the local well posedness and global solution for a modified periodic two component camassa holm system abstract the local well posedness and global solution for a modified periodic two component camassa holm system on the circle s are established in the sobolev space h s h s 2 with s 7 2 which are different from that of the two component camassa holm system

refined iterative algorithms based on arnoldi s process for large unsymmetric eigenproblems abstract arnoldi s method has been popular for computing the small number of selected eigenvalues and the associated eigenvectors of large unsymmetric matrices however the approximate eigenvectors or ritz vectors obtained by arnoldi s method cannot be guaranteed to converge in theory even if the approximate eigenvalues or ritz values do in order to circumvent this potential danger a new strategy is proposed that computes refined approximate eigenvectors by small sized singular value decompositions it is shown that refined approximate eigenvectors converge to eigenvectors if ritz values do moreover the resulting refined algorithms converge more rapidly we report some numerical experiments and compare the refined algorithms with their counterparts the iterative arnoldi and arnoldi chebyshev algorithms the results show that the refined algorithms are considerably more efficient than their counterparts

the calculation of eigenvalues for the stationary perturbation of poiseuille flow abstract the problem considered is that of two dimensional viscous flow in a straight channel the decay of a stationary perturbation from the poiseuille flow in both the upstream and downstream directions is sought a differential eigenvalue equation resembling the orr sommerfeld equation is solved using a spectral method for values of the reynolds number r between 0 and 2000 although the decay downstream has received considerable previous attention the situation upstream has not been studied in detail it is found that for r greater than about 250 the upstream disturbances decay more rapidly by an order of magnitude greater than the downstream disturbances the eigen functions are used to obtain an asymptotic solution for two dimensional channel flow both upstream and downstream and comparison is made with a previous numerical solution for flow in a channel with a step change in width

numerical enclosure for each eigenvalue in generalized eigenvalue problem an algorithm for enclosing all eigenvalues in generalized eigenvalue problem email protected is proposed this algorithm is applicable even if email protected c n x n is not hermitian and or email protected c n x n is not hermitian positive definite and supplies nerror bounds while the algorithm previously developed by the author supplies a single error bound it is proved that the error bounds obtained by the proposed algorithm are equal or smaller than that by the previous algorithm computational cost for the proposed algorithm is similar to that for the previous algorithm numerical results show the property of the proposed algorithm

improving the accuracy of computed eigenvalues and eigenvectors this paper describes a computational method for improving the accuracy of a given eigenvalue and its associated eigenvector the method is analogous to iterative improvement for the solution of linear systems an iterative algorithm using working precision arithmetic is applied to increase the accuracy of the eigenpair the only extended precision computation is the residual calculation the method is related to inverse iteration and to newton s method applied to the eigenvalue problem

existence and instability of steady states for a triangular cross diffusion system a computer assisted proof abstract in this paper we present and apply a computer assisted method to study steady states of a triangular cross diffusion system our approach consist in an a posteriori validation procedure that is based on using a fixed point argument around a numerically computed solution in the spirit of the newton kantorovich theorem it allows to prove the existence of various non homogeneous steady states for different parameter values in some situations we obtain as many as 13 coexisting steady states we also apply the a posteriori validation procedure to study the linear stability of the obtained steady states proving that many of them are in fact unstable

blow up and global solutions to a new integrable model with two components abstract we will discuss a new integrable model which describes the motion of fluid the present work is mainly concerned with global existence and blow up phenomena which are largely due to the application of conservation laws for this integrable equations moreover a new blow up criterion for nonperiodic case is also established via the associated potential some interesting examples are also given to illustrate the application of our results the precise blow up rate is also investigated finally we will emphasize the relations of classical camassa holm equation and our model by analyzing the existence of global solutions

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

error estimation with guaranteed accuracy of finite element method in nonconvex polygonal domains a new method to estimate errors of the finite element method fem for nonconvex polygonal domain is proposed it gives mathematically rigorous upper bounds for the errors using calculations with guaranteed accuracy numerical examples are shown and their orders concerning mesh sizes are compared with theoretical orders

