fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation abstract in this paper we mainly investigate the fractional spectral collocation discretization of optimal control problem governed by a space fractional diffusion equation existence and uniqueness of the solution to optimal control problem is proved the continuous first order optimality condition is derived the eigenfunctions of two classes of fractional strum liouville problems are used as basis functions to approximate state variable and adjoint state variable respectively the fractional spectral collocation scheme for the control problem is constructed based on first optimize then discretize approach note that the solutions of fractional differential equations are usually singular near the boundary a generalized fractional spectral collocation scheme for the control problem is proposed based on first optimize then discretize approach a projected gradient algorithm is designed based on the discrete optimality condition numerical experiments are carried out to verify the effectiveness of the proposed numerical schemes and algorithm

a two step matrix splitting iteration for computing pagerank the pagerank algorithm plays an important role in determining the importance of web pages the core of this algorithm involves using the classical power method to compute the pagerank vector which is the principal eigenvector of the matrix representing the web link graph nevertheless it is well known that the power method may perform poorly when the second largest eigenvalue is close to the dominant one in this article we present a new approach that is based on the two step splitting iteration framework the description and convergence of the new algorithm are discussed in detail numerical examples are given to illustrate the performance of this algorithm

accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

a note on the two step matrix splitting iteration for computing pagerank computing pagerank plays an important part in determining the importance of web pages based on the classical power method and the inner outer iteration proposed by gleich et al 2010 gu et al 2015 presented a two step splitting iteration i e the power inner outer pio iteration for the computation of pagerank in this paper we develop a variant of the pio iteration by applying multi step power method to combine with the inner outer iteration the new method is denoted as the mpio iteration its convergence is analyzed in detail numerical experiments on several pagerank problems are used to illustrate the effectiveness of the mpio iteration

parallel multisplitting iteration methods based on m splitting for the pagerank problem a class of iterative schemes based on the regular splitting is proposed for pagerank computation by some specific splitting these methods will reduce to some well known iterative schemes like jacobi method and successive overrelaxation method to suit computational requirements of the modern high speed multiprocessor environments we further study synchronous parallel counterparts for the new splitting iteration methods by using the m splitting of the coefficient matrix we establish the convergence theory of these synchronous multisplitting iteration methods finally some numerical experiments are presented to illustrate the feasibility stability and effectiveness of the proposed algorithms

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

rational extrapolation for the pagerank vector an important problem in web search is to determine the importance of each page from the mathematical point of view this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1 since this matrix is neither stochastic nor irreducible the power method has convergence problems so the matrix is replaced by a convex combination depending on a parameter c with a rank one matrix its left principal eigenvector now depends on c and it is the pagerank vector we are looking for however when c is close to 1 the problem is ill conditioned and the power method converges slowly so the idea developed in this paper consists in computing the pagerank vector for several values of c and then to extrapolate them by a conveniently chosen rational function at a point near 1 the choice of this extrapolating function is based on the mathematical expression of the pagerank vector as a function of c numerical experiments end the paper

a computational method for solving variable order fractional nonlinear diffusion wave equation abstract in this paper we generalize a one dimensional fractional diffusion wave equation to a one dimensional variable order space time fractional nonlinear diffusion wave equation v os tfnd we where the variable order fractional derivatives are considered in the caputo type to solve this introduced equation an easy to follow method is proposed which is based on the chebyshev cardinal functions coupling with the tau and collocation methods to carry out the method an operational matrix of variable order fractional derivative omv ofd is derived for the chebyshev cardinal functions to be employed for expanding the unknown function the proposed method can provide highly accurate approximate solutions by reducing the problem under study to a system of nonlinear algebraic equations which is technically simpler for handling the experimental results confirm the applicability and effectiveness of the method

