determination of eigenstates via lanczos based forward substitution and filter diagonalization a filter diagonalization scheme based on the lanczos tridiagonalization algorithm is proposed for the determination of highly excited eigenstates in a prespecified energy range filtered states localized in the energy domain can be generated during the lanczos iteration by forward substitution which formally corresponds to a green function filter eigenstates in the energy range of interest are then obtained by diagonalizing a small hamiltonian matrix in a subspace spanned by the filtered states it is shown that the accuracy of this approach is affected neither by the loss of orthogonality among the lanczos states nor by the instability in the forward substitution since the filtered states are assembled while the lanczos states are generated no storage of the lanczos states is needed the computational savings can be significant in calculating high energy eigenstates of large systems the proposed scheme is shown numerically to have comparable accuracy with the basic lanczos method

asymptotic distributions of the zeros of certain classes of gauss hypergeometric polynomials abstract we study the asymptotic distribution of the zeros of certain classes of gauss hypergeometric polynomials classical analytic methods such as watson s lemma and the method of steepest descents are used to analyze the large n behavior of the zeros of the gauss hypergeometric polynomials 2 f 1 n a kn b z 2 f 1 n kn c kn c z and 2 f 1 n pn z where n is a nonnegative integer the constants k a 0 p 1 and b c r owing to the connection between jacobi polynomials and gauss hypergeometric polynomials we prove a special case of a conjecture made by martinez finkelshtein martinez gonzalez and orive numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by mathematica version 4 0

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

real zeros of 2f1 hypergeometric polynomials we use a method based on the division algorithm to determine all the values of the real parameters b and c for which the hypergeometric polynomials 2f 1 n b c z have n real simple zeros furthermore we use the quasi orthogonality of jacobi polynomials to determine the intervals on the real line where the zeros are located

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

distribution and fluctuation properties of transition probabilities in a system between integrability and chaos the authors study the statistical properties of transition probabilities or generalized intensities equal to the squared matrix elements of operators having a classical limit they first generalize the shnirelman theorem to include semiclassically integrable states and use this to generalize the feingold peres formula for the average value of generalized intensities they perform an unfolding procedure to separate the smooth mean part of the intensities as a function of frequency from its fluctuating part by applying the generalized feingold peres formula this formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos the average values of the intensities as a function of frequency are excellently described by the generalized feingold peres formula

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

conjugate gradient eigenstructure tracking for adaptive spectral estimation a conjugate gradient iteration is derived that converges to the set of r dominant subdominant eigenpairs this iteration is used to construct two eigenstructure tracking algorithms that track the r dimensional dominant or subdominant subspaces of time varying data or data covariance matrices the two eigenstructure tracking algorithms have update complexities o m sup 2 r and the other o mr sup 2 where m is the data dimension the algorithms are customized to solve high resolution temporal and spatial frequency tracking problems they are compared with existing techniques by tying into published simulation based performance tests the algorithms demonstrate rapid convergence and tracking characteristics at a competitive cost

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

separation theorems for the zeros of certain hypergeometric polynomials we study interlacing properties of the zeros of two contiguous 2f1 hypergeometric polynomials we use the connection between hypergeometric 2f1 and jacobi polynomials as well as a monotonicity property of zeros of orthogonal polynomials due to markoff to prove that the zeros of contiguous hypergeometric polynomials separate each other we also discuss interlacing results for the zeros of 2f1 and those of the polynomial obtained by shifting one of the parameters of 2f1 by t where 0 t 1

general poissonian model of diffusion in chaotic components we extend our recent study of diffusion in strongly chaotic systems the random model to the general systems of mixed type dynamics including especially kam systems regarding the diffusion in chaotic components we do this by introducing a poissonian model as in our previous random model describing the strongly chaotic systems except that now we allow for different a priori probabilities in different cells of the discretized phase space surface of section thus the concept of greyness of cells denoted by g such that is introduced as is its distribution w g we derive the relationship between the dynamical property namely the normalized fraction of chaotic component as a function of discrete time j and w g we predict again the universal scaling law namely that for any w g the chaotic fraction is a function of only and not separately of j and n where n is the number of cells of equal size 1 n the random model of exponential is reproduced if all cells have g 1 i e we argue that in two dimensional systems at any finite n w g is non trivial due to the fractal dimension of the boundary of the chaotic component but is such that it goes to as whilst in systems with three or more degrees of freedom w g has a well defined limit with non zero values also at g 1 this is due to the existence of the arnold web we suggest how through our formalism one can calculate the lebesgue measure of the chaotic component at each finite discretization whose limit exists for our findings are verified and illustrated for two and three dimensional billiards