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

application of the iterative approach to modal methods for the solution of maxwell s equations in this work we discuss the possibility to reduce the computational complexity of modal methods i e methods based on eigenmodes expansion from the third power to the second power of the number of eigenmodes by applying the iterative technique the proposed approach is based on the calculation of the eigenmodes part by part by using shift and invert iterative procedure and by utilizing the iterative approach to solve linear equations to compute eigenmodes expansion coefficients as practical implementation the iterative modal methods based on polynomials and trigonometric functions as well as on finite difference scheme are developed alternatives to the scattering matrix s matrix technique which are based on pure iterative or mixed direct iterative approaches allowing to markedly reduce the number of required numerical operations are discussed additionally the possibility of diminishing the memory demand of the whole algorithm from second to first power of the number of modes by implementing the iterative approach is demonstrated this allows to carry out calculations up to hundreds of thousands eigenmodes without using a supercomputer modal methods computational complexity reduced to 2nd power of eigenmodes number memory demand reduced to the 1st power of the number of eigenmodes eigenmodes are found by batches using arnoldi method and shift and invert technique an iterative procedure is used instead of s or r matrix formalism calculations up to hundreds of thousands eigenmodes without using supercomputers

an iterative method for calculating several of the extreme eigensolutions of large real non symmetric matrices abstract a new method is described for calculating several of the extreme eigensolutions of large real non symmetric matrices the matrices considered are assumed to have real eigenvectors and eigenvaluesw an algorithm is presented for both single and multiple vector optimization the efficiency of the two versions are compared using some test calculations the method is a generalization of the davidson approach

fast enclosure of matrix eigenvalues and singular values via rounding mode controlled computation abstract modifications of bauer fike type and weyl type perturbation theorems are presented for matrix eigenvalue and singular value problems it is shown that the conditions of the presented theorems can be rigorously checked by floating point computation with rounding mode control it is stressed that verification programs can be easily constructed on usual numerical softwares like matlab computational cost of obtaining rigorous error bounds for computed eigenvalues is shown to be 6n 3 flops for a real symmetric n n matrix

the cauchy problem for a two component generalized θ equations abstract in this paper we investigate a two component system which arises in shallow water theory the present work is mainly concerned with blow up phenomenon persistence properties and infinite propagation speed for this new kind of system firstly we establish sufficient conditions on the initial data to guarantee wave breaking for this system moreover the persistence properties of the strong solutions are also analyzed finally we intend to investigate infinite propagation speed in the sense that the corresponding solution u x t does not have compact spatial support for t 0 though u 0 c 0 r while the other one x t is always compactly supported under the assumption of its initial compact support condition

error bounds on complex floating point multiplication given floating point arithmetic with t digit base beta significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value we prove that the product of complex values z 0 and z 1 can be computed with maximum absolute error abs z 0 abs z 1 frac 1 2 beta 1 t sqrt 5 in particular this provides relative error bounds of 2 24 sqrt 5 and 2 53 sqrt 5 for ieee 754 single and double precision arithmetic respectively provided that overflow underflow and denormals do not occur we also provide the numerical worst cases for ieee 754 single and double precision arithmetic

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

the elr method for computing the eigenvalues of a general matrix this paper presents an algorithm for computing all the eigenvalues of a full nonsymmetric matrix a all the current algorithms begin with the reduction of a to a hessenberg matrix h it is widely believed that the qr algorithm is the best method for computing the eigenvalues of h in this paper we suggest an alternative method which based on the fact that the eigenvalues of a tridiagonal matrix can be computed much faster than those of a hessenberg matrix we first reduce h to a tridiagonal matrix t and then compute the eigenvalues of t the reduction of h to t is performed by the elimination method 3 while the eigenvalues of t are computed by the lr method although the above two methods have been well known for a long time their use in this way has traditionally been rejected because of the possibility of numerical instability we show here that under certain conditions the elimination method is quite stable furthermore we describe a strategy for the lr algorithm that again results in stability

a newton based method for the calculation of the distance to instability in this paper a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided the method uses newton s method to find a two dimensional jordan block corresponding to a pure imaginary eigenvalue in a certain two parameter hamiltonian eigenvalue problem introduced by byers r byers a bisection method for measuring the distance of a stable matrix to the unstable matrices siam j sci statist comput 9 1988 875 881 this local method is augmented by a test step previously used by other authors to produce a global method numerical results are presented for several examples and comparison is made with the methods of boyd and balakrishnan s boyd v balakrishnan a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm systems control lett 15 1990 1 7 and he and watson c he g a watson an algorithm for computing the distance to instability siam j matrix anal appl 20 1999 101 116