chebyshev filtered subspace iteration method free of sparse diagonalization for solving the kohn sham equation first principles density functional theory dft calculations for the electronic structure problem require a solution of the kohn sham equation which requires one to solve a nonlinear eigenvalue problem solving the eigenvalue problem is usually the most expensive part in dft calculations sparse iterative diagonalization methods that compute explicit eigenvectors can quickly become prohibitive for large scale problems the chebyshev filtered subspace iteration chefsi method avoids most of the explicit computation of eigenvectors and results in a significant speedup over iterative diagonalization methods for the dft self consistent field scf calculations however the original formulation of the chefsi method utilizes a sparse iterative diagonalization at the first scf step to provide initial vectors for subspace filtering at latter scf steps this diagonalization is expensive for large scale problems we develop a new initial filtering step to avoid completely this diagonalization thus making the chefsi method free of sparse iterative diagonalizations at all scf steps our new approach saves memory usage and can be two to three times faster than the original chefsi method

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

a priori error bounds on invariant subspace approximations by block krylov subspaces abstract we derive a priori error bounds for the block krylov subspace methods in terms of the sine between the desired invariant subspace and the block krylov subspace the obtained results can be seen as the block analogue of the classical a priori estimates for standard projection methods

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

a gmres power algorithm for computing pagerank problems abstract the pagerank algorithm for determining the importance of web pages has become a central technique in web search we propose a new method to speed up the convergence performance for computing pagerank when the damping factor is close to one called as gmres power which is based on a periodic combination of the power method with the gmres algorithm the description and convergence analysis of the new algorithm are discussed in detail numerical results are reported to confirm the efficiency of the new algorithm

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

extrapolation techniques for ill conditioned linear systems in this paper the regularized solutions of an ill conditioned system of linear equations are computed for several values of the regularization parameter lambda then these solutions are extrapolated at lambda 0 by various vector rational extrapolations techniques built for that purpose these techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition numerical results illustrate the effectiveness of the procedures

a space time fractional optimal control problem analysis and discretization we study a linear quadratic optimal control problem involving a parabolic equation with fractional diffusion and caputo fractional time derivative of orders s in 0 1 and gamma in 0 1 respectively the spatial fractional diffusion is realized as the dirichlet to neumann map for a nonuniformly elliptic operator thus we consider an equivalent formulation with a quasi stationary elliptic problem with a dynamic boundary condition as state equation the rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation we consider a fully discrete scheme piecewise constant functions for the control and for the state first degree tensor product finite elements in space and a finite difference discretization in time we show convergence of this scheme and under additional data regularity derive a priori error estimates for the case s in 0 1 and gamma 1

numerical solution of a class of fractional optimal control problems via the legendre orthonormal basis combined with the operational matrix and the gauss quadrature rule abstract a numerical direct method for solving a general class of fractional optimal control problems focps is presented in the discussed focp the fractional derivative in the dynamical system is considered in the caputo sense to solve the problem first the focp is transformed into an equivalent variational problem then using the legendre orthonormal basis the problem is reduced to the problem of solving a system of algebraic equations with the aid of an operational matrix of riemann liouville fractional integration gauss quadrature formula and newton s iterative method for solving a system of algebraic equations the problem is solved approximately approximations achieved by this method satisfy all the initial conditions of the problem which is an important property the convergence of the method is extensively discussed and finally some illustrative examples are included to demonstrate the applicability of the new technique

a fem for an optimal control problem of fractional powers of elliptic operators we study solution techniques for a linear quadratic optimal control problem involving fractional powers of elliptic operators these fractional operators can be realized as the dirichlet to neumann map for a nonuniformly elliptic problem posed on a semi infinite cylinder in one more spatial dimension thus we consider an equivalent formulation with a nonuniformly elliptic operator as the state equation the rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation we discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes for the control problem we analyze two approaches one that is semidiscrete based on the so called variational approach where the control is not discretized and the other one that is fully discrete via the discretization of the control by piecewise constant functions for both approaches we derive a priori error estimates with respect to degrees of freedom numerica