approximation accuracy of the krylov subspaces for linear discrete ill posed problems abstract for the large scale linear discrete ill posed problem min a x b or a x b with b contaminated by gaussian white noise the lanczos bidiagonalization based krylov solver lsqr and its mathematically equivalent cgls the conjugate gradient cg method implicitly applied to a t a x a t b are most commonly used and cgme the cg method applied to min a a t y b or a a t y b with x a t y and lsmr which is equivalent to the minimal residual minres method applied to a t a x a t b have also been choices these methods exhibit typical semi convergence feature and the iteration number k plays the role of the regularization parameter however there has been no definitive answer to the long standing fundamental question can lsqr and cgls find 2 norm filtering best possible regularized solutions the same question is for cgme and lsmr too at iteration k lsqr cgme and lsmr compute different iterates from the same k dimensional krylov subspace a first and fundamental step towards answering the above question is to accurately estimate the accuracy of the underlying k dimensional krylov subspace approximating the k dimensional dominant right singular subspace of a assuming that the singular values of a are simple we present a general sin theorem for the 2 norm distances between these two subspaces and derive accurate estimates on them for severely moderately and mildly ill posed problems we also establish some relationships between the smallest ritz values and these distances numerical experiments justify the sharpness of our results

efficient approximate solution of sparse linear systems abstract we consider the problem of approximate solution x of of a linear system ax b over the reals such that a x b e b for a given e 0 e a a a 1 be the condition number of the n n input matrix a sparse diagonally dominant dd linear systems appear very frequently in the solution of linear systems associated with pdes and stochastic systems and generally have polynomial condition number while there is a vast literature on methods for approximate solution of sparse dd linear systems most of the results are empirical and to date there are no known proven linear bounds on the complexity of this problem using iterative algorithms and building on the work of vaidya 1 and gremban et al 2 4 we provide the best known sequential work bounds for the solution of a number of major classes of dd sparse linear systems let log a e the sparsity graph of a is a graph whose nodes are the indices and whose edges represent pairs of indices of a with nonzero entries the following results hold for a dd matrix a with nonzero off diagonal entries of bounded magnitude 1 1 if a has a sparsity graph which is a regular d dimensional grid for constant d then our work is o n 2 2 2 if a is a stochastic matrix with fixed s n separable graph as its sparsity graph then our work is o n s n 2 the following results hold for a dd matrix a with entries of unbounded magnitude 1 3 if a is sparse i e o n nonzeros our work is less than o n log n 1 5 2 4 if a has a sparsity graph in a family of graphs with constant size forbidden graph minors e g planar graphs then our work is bounded by o n log n 1 o 1 in the case log n o log and o n log n 1 o 1 in the case log o log n we use approximate preconditioned iterations to compute a sequence of iterative improvements to the preconditioned linear system for class 1 of matrices and class 2 with s n o n we construct in n work preconditioners which reduce the condition number of the resulting preconditioned linear system condition number to o 1 and our resulting total work bounds to approximately solve the linear systems are o n if o 1 for class 4 we are within a small increasing factor of these bounds

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

polya frequency sequences and real zeros of some f23 polynomials we prove that the zeros of some families of f23 hypergeometric polynomials are all real and negative this result has a connection with the theory of polya frequency sequences and functions as a consequence we establish the asymptotic distribution of these zeros when the degree of the polynomials tends to infinity

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

an evaluation of limited memory sparse linear solvers for thermo mechanical applications we consider the performance of sparse linear solvers for problems that arise from thermo mechanical applications such problems have been solved using sparse direct schemes that enable robust solution at the expense of memory requirements that grow non linearly with the dimension of the coefficient matrix in this paper we consider a class of preconditioned iterative solvers as a limited memory alternative to direct solution schemes however such preconditioned iterative solvers typically exhibit complex trade offs between reliability and performance we therefore characterize such trade offs for systems from thermo mechanical problems by considering several preconditioning schemes including multilevel methods and those based on sparse approximate inversion and incomplete matrix factorization we provide an analysis of computational costs and memory requirements for model thermo mechanical problems indicating that certain incomplete factorization schemes can achieve good performance we also provide empirical evaluations that corroborate our analysis and indicate the relative effectiveness of different solution schemes our results indicate that our drop threshold incomplete cholesky preconditioning is more robust efficient and flexible than other popular preconditioning schemes in addition we propose preconditioner reuse to amortize preconditioner construction cost over a sequence of linear systems that arise from non linear solutions in a plastic regime copyright 2007 john wiley sons ltd

a parallel alternating direction implicit preconditioning method abstract the alternating direction implicit adi iterative method is an efficient iterative method to solve systems of linear equations due to its extremely fast convergence the adi method has also been used successfully as a preconditioner in some other iterative methods such as the preconditioned conjugate gradient in this paper a parallel algorithm for the adi preconditioning is proposed in this algorithm several steps of the adi iteration are computed simultaneously this means that several tridiagonal systems that are traditionally solved sequentially are now solved concurrently the high performance of this algorithm is achieved by increasing the degree of parallelism and reducing memory contention the algorithm can easily be implemented in a multiprocessor architecture experiments have been conducted on the myrias sps 2 computer with 64 processors and good performance of this algorithm is observed