wavelets associated with representations of the affine weyl heisenberg group the affine weyl heisenberg group generated by time and frequency translations and time dilations is considered and some associated resolutions of the identity are derived as a result it will be shown that they can all be obtained from those associated with the affine group and the weyl heisenberg group by frequency translations of the analyzed function and the analyzing and reconstructing wavelets

a block arnoldi chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices in this paper we describe a block version of arnoldi s method for computing a few eigenvalues with largest or smallest real parts the method is accelerated via chebyshev iteration and a procedure is developed to identify the optimal ellipse which encloses the spectrum a parallel implementation of this method is investigated on the eight processor alliant fx 80 numerical results and comparisons with simultaneous iteration on some harwell boeing matrices are reported

on lower bounds for the smallest eigenvalue of a hermitian positive definite matrix presents an improvement to demho s 1988 lower bound on the smallest eigenvalue of a hermitian positive definite matrix unlike dembo s bound the improved bound is always positive

a new method for diagonalising large matrices the structure and implementation of a new general iterative method for diagonalising large matrices the residual minimisation direct inversion in the iterative subspace method of bendt and zunger are described and contrasted with other more commonly used iterative techniques the method requires the direct diagonalisation of only a small submatrix does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure numerical results for two rather different matrices a real 50 50 non diagonally dominant matrix and a complex hermitian 181 181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set are used to compare the new method with the competing methods the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements it is quite insensitive to the properties of the matrices used this technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required

computation of eigenpair partial derivatives by rayleigh ritz procedure based on rayleigh ritz procedure a new method is proposed for a few eigenpair partial derivatives of large matrices this method simultaneously computes the approximate eigenpairs and their partial derivatives the linear systems of equations that are solved for eigenvector partial derivatives are greatly reduced from the original matrix size and the left eigenvectors are not required moreover errors of the computed eigenpairs and their partial derivatives are investigated hausdorff distance and containment gap are used to measure the accuracy of approximate eigenpair partial derivatives error bounds on the computed eigenpairs and their partial derivatives are derived finally numerical experiments are reported to show the efficiency of the proposed method

generalized epsilon pseudospectra normal matrices have complete sets of orthonormal eigenvectors and therefore the spectral decomposition is useful in studying the properties of normal operators in contrast the eigenvectors of nonnormal matrices can be nearly linearly dependent and the eigenvalue problem may by highly ill conditioned thus additional concepts and analysis techniques are useful in examining nonnormal operators the concepts of epsilon pseudospectra introduced by l n trefethen has proven to be a powerful tool in the analysis of nonnormal operators the author generalizes epsilon pseudospectra and the associated computational algorithms to the generalized eigenvalue problem rank 1 perturbations are used to determine the epsilon pseudospectra

a two component generalization of the degasperis procesi equation we present two different hamiltonian extensions of the degasperis procesi equation to two component equations the construction is based on the observation that the second hamiltonian operator of the degasperis procesi equation could be considered as the dirac reduced poisson tensor of the second hamiltonian operator of the boussinesq equation the first extension is generated by the hamiltonian operator which is a dirac reduced operator of the generalized but degenerated second hamiltonian operator of the boussinesq equation the second one is obtained by the n 2 supersymmetric extension of the aforementioned method as the byproduct of this procedure we obtain the hamiltonian system of interacting equations which contains the camassa holm and degasperis procesi equations

computing eigenvalues of very large symmetric matrices an implementation of a lanczos algorithm with no reorthogonalization many engineering and scientific applications require the computation of eigenvalues and eigenvectors of very large symmetric or hermitian matrices we describe a lanczos procedure which allows us to compute either few or many eigenvalues of such matrices in any intervals specified by the user this procedure can even be used to compute all of the eigenvalues the desired eigenvalues are computed as eigenvalues of an associated symmetric tridiagonal matrix tm whose order depends upon the distribution of the eigenvalues in the given matrix a and upon which portions of the spectrum of a are desired the storage requirements depend linearly upon the order of a if the the storage required to generate the products ax is also linear in the order the amount of computation required depends directly upon the distribution of the desired eigenvalues and upon the cost of computing ax numerical results for a very large matrix of order 4900 demonstrate that this procedure can be used on very large matrices

an a posteriori verification method for generalized real symmetric eigenvalue problems in large scale electronic state calculations abstract an a posteriori verification method is proposed for the generalized real symmetric eigenvalue problem and is applied to densely clustered eigenvalue problems in large scale electronic state calculations the proposed method is realized by a two stage process in which the approximate solution is computed by existing numerical libraries and is then verified in a moderate computational time the procedure returns intervals containing one exact eigenvalue in each interval test calculations were carried out for organic device materials and the verification method confirms that all exact eigenvalues are well separated in the obtained intervals this verification method will be integrated into eigenkernel https github com eigenkernel which is middleware for various parallel solvers for the generalized eigenvalue problem such an a posteriori verification method will be important in future computational science

the rayleigh ritz method still competitive abstract an extended rayleigh ritz method for computing two sided eigenvalue bounds of the one dimensional schrodinger equation is presented the method is based on the b spline approximation over the truncated domain for the low lying eigenvalues and problems with rapidly increasing spectrum the corresponding lower bounds are much more accurate than the known temple bounds numerical results for problems exhibiting different kinds of the eigenfunction behaviour are presented

a breakdown free variation of the nonsymmetric lanczos algorithms the nonsymmetric lanczos tridiagonalization algorithm is essentially the gram schmidt biorthogonalization method for generating biorthogonal bases of a pair of krylov subspaces it suffers from breakdown and instability when a pivot at some step is zero or nearly zero which is often the result of mismatch of the two krylov subspaces in this paper we propose to modify one of the two krylov subspaces by introducing a new start vector when a pivot is small this new start vector generates another krylov subspace which we add to the old one in an appropriate way so that the gram schmidt method for the modified subspaces yields a recurrence similar to the lanczos algorithm our method enforces the pivots to be above a certain threshold and can handle both exact breakdown and near breakdown in particular we recover lookahead lanczos algorithms and arnoldi s algorithm as two special cases we also discuss theoretical and practical issues concerning the new start procedure and present a convergence analysis as well as some numerical examples

an algorithm for the symmetric generalized eigenvalue problem abstract a new method is presented for the solution of the matrix eigenvalue problem ax bx where a and b are real symmetric square matrices and b is positive semidefinite it reduces a and b to diagonal form by congruence transformations that preserve the symmetry of the problem this method is closely related to the qr algorithm for real symmetric matrices

parallel computation of polynomials with minimal uniform norm and its application to large eigenproblems we investigate the parallelization of an algorithm that computes the polynomial with minimal uniform norm on polygonal domains the obtained polynomial is used to accelerate large non hermitian eigenvalue problems we report the performance results obtained on the machine paragon and compare this method with the one based on chebyshev acceleration techniques

the matrix equation xa atx and an associated algorithm for solving the inertia and stability problems abstract we develop a constructive procedure for generating nonsingular solutions of the matrix equation xa a t x and establish an interesting relationship between a given solution x of the above equation and the associated matrix polynomial p a the latter is then used to develop an algorithm for computing the inertia of a matrix the algorithm is more efficient than the other common procedures

generalized block lanczos methods for large unsymmetric eigenproblems generalized block lanczos methods for large unsymmetric eigenproblems are presented which contain the block arnoldi method and the block arnoldi algorithms are developed the convergence of this class of methods is analyzed when the matrix a is diagonalizable upper bounds for the distances between normalized eigenvectors and a block krylov subspace are derived and a priori theoretical error bounds for ritz elements are established compared with generalized lanczos methods which contain arnoldi s method the convergence analysis shows that the block versions have two advantages first they may be efficient for computing clustered eigenvalues second they are able to solve multiple eigenproblems however a deep analysis exposes that the approximate eigenvectors or ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix a even if corresponding approximate eigenvalues or ritz values do since the convergence of ritz vectors needs more sufficient conditions which may be impossible to satisfy theoretically than that of ritz values does the issues of how to restart and to solve multiple eigenproblems are addressed and some numerical examples are reported to confirm the theoretical analysis

on error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains numerical verification methods so called nakao s methods on existence or uniqueness of solutions to pdes have been developed by nakao and his group including the authors they are based on the error estimation of approximate solutions which are mainly computed by fem it is a standard way of the error estimation of fem to estimate the projection errors by elementwise interpolation errors there are some constants in the error estimation which depend on the mesh size parameters h the explicit values of the constants are necessary in order to use nakao s method however there were not so many researches for the computation of the explicit values of the constants then we had to develop the computation by ourselves especially with guaranteed accuracy note that the methods of the computation depend on the dimension the degree of bases and the shape of the domain etc the present paper shows how we have developed the methods to calculate the constants and describes new results for nonconvex domains

a generalization of the davidson s method to large nonsymmetric eigenvalue problems abstract the davidson s algorithm for solving large symmetric matrices is generalized to nonsymmetric cases the method is based on the non variational eigenvalue equation the algorithm can be expected to converge particularly well for the eigenvalue whose eigenvector has a desired structure excited states as well as the ground state the method has been applied successfully to some nonsymmetric eigenvalue problems in our cluster expansion theory

verified partial eigenvalue computations using contour integrals for hermitian generalized eigenproblems abstract we propose a verified computation method for partial eigenvalues of a hermitian generalized eigenproblem the block sakurai sugiura hankel method a contour integral type eigensolver can reduce a given eigenproblem into a generalized eigenproblem of block hankel matrices whose entries consist of complex moments in this study we evaluate all errors in computing the complex moments we derive a truncation error bound of the quadrature then we take numerical errors of the quadrature into account and rigorously enclose the entries of the block hankel matrices each quadrature point gives rise to a linear system and its structure enables us to develop an efficient technique to verify the approximate solution numerical experiments show that the proposed method outperforms a standard method and infer that the proposed method is potentially efficient in parallel

differential eigenvalue problems in which the parameter appears nonlinearly several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly the equations are the result of the discretization of differential eigenvalue problems using a finite chebyshev series two global methods are considered which determine the spectrum of eigenvalues without an initial estimate a local iteration scheme with cubic convergence is presented calculations are performed for a model second order differential problem and the orr sommerfeld problem for plane poiseuille flow

modification of the liu davidson method for obtaining one or simultaneously several eigensolutions of a large real symmetric matrix a new and efficient iteration method for obtaining simultaneously several eigensolutions and even for obtaining only one solution of a large real symmetric matrix is presented by modifying the simultaneous expansion method by davidson and liu the method is basically the ritz iteration method to correct trial vector s simultaneously using correction vectors however the number of the correction vectors determined in each iteration need not be the same as the number of the desired solutions it is advantageous for the former number to be smaller than the latter when many eigensolutions are sought in addition trial vectors which need not be obtained exactly nor be corrected are included so as to bound the desired solutions which are to be obtained exactly from above and or below when interior solutions are sought the correction vector space may be kept constant throughout the iterations when the correction vector space has room enough to include the trial vectors of the previous iterations the old trial vectors can serve as expansion vectors the performance of the algorithm is tested for five matrices and is compared with the original liu davidson algorithm the results demonstrate that the present algorithm attains efficient use of memory space and reduction of iteration cycles arithmetic operations and i o processings

a method for separating nearly multiple eigenvalues for hermitian matrix in this paper we propose a numerical method to verify for nearly multiple eigenvalues of a hermitian matrix not being strictly multiple eigenvalues from approximate eigenvalues computed it seems to be difficult to distinguish whether they are strictly multiple eigenvalues or simple ones and if they are very close each other the verification method for simple eigenvalues may fail to enclose them separately because of singularity of the system in the verification there are several methods for enclosing multiple and nearly multiple eigenvalues e g rump computational error bounds for multiple or nearly multiple eigenvalues linear algebra appl 324 2001 209 226 for such cases there is no result to decide the enclosed eigenvalues are nearly multiple or strictly multiple up to now so for enclosed eigenvalues we propose a numerical method to separate nearly multiple eigenvalues

the self organizing field kohonen maps many of the properties of the well known kohonen map algorithm are not easily derivable from its discrete formulation for instance the projection implemented by the map from a high dimensional input space to a lower dimensional map space must be properly regarded as a projection from a smooth manifold to a lattice and in this framework some of its properties are not easily identified this paper describes the self organizing field a continuous embedding of a smooth manifold the map into another the input manifold that implements a topological map by self organization the adaptation of the self organizing field is governed by a set of differential equations analogous to the difference equations that determine weights updates in the kohonen map this paper derives several properties of the self organizing field and shows that the emergence of certain structures on the brain like the columnar organization in the primary visual cortex arise naturally in the new model

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

verified solutions of delay eigenvalue problems a numerical algorithm for enclosing solutions of nonlinear eigenvalue problems arising in delay differential equations is proposed this algorithm verifies the uniqueness of the enclosed eigenvalue when it is simple and is applicable even for multiple eigenvalues numerical results show properties of the algorithm

asymptotic profiles of solutions to the two component camassa holm system abstract we discuss the two component camassa holm system which arises in shallow water theory this paper is concerned with the asymptotic description of solutions in the following sense the corresponding solution to initial data with algebraic decay at infinity will retain this property at infinity in its lifespan

a simple method for error bounds of eigenvalues of symmetric matrices abstract we propose a simple method for validated computation of eigenvalues of symmetric matrices the method is based on ldl t decomposition and its error estimation the indices of eigenvalues with respect to magnitude can also be obtained by this method

a new splitting to solve a large hermitian eigenproblem abstract a particular case of rohe s convergent splitting is presented which gives a rapidly converging method of estimating the lowest eigenvalues and corresponding vectors of a large hermitian matrix when a leading principal minor of that matrix provides a good approximation to the desired eigensolution this is particularly relevant for example to the total energy calculations of solids

a survey of lanczos procedures for very large real symmetric eigenvalue problems abstract first we identify five options which can be used to distinguish one lanczos eigenelement procedure from another then using these options to characterize the various procedures which have been proposed we briefly survey the recent research on lanczos methods for solving real symmetric eigenvalue problems

computational error bounds for multiple or nearly multiple eigenvalues abstract in this paper bounds for clusters of eigenvalues of non selfadjoint matrices are investigated we describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues and for a basis of the corresponding invariant subspaces the input matrix may be real or complex dense or sparse the method is based on a quadratically convergent newton like method it includes the case of defective eigenvalues uncertain input matrices and the generalized eigenvalue problem computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

asymptotic level density for a class of vector quantization processes it is shown that for a class of vector quantization processes related to neural modeling that the asymptotic density q x of the quantization levels in one dimension in terms of the input signal distribution p x is a power law q x c p x sup alpha where the exponent alpha depends on the number n of neighbors on each side of a unit and is given by alpha 2 3 1 3n sup 2 3 n 1 sup 2 the asymptotic level density is calculated and monte carlo simulations are presented

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

a lanczos type algorithm for the generalized eigenvalue problem ax λbx abstract this paper describes a generalization of the lanczos algorithm to make it applicable directly to the generalized matrix eigenvalue problem ax bx where a b are real symmetric matrices of high order n n and b is positive definite the algorithm is based on the construction of a sequence of b orthonormal vectors b i if u p b 1 b p then the eigen values of the p p matrix u p t au p p n approximate some of the extreme eigenvalues of the original problem ax bx

fast enclosure for solutions of sylvester equations abstract fast algorithms for enclosing solutions of sylvester equations ax xb c a c m m b c n n x c c m n are proposed the results obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account for developing these algorithms theories which directly supply error bounds for numerical solutions are established the proposed algorithms require only o m 3 n 3 operations if a and b are diagonalizable techniques for accelerating the enclosure and obtaining smaller error bounds are introduced numerical results show the properties of the proposed algorithms

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

fast enclosure for all eigenvalues in generalized eigenvalue problems a fast method for enclosing all eigenvalues in generalized eigenvalue problems email protected is proposed firstly a theorem for enclosing all eigenvalues which is applicable even if a is not hermitian and or b is not hermitian positive definite is presented next a theorem for accelerating the enclosure is presented the proposed method is established based on these theorems numerical examples show the performance and property of the proposed method as an application of the proposed method an efficient method for enclosing all eigenvalues in polynomial eigenvalue problems is also sketched