spectral isolation of naturally reductive metrics on simple lie groups we show that within the class of left invariant naturally reductive metrics mathcal m rm nat g on a compact simple lie group g every metric is spectrally isolated we also observe that any collection of isospectral compact symmetric spaces is finite this follows from a somewhat stronger statement involving only a finite part of the spectrum

google pageranking problem the model and the analysis the spectral and jordan structures of the web hyperlink matrix g c cg 1 c ev t have been analyzed when g is the basic stochastic google matrix c is a real parameter such that 0 1y c nv for all v this limit may fail to exist for some v if l is not semisimple as a special case of our results we obtain a complex analog of pagerank for the web hyperlink matrix g c with a complex parameter c we study regularity limits expansions and conditioning of y c and we propose algorithms e g complex extrapolation power method on a modified matrix etc that may provide an efficient way to compute pagerank also with c close or equal to 1 an interpretation of the limit vector nv and a related critical discussion on the model on its adherence to reality and possible ways for its improvement represent the contribution of the paper on modeling issues

bounding the spectrum of large hermitian matrices abstract estimating upper bounds of the spectrum of large hermitian matrices has long been a problem with both theoretical and practical significance algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas we present a practical algorithm that exploits k step lanczos iteration with a safeguard step the k is generally very small say 5 8 regardless of the large dimension of the matrices this makes the lanczos iteration economical the safeguard step can be realized with marginal cost by utilizing the theoretical bounds developed in this paper the bounds establish the theoretical validity of a previous bound estimator that has been successfully used in various applications moreover we improve the bound estimator which can now provide tighter upper bounds with negligible additional cost

continuous families of isospectral metrics on simply connected manifolds we construct continuous families of riemannian metrics on certain simply connected manifolds with the property that the resulting riemannian manifolds are pairwise isospectral for the laplace operator acting on functions these are the first examples of simply connected riemannian manifolds without boundary which are isospectral but not isometric for example we construct continuous isospectral families of metrics on the product of spheres s4 x s3 x s3 the metrics considered are not locally homogeneous for a big class of such families the set of critical values of the scalar curvature function changes during the deformation moreover the manifolds are in general not isospectral for the laplace operator acting on 1 forms

the general inner outer iteration method based on regular splittings for the pagerank problem abstract in this paper combined the regular splittings of the coefficient matrix i p with the inner outer iteration framework 9 a general inner outer gio iteration method is presented for solving the pagerank problem firstly the aor and modified aor maor methods for solving the pagerank problem are constructed and several comparison results are also given next the gio iteration scheme is developed and its overall convergence is analyzed in detail furthermore the preconditioner derived from the gio iteration can be used to accelerate the krylov subspace methods such as gmres method finally some numerical experiments on several pagerank problems are provided to illustrate the efficiency of the proposed algorithm

uncovering disassortativity in large scale free networks mixing patterns in large self organizing networks such as the internet the world wide web and social and biological networks are often characterized by degree degree dependencies between neighboring nodes in this paper we propose a new way of measuring degree degree dependencies one of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size we mathematically explain this phenomenon and validate the results on synthetic graphs and real world network data as an alternative we suggest to use rank correlation measures such as spearman s rho our experiments convincingly show that spearman s rho produces consistent values in graphs of different sizes but similar structure and it is able to reveal strong positive or negative dependencies in large graphs in particular we discover much stronger negative degree degree dependencies in web graphs than was previously thought rank correlations allow us to compare the assortativity of networks of different sizes which is impossible with the assortativity coefficient due to its genuine dependence on the network size we conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns

parallel self consistent field calculations via chebyshev filtered subspace acceleration solving the kohn sham eigenvalue problem constitutes the most computationally expensive part in self consistent density functional theory dft calculations in a previous paper we have proposed a nonlinear chebyshev filtered subspace iteration method which avoids computing explicit eigenvectors except at the first self consistent field scf iteration the method may be viewed as an approach to solve the original nonlinear kohn sham equation by a nonlinear subspace iteration technique without emphasizing the intermediate linearized kohn sham eigenvalue problems it reaches self consistency within a similar number of scf iterations as eigensolver based approaches however replacing the standard diagonalization at each scf iteration by a chebyshev subspace filtering step results in a significant speedup over methods based on standard diagonalization here we discuss an approach for implementing this method in multi processor parallel environment numerical results are presented to show that the method enables to perform a class of highly challenging dft calculations that were not feasible before

a robust numerical method for a control problem involving singularly perturbed equations we consider an unconstrained linear quadratic optimal control problem governed by a singularly perturbed convection reaction diffusion equation we discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by duran and lombardi in du an and lombardi 2005 2006 we prove convergence of this scheme in addition when the state equation is a singularly perturbed reaction diffusion equation we derive quasi optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes we present several numerical experiments when the state equation is both a reaction diffusion and a convection reaction diffusion equation these numerical experiments reveal a competitive performance of the proposed solution technique

a preconditioned and shifted gmres algorithm for the pagerank problem with multiple damping factors google has become one of the most popular and successful search engines in recent years google s success can be attributed to its simple and elegant algorithm pagerank in practice one often needs to solve the pagerank problem with multiple damping factors or with multiple damping factors and multiple personalization vectors the conventional pagerank algorithm has to solve these problems one by one the shifted gmres m algorithm can be used to solve them in the same search subspace however there are two disadvantages to this algorithm the first is near singularity and the second is stagnation in this paper we first present a modified and shifted gmres m algorithm to deal with the problem of near singularity in order to overcome the drawback of stagnation and to improve convergence we propose a polynomial preconditioner for the modified algorithm we show that the resulting algorithm can circumvent the drawbacks of near singularity and stagnation that occur in its original counterpart

krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices abstract in this paper we propose a general approach by which eigenvalues with a special property of a given matrix a can be obtained in this approach we first determine a scalar function c c whose modulus is maximized by the eigenvalues that have the special property next we compute the generalized power iterations uinj 1 a u j j 0 1 where u 0 is an arbitrary initial vector finally we apply known krylov subspace methods such as the arnoldi and lanczos methods to the vector u n for some sufficiently large n we can also apply the simultaneous iteration method to the subspace span x n 1 x n k with some sufficiently large n where x j 1 m a x j m j 0 1 m 1 k in all cases the resulting ritz pairs are approximations to the eigenpairs of a with the special property we provide a rather thorough convergence analysis of the approach involving all three methods as n for the case in which a is a normal matrix we also discuss the connections and similarities of our approach with the existing methods and approaches in the literature

a numerical technique for solving fractional optimal control problems this paper presents a numerical method for solving a class of fractional optimal control problems focps the fractional derivative in these problems is in the caputo sense the method is based upon the legendre orthonormal polynomial basis the operational matrices of fractional riemann liouville integration and multiplication along with the lagrange multiplier method for the constrained extremum are considered by this method the given optimization problem reduces to the problem of solving a system of algebraic equations by solving this system we achieve the solution of the focp illustrative examples are included to demonstrate the validity and applicability of the new technique

computing eigenpairs of hermitian matrices in augmented krylov subspace produced by rayleigh quotient iterations abstract in this paper we first introduce a class of augmented krylov subspace that is a union of the standard krylov subspace and another low dimension subspace and then construct the corresponding augmented krylov subspace method for computing some extreme eigenvalues and their associated eigenvectors of hermitian matrices in order to improve the overall performance of this method we present some practical implementations with preconditioning which then results in the preconditioned augmented krylov subspace method in addition through convergence analysis it indicates that the preconditioned augmented krylov subspace method attains cubic convergence rate locally in the last we use some numerical examples to demonstrate the convergence property and the superiority of this method