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

an augmented analysis of the perturbed two sided lanczos tridiagonalization process abstract we generalize an augmented rounding error result that was proven for the symmetric lanczos process in c c paige an augmented stability result for the lanczos hermitian matrix tridiagonalization process siam j matrix anal appl 31 2010 2347 2359 to the two sided usually unsymmetric lanczos process for tridiagonalizing a square matrix we extend the analysis to more general perturbations than rounding errors in order to provide tools for the analysis of inexact and related methods the aim is to develop a deeper understanding of the behavior of all these methods our results take the same form as those for the symmetric lanczos process except for the bounds on the backward perturbation terms the generalizations of backward rounding errors for the augmented system in general we cannot derive tight a priori bounds for these terms as was done for the symmetric process but a posteriori bounds are feasible while bounds related to certain properties of matrices would be theoretically desirable

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

box spline interpolation a computational study abstract we present a computational study of box spline interpolation in two space dimensions we are in particular concerned with the problem of computing the box spline coefficients in the presence of large data sets the issues of convergence and regularity of the interpolants are also considered from a computational point of view

fractal weyl law for chaotic microcavities fresnel s laws imply multifractal scattering we demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities as an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium we show that the conjectured fractal weyl law for open chaotic systems lu et al phys rev lett 91 154101 2003 is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating fresnel s laws

dynamical localization in chaotic systems spectral statistics and localization measure in the kicked rotator as a paradigm for time dependent and time independent systems we study the kicked rotator in the classically fully chaotic regime using izrailev s n dimensional model for various n 4000 which in the limit n tends to the quantized kicked rotator we do treat not only the case k 5 as studied previously but also many different values of the classical kick parameter 5 k 35 and many different values of the quantum parameter ke 5 60 we describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time periodic and time independent autonomous fully chaotic or and mixed type hamilton systems we generalize the scaling variable l n to the case of anomalous diffusion in the classical phase space by deriving the localization length l for the case of generalized classical diffusion we greatly improve the accuracy and statistical significance of the numerical calculations giving rise to the following conclusions 1 the level spacing distribution of the eigenphases or quasienergies is very well described by the brody distribution systematically better than by other proposed models for various brody exponents br 2 we study the eigenfunctions of the floquet operator and characterize their localization properties using the information entropy measure which after normalization is given by loc in the interval 0 1 the level repulsion parameters br and loc are almost linearly related close to the identity line 3 we show the existence of a scaling law between loc and the relative localization length now including the regimes of anomalous diffusion the above findings are important also for chaotic eigenstates in time independent systems batistic and robnik j phys a math gen 43 215101 2010 arxiv 1302 7174 2013 where the brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates even in the mixed type systems after separation of regular and chaotic eigenstates

anomaly preserving ell _ scriptscriptstyle 2 infty optimal dimensionality reduction over a grassmann manifold in this paper we address the problem of redundancy reduction of high dimensional noisy signals that may contain anomaly rare vectors which we wish to preserve since anomaly data vectors contribute weakly to the l2 norm of the signal as compared to the noise l2 based criteria are unsatisfactory for obtaining a good representation of these vectors as a remedy a new approach named min max svd mx svd was recently proposed for signal subspace estimation by attempting to minimize the maximum of data residual l2 norms denoted as l2 l and designed to represent well both abundant and anomaly measurements however the mx svd algorithm is greedy and only approximately minimizes the proposed l2 l norm of the residuals in this paper we develop an optimal algorithm for the minization of the l2 l norm of data misrepresentation residuals which we call maximum orthogonal complements optimal subspace estimation moose the optimization is performed via a natural conjugate gradient learning approach carried out on the set of n dimensional subspaces in irm m n which is a grassmann manifold the results of applying moose mx svd and l2 based approaches are demonstrated both on simulated and real hyperspectral data

an application of the discrete time toda lattice to the progressive algorithm by lanczos and related problems the finite or the semi infinite discrete time toda lattice has many applications to various areas in applied mathematics the purpose of this paper is to review how the toda lattice appears in the lanczos algorithm through the quotient difference algorithm and its progressive form pqd then a multistep progressive algorithm mpa for solving linear systems is presented the extended lanczos parameters can be given not by computing inner products of the extended lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost the asymptotic behavior of the pqd algorithm brings us some applications of mpa related to eigenvectors

large parameter cases of the gauss hypergeometric function we consider the asymptotic behavior of the gauss hypergeometric function when several of the parameters a b c are large we indicate which cases are of interest for orthogonal polynomials jacobi but also meixner krawtchouk etc which results are already available and which cases need more attention we also consider a few examples of 3f2 functions of unit argument to explain which difficulties arise in these cases when standard integrals or differential equations are not available

