feti variants furthermore two alternative formulations of the balancing domain decomposition method are described while interrelations between the introduced and existing methods are established finally comparative numerical tests demonstrate the differences in the computational performance of the methods in question

a note on the axioms for differentially closed fields of characteristic zero abstract we rework the foundations of the theory of differentially closed fields of characteristic zero in a geometric setting the new axioms will say that ifvis an irreducible variety andwis an irreducible subvariety of the appropriate torsor v projecting generically ontov thenwhas a generic point of the form a d a

the uniform companion for large differential fields of characteristic 0 we show that there is a theory uc of differential fields in several commuting derivatives of characteristic 0 which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields as an application we introduce differentially closed ordered fields differentially closed p adic fields and differentially closed pseudo finite fields

perturbation analysis of the matrix sign function perturbation expansions and new perturbation bounds for the matrix sign function are derived by using a simple technique the perturbation expansions are then used to derive frechet derivatives th

quantifier elimination for algebraic d groups we prove that if g is an algebraic d group in the sense of buium over a differentially closed field k of characteristic 0 then the first order structure consisting of g together with the algebraic d subvarieties of g g x g has quantifier elimination in other words the projection on g of a d constructible subset of g n 1 is d constructible among the consequences is that any finite dimensional differential algebraic group is interpretable in an algebraically closed field

structured backward errors for kkt systems karush kuhn tucker kkt systems are linear systems with coefficient matrices of the form graphics where h is symmetric a normwise structured backward error for kkt systems is defined and a com

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

domain decomposition and parallel processing of a finite element model of the shallow water equations abstract we present nonoverlapping domain decomposition techniques applied to a two stage numerov galerkin finite element model of the shallow water equations over a limited area domain the schur complement matrix formulation is employed and a modified interface matrix approach is proposed to handle the coupling between subdomains the resulting nonsymmetric schur complement matrices modified interface matrices as well as the subdomain coefficient matrices are solved using preconditioned conjugate gradient squared pcgs non symmetric iterative solvers various stages of the finite element solution are parallelized and the code is implemented ona four processor cray y mp supercomputer applying multitasking techniques in a dedicated environment

on estimating the separation of two regular matrix pairs let a b and c d be two n times n n n and m times m m m regular matrix pairs respectively the main contribution of this paper is to derive lower and upper bounds of the separation between a b and c d in terms of their generalized schur decompositions and weierstrass canonical forms the results are illustrated by numerical examples

the influence of orthogonality on the arnoldi method abstract many algorithms for solving eigenproblems need to compute an orthonormal basis the computation is commonly performed using a qr factorization computed using the classical or the modified gram schmidt algorithm the householder algorithm the givens algorithm or the gram schmidt algorithm with iterative reorthogonalization for the eigenproblem although textbooks warn users about the possible instability of eigensolvers due to loss of orthonormality few theoretical results exist in this paper we prove that the loss of orthonormality of the computed basis can affect the reliability of the computed eigenpair when we use the arnoldi method we also show that the stopping criterion based on the backward error and the value computed using the arnoldi method can differ because of the loss of orthonormality of the computed basis of the krylov subspace we also give a bound which quantifies this difference in terms of the loss of orthonormality

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

structured matrices and newton s iteration unified approach abstract recent progress in the study of structured matrices shows advantages of unifying the treatment of various classes of such matrices we recall some fundamental techniques for such a unification and then specify it in full details for newton s iteration which rapidly improves an initial approximation to the inverse matrix by performing two matrix multiplications per recursive step the iteration is particularly suitable for n n structured matrices represented with o n entries of their short generators rather than with their own n 2 entries based on such a representation matrix operations are performed much more rapidly and use much less memory space a major problem is to control the length of the generators which tends to grow quite rapidly in the iterative process two known methods solve this problem for toeplitz like and cauchy like matrices we extend both methods to a more general class of structured matrices and estimate the convergence rate as well as the computational complexity some novel techniques are introduced in this study in particular for the estimation of the norms of the inverse displacement operators

palindromic matrix polynomials matrix functions and integral representations abstract we study the properties of palindromic quadratic matrix polynomials z p qz pz 2 i e quadratic polynomials where the coefficients p and q are square matrices and where the constant and the leading coefficients are equal we show that for suitable choices of the matrix coefficients p and q it is possible to characterize by means of z well known matrix functions namely the matrix square root the matrix polar factor the matrix sign and the geometric mean of two matrices finally we provide some integral representations of these matrix functions

the orthogonal rayleigh quotient iteration orqi method abstract this paper presents a new method for computing all the eigenvectors of a real n n symmetric band matrix t the algorithm computes an orthogonal matrix q q 1 q n and a diagonal matrix diag 1 n such that tq q the basic ideas are rather simple assume that q 1 q k 1 and 1 k 1 have already been computed then q k is obtained via the rayleigh quotient iteration rqi method starting from an arbitrary vector u 0 the rqi method generates a sequence of vectors u l l 1 2 and a sequence of scalars l l 0 1 2 the theory tells us that these two sequences converge almost always to an eigenpair u the appeal of the rqi method comes from the observation that the final rate of convergence is cubic furthermore if the starting point is forced to satisfy q t i u 0 0 for i 1 k 1 as our method does then all the coming vectors u l l 1 2 and their limit point u should stay orthogonal to q 1 q k 1 in practice orthogonality is lost because of rounding errors this difficulty is resolved by successive orthogonalization of u l against q 1 q k 1 the key for effective implementation of the algorithm is to use a selective orthogonalization scheme in which u l is orthogonalized only against close eigenvectors that is u l is orthogonalized against q i only if l i where is a small threshold value e g t 1000 an essential feature of the proposed orthogonalization scheme is the use of reorthogonalization the orqi method is supported by forward and backward error analysis preliminary experiments on medium size problems n 1000 are quite encouraging the average number of iterations per eigenvector was less than 13 while the overall number of flops required for orthogonalizations is often below n 3 2

a sixth order iterative method for approximating the polar decomposition of an arbitrary matrix a new iterative method for computing the polar decomposition of any rectangular complex matrix is presented and analyzed the study of the convergence shows that this method has order of convergence six some numerical tests confirm the theoretical results and allow us to compare the proposed iterative scheme with other known ones

on a class of alternating coefficient matrices quadratic eigenvalue problem in this paper we consider a class of alternating coefficient matrices quadratic eigenvalue problem aqep appearing in a wide range of applications at first the aqep is reformulated as a generalized eigenvalue problem gep by applying tisseur s linearization technique then we point out a property about skew symmetric tridiagonal matrix eigenvalue problem consequently we give that the corresponding gep can be solved by applying the cholesky ql algorithm or lanczos algorithm without complex arithmetic the structured backward error for qep would be discussed because of the special structure of the polynomial eigenvalue problems we develop structured condition number the upper bound and the lower bound are given some computational experiments and some concluding remarks are provided

interface relaxation algorithms for bem bem coupling and fem bem coupling this paper presents several interface relaxation algorithms for boundary element boundary element coupling bem bem and for finite element boundary element coupling fem bem the domain of the original problem is sub divided into sub domains which are modeled by the finite element or boundary element methods the multi domain system is coupled using smoothing operators on the inter domain boundaries separate computations for the bem and fem sub domains and successive update of the boundary conditions at the interfaces are performed until convergence is achieved the interface relaxation coupling algorithms preserve the nature of the fem and bem further they do not require any access to the matrices generated by the fem or bem and make it easier to utilize different software in different sub domains

structured backward errors for generalized saddle point systems abstract in this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its 2 2 block is not zero and has perturbations computable formula of backward errors are derived the expressions are useful for testing the stability of practical algorithms

convex invertible sets and matrix sign function it is proved that in two dimensional unital real algebras a step by step construction of minimal convex invertible sets terminates in two steps an analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps and the complex field is the only finite dimensional complex algebra with this property connections are made with the matrix sign function the matrix sign function of a matrix belongs to the convex invertible set generated by the matrix an example is given to show that the matrix sign function cannot always be obtained in two steps by following the step by step construction

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

the sign matrix and the separation of matrix eigenvalues abstract the sign matrices uniquely associated with the matrices m j i 2 where j are the corners of a rectangle oriented at 4 to the axes of a cartesian coordinate system may be used to compute the number of eigenvalues of the arbitrarily chosen matrix m which lie within the rectangle and to determine the left and right invariant subspaces of m associated with these eigenvalues this paper is concerned with the proof of this statement and with the details of the computation of the required sign matrices

a newton squaring algorithm for computing the negative invariant subspace of a matrix by combining newton s method for the matrix sign function with a squaring procedure a basis for the negative invariant subspace of a matrix can be computed efficiently the algorithm presented is a variant of multiplication rich schemes for computing the matrix sign function such as the well known inversion free schulz method which requires two matrix multiplications per step however by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace the final newton steps can be replaced by steps which require only one matrix squaring each this efficiency is attained without sacrificing the quadratic convergence of newton s method

eigenvector free solutions to ax b with px xp and xh sx constraints abstract the matrix equation ax b with px xp and xh sx constraints is considered where p is a given hermitian involutory matrix and s 1 by an eigenvalue decomposition of p we equivalently transform the constrained problem to two well known constrained problems and represent the solutions in terms of the eigenvectors of p using moore penrose generalized inverses of the products generated by matrices a b and p the involved eigenvectors can be released and eigenvector free formulas of the general solutions are presented similar strategy is applied to the equations ax b xc d with the same constraints

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question