block arnoldi and davidson methods for unsymmetric large eigenvalue problems we present two methods for computing the leading eigenpairs of large sparse unsymmetric matrices namely the block arnoldi method and an adaptation of the davidson method to unsymmetric matrices we give some theoretical results concerning the convergence and discuss implementation aspects of the two methods finally some results of numerical tests on a variety of matrices in which we compare these two methods are reported

on perturbations of principal eigenvectors of substochastic matrices in this report we consider substochastic matrices with some zero rows and study the sensitivity of their eigenvectors to the modifications of zero entries the analysis is prompted by and directly applied to the google method of ranking the web sites which substitutes the pages without outlinks dangling nodes in the original web link matrix by non zero rows dangling vectors we present an analysis of the influence of artificial links attributed to the dangling nodes on the principal eigenvectors of the web matrix we clarify when the choice of the dangling vector does not change the original eigenvectors and give an evaluation for perturbations of the principal eigenvectors when they are subject to modification

an arnoldi inout algorithm for computing pagerank problems the pagerank algorithm plays an important role in the web search engines to speed up the convergence behavior for computing pagerank we propose a new method called as arnoldi inout which is the inner outer iteration method modified with the thick restarted arnoldi method the description and convergence of the new algorithm are discussed in detail numerical results are given to illustrate the efficiency of the new algorithm

an iteration method for solving the linear system ax b abstract in this paper based on a convergence splitting of the matrix a we present an inner outer iteration method for solving the linear system a x b we analyze the overall convergence of this method without any other restriction on its parameters moreover we give the accelerated inner outer iteration method and discuss how to apply the inner outer iterations as a preconditioner for the krylov subspace methods the inner outer iteration method can also be used for the solution of a x b c finally several numerical examples are given to validate the performance of our proposed algorithms

projection and deflation method for partial pole assignment in linear state feedback two projection methods are proposed for partial pole placement in linear control systems these methods are of interest when the system is very large and only a few of its poles must be assigned the first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace the second method can be regarded as a variant of the weilandt deflation technique used in eigenvalue methods

a partition property of simplices in euclidean space in a series of papers erdos et al e have investigated this property they have shown that all ramsey sets are spherical that is every ramsey set is contained in an appropriate sphere on the other hand they have shown that the vertex set and therefore all its subsets of bricks d dimensional parallelepipeds is ramsey the simplest sets that are spherical but cannot be embedded into the vertex set of a brick are the sets of obtuse triangles in fr1 it is shown that they are indeed ramsey using ramsey s theorem cf g2 and the product theorem of e the aim of the present paper is twofold first we want to show that the vertex set of every nondegenerate simplex in any dimension is ramsey second we want to show that for both simplices and bricks and even for their products one can in fact choose n r b c b logr where c b is an appropriate positive constant the paper is organized as follows in 2 super ramsey property is introduced this notion is stronger than being ramsey it is shown that the direct product of super ramsey sets is super ramsey in 3 it is shown that if every edge of an n dimensional simplex is between 1 e and 1 e with e e n being a sufficiently small positive number then it can be embedded into a brick i e into the direct product of two element sets in 4 it is proved that given a nondegenerate simplex with edge lengths aij 1 0 there exists some super ramsey simplex whose edge lengths vi verify la 2 v2ji d5 for all 1 i n intheproofofthisresult an earlier result

fom accelerated by an extrapolation method for solving pagerank problems this paper formulates the pagerank problem a x x into a consistent singular linear system i a x 0 and applies the full orthogonalization method fom to solve it this singular system is characterized by index one namely i n d e x i a 1 we analyze the breakdown performance of fom on a general singular linear system and conclude that fom can determine a solution if it converges without any unfortunate breakdowns for our target problem then we propose to use a vector extrapolation method to speed up the convergence performance of fom this extrapolation procedure is based on ritz values which directly stems from the arnoldi extrapolation algorithm wu and wei 2010 eventually numerical experiments are presented to illustrate the effectiveness of our approaches

a new extrapolation method for pagerank computations abstract the pagerank algorithm is widely considered these years because of its great significance in search engine technology and other scientific domains though the power method is the initial measure to settle the pagerank problem it gives poor convergence when the damping factor is sufficiently close to 1 this difficulty encourages researchers to present improved iterative methods for accelerating pagerank computations in this paper a cheap and practical extrapolation approach which is determined by the trace of the google matrix is proposed and it is more attractive when combined with the well known arnoldi type algorithm convergence analysis of our algorithms is given numerical examples illustrate the efficiency of these accelerated schemes

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

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

a shift and invert reorthogonalization arnoldi algorithm for solving the chemical master equation abstract the shift and invert arnoldi sia method is a numerical algorithm for approximating the product of toeplitz matrix exponential with a vector in this paper we extend the sia method to chemical master equation cme and propose a sia algorithm based on the strategy of reorthogonalization sira we establish a theoretical error of the resulting approximation of sira algorithm numerical experiments show that the sira algorithm is more efficient than the krylov fsp algorithm in terms of finite models and the error estimate can be used to determine whether this result obtained by sira algorithm is acceptable or not

arnoldi chebyshev approach for convection diffusion computations abstract in a convection diffusion equation if the convection term is very dominant the linear system of equations which result from either finite differencing or finite elementing will not have a strictly diagonally dominant coefficient matrix so if one tries a conventional iteration method jacobi or gauss seidel to solve the linear system of equations the iteration matrix may not satisfy the spectral radius condition for convergence and hence no converging solution may be obtained the problem can be overcome under certain conditions if one uses a two step iteration procedure involving the spectral enveloping ellipse for the iteration matrix in this paper we present such a two step method that combines an arnoldi chebyshev approach for convection diffusion computations

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

practical acceleration for computing the hits expertrank vectors a meaningful rank as well as efficient methods for computing such a rank are necessary in many areas of applications major methodologies for ranking often exploit principal eigenvectors kleinberg s hits model is one of such methodologies the standard approach for computing the hits rank is the power method unlike the pagerank calculations where many acceleration schemes have been proposed relatively few works on accelerating hits rank calculation exist this is mainly because the power method often works quite well in the hits setting however there are cases where the power method is ineffective moreover a systematic acceleration over the power method is desirable even when the power method works well we propose a practical acceleration scheme for hits rank calculations based on the filtered power method by adaptive chebyshev polynomials for cases where the gap ratio is below 0 85 for which the power method works well our scheme is about twice faster than the power method for cases where gap ratio is unfavorable for the power method our scheme can provide significant speedup when the ranking problems are of very large scale even a single matrix vector product can be expensive for which accelerations are highly necessary the scheme we propose is desirable in that it provides consistent reduction in number of matrix vector products as well as cpu time over the power method with little memory overhead

comparison of krylov subspace methods on the pagerank problem pagerank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions the problem incorporates a parameter a that determines the difficulty of the problem in this paper the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of a we see that stationary methods are very reliable and more competitive when the problem is well conditioned that is for small values of a however for large values of the parameter a the problem becomes more difficult and methods such as preconditioned bicgstab or restarted preconditioned gmres become competitive with stationary methods in terms of mflops count as well as in number of iterations necessary to reach convergence

an arnoldi extrapolation algorithm for computing pagerank the arnoldi type algorithm proposed by golub and greif g golub c greif an arnoldi type algorithm for computing pagerank bit 46 2006 759 771 is a restarted krylov subspace method for computing pagerank however this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small in this paper we first develop an extrapolation method based on ritz values we then consider how to periodically knit this extrapolation method together with the arnoldi type algorithm the resulting algorithm is the arnoldi extrapolation algorithm the convergence of the new algorithm is analyzed numerical experiments demonstrate the numerical behavior of this algorithm

numerical solutions for large sparse quadratic eigenvalue problems abstract we study the quadratic eigenvalue problem a b 2c x 0 where a b and c are symmetric real n n matrices and a c are positive definite we propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors the new algorithm includes two parts the first part gives iterative methods which can be used to compute the smallest positive eigenvalue we develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations the second part uses the nonequivalence deflation technique this technique allows us to transform the original problem to a new problem with different a b and c the new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero therefore the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue proceeding in this way we can find out the 3rd 4th smallest positive eigenvalues our algorithm utilizes the symmetry and positivity of the given matrices and avoids computing the undesired complex conjugate eigenvalues under some mild conditions is efficient and reliable the above process can also be used to find a few of the largest negative eigenvalues

moduli space heights and isospectral sets of plane domains let q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type then there exists a flat metric g on 2 such that q is isometric with 29g in what follows we do not distinguish between isometric domains by the spectrum of 29 we mean the spectrum of the laplace beltrami operator ag on 29 with dirichlet boundary conditions the height h eg log det ag is a spectral invariant and plays a central role in this paper among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum this metric is characterized by the fact that d 2 has constant geodesic curvature we call such a metric uniform and denote it by u the set of all such metrics is denoted by u 2 we can therefore identify with the moduli space 2 of conformal structures on e for n 3 we introduce a special parametrization for mu 2 by means of which we show that h u oo as u approaches the boundary of fu 2 using this along with the heat invariants for the laplacian we then show that any isospectral set of plane domains is compact in the c topology similar results hold for n 1 and 2

isospectral deformations of metrics on spheres we construct non trivial continuous isospectral deformations of riemannian metrics on the ball and on the sphere in r n for every n 9 the metrics on the sphere can be chosen arbitrarily close to the round metric in particular they can be chosen to be positively curved the metrics on the ball are both dirichlet and neumann isospectral and can be chosen arbitrarily close to the flat metric

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

self consistent field calculations using chebyshev filtered subspace iteration the power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each self consistent field scf iteration the method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invariant subspace obtained with the help of well selected chebyshev polynomial filters in this approach only the initial scf iteration requires solving an eigenvalue problem in order to provide a good initial subspace in the remaining scf iterations no iterative eigensolvers are involved instead chebyshev polynomials are used to refine the subspace the subspace iteration at each step is easily five to ten times faster than solving a corresponding eigenproblem by the most efficient eigen algorithms moreover the subspace iteration reaches self consistency within roughly the same number of steps as an eigensolver based approach this results in a significantly faster scf iteration

a mesh free pseudospectral approach to estimating the fractional laplacian via radial basis functions abstract this paper investigates the use of radial basis function rbf interpolants to estimate a function s fractional laplacian of a given order through a mesh free pseudospectral method the mesh free approach yields an algorithm that can be implemented in high dimensional settings without adjustment moreover the fractional laplacian is defined in terms of the fourier transform and the symmetry of rbfs can be exploited to simplify the estimation problem convergence rates are established for rbfs when the function whose fractional laplacian to be estimated is compactly supported further results demonstrate convergence when a function is in the native space for a wendland rbf i e a sobolev space and satisfies a certain l 1 condition numerical experiments demonstrate the developed method by estimating the fractional laplacian of several functions and by solving a fractional poisson equation with extended dirichlet condition in one and two dimensions

an inner outer iteration for computing pagerank we present a new iterative scheme for pagerank computation the algorithm is applied to the linear system formulation of the problem using inner outer stationary iterations it is simple can be easily implemented and parallelized and requires minimal storage overhead our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved the same idea can be used as a preconditioning technique for nonstationary schemes numerical examples featuring matrices of dimensions exceeding 100 000 000 in sequential and parallel environments demonstrate the merits of our technique our code is available online for viewing and testing along with several large scale examples

two stage iterations based on composite splittings for rectangular linear systems abstract in this paper we introduce a two stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods under suitable conditions we prove convergence of the new method the number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations further we perform a comparison analysis and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix we also discuss the uniqueness of a proper splitting and illustrate different comparison theorems for different subclasses of proper splittings