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

distribution of resonance strengths in microwave billiards of mixed and chaotic dynamics a new measure for statistical properties of the wave function components of quantum systems the distribution of the product of two partial widths is introduced it is tested with data obtained in analog experiments with microwave billiards where the product of two partial widths equals the resonance strengths in the microwave spectra the billiards are from the family of the limacons one with chaotic and two with mixed classical dynamics for completely chaotic systems the partial widths generically obey a porter thomas distribution we show that in this case the distribution of their product equals a k0 distribution while we find deviations of the experimental strength distribution from the k0 distribution for the billiards with mixed dynamics the distributions agree perfectly for the chaotic billiard when taking into account the experimental threshold of detection in the theoretical description hence the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory

an incomplete inverse as a preconditioner for the conjugate gradient method abstract this paper introduces a new preconditioner with a super convergence for the conjugate gradient method for solving a system of linear equations in a banded structure the idea is based upon an innovative method of the author to decompose a symmetric matrix into its inverse and the preconditioner is the result of an incomplete decomposition i e an approximation to the inverse by the presented preconditioner the rate of convergence can be significantly improved examples to demonstrate the improvement of rate of convergence and the speedup implemented on an alliant fx8 computer are included

on the implementation of the gmres m method to elliptic equations in meteorology in this paper we consider the relatively new preconditioned generalized minimal residual method restarted every m iterations gmres m for the solution of three dimensional elliptic equations large sparse non symmetric matrices are involved the particular equation of interest is the quasi geostrophic omega equation often used in meteorology to compute vertical motion the gmres m method is tested with different preconditioners for the solution of two and three dimensional elliptic equations the method requires no relaxation parameters and has no restrictions on the size of the 3d grid gmres can be used for more types of matrices than other methods such as sor numerical results show that jacobi preconditioned gmres m performs best for 3d and high resolution problems among five different preconditioners tested while ilij factorization of the partial or whole matrix a as a preconditioner is good for 2d and low resolution problems the sor preconditioners for the gmres m method with optimal relaxation parameters are not as efficient and the best choice for the relaxation parameter in sor preconditioning is not the same as the best choice for the simple sor method an algorithm for using the preconditioned gmres m method is presented

convergence rate results for steepest descent type method for nonlinear ill posed equations convergence rate result for a modified steepest descent method and a modified minimal error method for the solution of nonlinear ill posed operator equation have been proved with noisy data to our knowledge convergence rate result for the steepest descent method and minimal error method with noisy data are not known we provide a convergence rate results for these methods with noisy data the result in this paper are obtained under less computational cost when compared to the steepest descent method and minimal error method we present an academic example which satisfies the assumptions of this paper

fourier analysis of iterative methods for elliptic boundary value problems his paper presents a fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems as in the von neumann stability analysis of hyperbolic and parabolic problems the approach is easier to apply reveals more details about convergence properties than about standard techniques and can be applied in a systematic way to a wide class of numerical methods although the analysis is applicable only to periodic problems the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions such as the dirichlet problem in addition they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the dirichlet case

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

superlinear convergence of krylov subspace methods for self adjoint problems in hilbert space the conjugate gradient and minimum residual methods for self adjoint problems in hilbert space are considered linear and superlinear convergence results with respect to both q and r rates are reviewed new results on ell step q superlinear and r superlinear convergence for the minimum residual method are provided and examples are considered to underscore the relevance of a hilbert space theory

breakdowns in the implementation of the lanczos method for solving linear systems abstract the lanczos method for solving systems of linear equations is based on formal orthogonal polynomials its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials a division by zero can occur in such recurrence relations thus causing a breakdown in the algorithm which has to be stopped in this paper two types of breakdowns are discussed the true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used among all the recurrence relationships which can be used and all the algorithms for implementing the lanczos method which came out from them the only reliable algorithm is lanczos orthodir which can only suffer from true breakdowns it is shown how to avoid true breakdowns in this algorithm other algorithms are also discussed and the case of near breakdown is treated the same treatment applies to other methods related to lanczos

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

on the discontinuities of the boundary in billiards abstract this work concerns billiards in domains of r 2 the set s of pre images of discontinuities of t the bouncing map induced by the flow is studied particularly those discontinuities due to points where the boundary is not c1 we prove that if the angle at the discontinuity is a divisor of there exists a power of t which is continuous around such a point and that under conditions of symmetry this power is locally differentiable it is assumed that continuity may often be extended to certain areas of phase space this property is used to show by means of numerical simulations of two billiards how s fashions the phase space in general if the angle is not a divisor of s separates the distinct areas of quasi interagrable motion one of the billiards is isomorphic to a mechanical model of three interacting particles on a line

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically