a dynamic thick restarted semi refined able algorithm for computing a few selected eigentriplets of large nonsymmetric matrices able method can be used to compute eigentriplets of large scale nonsymmetric matrices however there is no guarantee for the ritz vectors obtained by this method to converge even if the subspace is good enough furthermore for the sake of storage limitation and amount of computations restarting techniques are often needed in order to deal with these problems we propose a dynamic thick restarted semi refined able algorithm sable in which we use semi refined ritz vectors to approximate the desired eigenvectors the relationship between the novel method and the classical one is given theoretical results show that the new method can circumvent the possible danger that exists in the standard one in some degree numerical experiments are made on real world problems and comparisons are drown on the thick restarted able algorithm table and the dynamic thick restarted semi refined able algorithm sable they show that the latter is often more powerful and attractive than its standard counterpart

analysis of directed networks via partial singular value decomposition and gauss quadrature abstract large scale networks arise in many applications it is often of interest to be able to identify the most important nodes of a network or to determine the ease of traveling between them we are interested in carrying out these tasks for directed networks these networks have a nonsymmetric adjacency matrix a benzi et al 6 recently proposed that these tasks can be accomplished by studying certain matrix functions such as hyperbolic cosine and sine of a t a and a a t for small to medium sized networks the required computations can be easily carried out by first computing the singular value decomposition of a however for large networks this is impractical we propose to first compute a partial singular value decomposition of a which allows us to determine a subset of nodes that contains the most important nodes or a subset of nodes between which it is easy to travel we then apply gauss quadrature to rank the nodes in these subsets several computed examples illustrate the performance of the approach proposed

schur complements of bezoutians and the inversion of block hankel and block toeplitz matrices abstract we derive a closed inversion formula for an np np square block hankel matrix hn 1 wi j with entries wi from the ring of the p p matrices over a field the representation of h 1n 1 relies upon a strong structure preserving property of the schur complements of the nonsingular leading principal submatrices of a certain generalized bezoutian of matrix polynomials these polynomials may be determined by computing iteratively the schur complements of the nonsingular leading principal submatrices of hn 1 in an augmented block hankel matrix h which is a bezoutian matrix up to multiplication by proper reverse matrices since the special structure of the bezoutian matrix is preserved at any step of the computation such method can be efficiently implemented at an overall computational cost of o p3n2 operations in the average case and o p3n3 operations in the worst case in the normal case a suitable modification which uses both divide and conquer strategies and fast polynomial arithmetic reaches the bound of o p3n log2 n arithmetical operations

bifurcations of corank greater than two abstract we study the solution branches of stable and unstable bifurcations in certain three dimensional semilinear elliptic eigenvalue problems with dirichlet boundary conditions numerically by the continuation methods recent fast linear solvers for large sparse continuation problems are briefly reviewed sample numerical results on corank 3 bifurcation points with computer graphic output are reported

generalized anti gauss quadrature rules gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis laurie proposed an n 1 point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n point gauss quadrature rule for all polynomials of degree up to 2 n 1 this rule is referred to as an anti gauss rule it is useful for the estimation of the error in the approximation of the desired integral furnished by the n point gauss rule this paper describes a modification of the n 1 point anti gauss rule that has n k nodes and gives an error of the same magnitude and of opposite sign as the associated n point gauss quadrature rule for all polynomials of degree up to 2 n 2 k 1 for some k 1 we refer to this rule as a generalized anti gauss rule an application to error estimation of matrix functionals is presented

parametrising the attractor of the two dimensional navier stokes equations with a finite number of nodal values abstract we consider the solutions lying on the global attractor of the two dimensional navier stokes equations with periodic boundary conditions and analytic forcing we show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely proving a conjecture due to foias and temam our results also hold for the complex ginzburg landau equation the kuramoto sivashinsky equation and reaction diffusion equations with analytic nonlinearities

existence time for the 3d navier stokes equations in a generalized gevrey class abstract gevrey class technique is a widely used tool for studying higher regularity properties of solutions to dissipative equations maximal radius in a gevrey class determines a small length scale associated to the decay of the fourier power spectrum and turbulence in this paper we consider existence theory for the three dimensional incompressible navier stokes equations in a certain generalized gevrey class of functions which contains all the analytic and sub analytic gevrey classes and is in turn contained in the space of all c functions this class has been in focus recently pertaining to the study of the attractor for the 2d navier stokes equations particularly in relation to a question posed by peter constantin as to whether or not zero is in the attractor of the 2d navier stokes equations we show that in the 3d case the differential inequality that one obtains in this class is almost linear and the corresponding existence time is better than the reciprocal of any algebraic power of the norm of the initial data in this class subsequently we compare the existence times with well known ones in sobolev classes for certain types of initial data we also obtain a lower bound for the maximal radius in the generalized gevrey class under consideration here the first two authors are delighted to dedicate this paper to professor edriss titi on the occasion of his 60th birthday they admire professor titi s enormous research contributions to the field of fluid dynamics as well as his mathematical acumen novel ideas and infectious enthusiasm for the subject as evidenced by his numerous lucid and enthralling lectures this research was completed after the passing of professor basil nicolaenko the first two authors are confident that he too would have shared their sentiments concerning professor titi

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

network analysis via partial spectral factorization and gauss quadrature large scale networks arise in many applications it is often of interest to be able to identify the most important nodes of a network or to ascertain the ease of traveling between nodes these and related quantities can be determined by evaluating expressions of the form mathbf u tf a mathbf w where a is the adjacency matrix that represents the graph of the network f is a nonlinear function such as the exponential function and mathbf u and mathbf w are vectors for instance axis vectors this paper describes a novel technique for determining upper and lower bounds for expressions mathbf u tf a mathbf w when a is symmetric and bounds for many vectors mathbf u and mathbf w are desired the bounds are computed by first evaluating a low rank approximation of a which is used to determine rough bounds for the desired quantities for all nodes these rough bounds indicate for which vectors mathbf u and mathbf w more accurate bounds should be computed with the aid

boldmath topological and epsilon entropy for large volume limits of discretized parabolic equations we consider semidiscrete and fully discrete approximations of nonlinear parabolic equations in the limit of unbounded domains which by a scaling argument is equivalent to the limit of vanishing viscosity we define the spatial density of varepsilon entropy topological entropy and dimension for the attractors and show that these quantities are bounded we also provide practical means of computing lower bounds on them the proof uses the property that solutions lie in gevrey classes of analyticity which we define in a way that does not depend on the size of the spatial domain as a specific example we discuss the complex ginzburg landau equation

analyticity for the kuramoto sivashinsky equation abstract we study the analyticity properties of solutions of the kuramoto sivashinsky equation tu x t 2x 4x u x t u x t xu x t for initial data which are periodic with period l numerical expers are presented which show that the solutions of the ks equation are analytic in a strip around the real axis whose width is independent of l a rigorous lower bound o l 16 25 is given for this width

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

anti gaussian quadrature formulas an anti gaussian quadrature formula is an n 1 point formula of degree 2n 1 which integrates polynomials of degree up to 2n 1 with an error equal in magnitude but of opposite sign to that of the n point gaussian formula its intended application is to estimate the error incurred in gaussian integration by halving the difference between the results obtained from the two formulas we show that an anti gaussian formula has positive weights and that its nodes are in the integration interval and are interlaced by those of the corresponding gaussian formula similar results for gaussian formulas with respect to a positive weight are given except that for some weight functions at most two of the nodes may be outside the integration interval the anti gaussian formula has only interior nodes in many cases when the kronrod extension does not and is as easy to compute as the n 1 point gaussian formula

on the domain of analyticity of solutions to semilinear klein gordon equations abstract we consider initial value problems for semilinear klein gordon equations u t t x u u f u 0 with periodic boundary conditions assuming that both the initial data and the nonlinear forcing term f u are analytic we provide explicit lower bounds on the decay of the radius t of analyticity of the solutions as a function of time in particular in one space dimension with u real valued and f u u 2 k 1 we prove that the decay of t is not worse than 1 t the results are given in a general framework including gevrey class solutions

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

robust pade approximation via svd pade approximation is considered from the point of view of robust methods of numerical linear algebra in particular the singular value decomposition this leads to an algorithm for practical computation that bypasses most problems of solution of nearly singular systems and spurious pole zero pairs caused by rounding errors for which a matlab code is provided the success of this algorithm suggests that there might be variants of pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to infty unlike traditional pade approximants which converge only in measure or capacity

complex shift and invert strategies for real matrices abstract when using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the form fu mu where f and m are real banded matrices it is often desirable to work with the shifted and inverted operator b f m 1 m in order to enhance the eigenvalue separation and improve efficiency unfortunately the shift is generally complex and so is the matrix b the question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair f m for the classical problem where m i and f is banded complex arithmetic can be avoided by using double shifts i e by working with the real matrix bb whose bandwidth is double that of f this satisfactory solution extends to the case where m is diagonal as well in the generalized case the answer to the above question is negative in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness one solution is to factor the shifted matrix f m in complex arithmetic but employ real arithmetic subsequently in the iterative procedure this paper examines several approaches and discusses their respective merits under various circumstances

gevrey regularity for the navier stokes in a half space abstract we consider the navier stokes equations posed on the half space with dirichlet boundary conditions we give a direct energy based proof for the instantaneous space time analyticity and gevrey class regularity of the solution uniformly up to the boundary of the half space

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

a look ahead algorithm for the solution of general hankel systems the solution of systems of linear equations with hankel coefficient matrices can be computed with onlyo n 2 arithmetic operations as compared too n 3 operations for the general cases however the classical hankel solvers require the nonsingularity of all leading principal submatrices of the hankel matrix the known extensions of these algorithms to general hankel systems can handle only exactly singular submatrices but not ill conditioned ones and hence they are numerically unstable in this paper a stable procedure for solving general nonsingular hankel systems is presented using a look ahead technique to skip over singular or ill conditioned submatrices the proposed approach is based on a look ahead variant of the nonsymmetric lanczos process that was recently developed by freund gutknecht and nachtigal we first derive a somewhat more general formulation of this look ahead lanczos algorithm in terms of formally orthogonal polynomials which then yields the look ahead hankel solver as a special case we prove some general properties of the resulting look ahead algorithm for formally orthogonal polynomials these results are then utilized in the implementation of the hankel solver we report some numerical experiments for hankel systems with ill conditioned submatrices

the symplectic eigenvalue problem the butterfly form the sr algorithm and the lanczos method abstract we discuss some aspects of the recently proposed symplectic butterfly form which is a condensed form for symplectic matrices any 2n 2n symplectic matrix can be reduced to this condensed form which contains 8n 4 nonzero entries and is determined by 4n 1 parameters the symplectic eigenvalue problem can be solved using the sr algorithm based on this condensed form the sr algorithm preserves this form and can be modified to work only with the 4n 1 parameters instead of the 4n2 matrix elements the reduction of symplectic matrices to symplectic butterfly form has a close analogy to the reduction of arbitrary matrices to hessenberg form a lanczos like algorithm for reducing a symplectic matrix to butterfly form is also presented

a new implementation of the cmrh method for solving dense linear systems the cmrh method h sadok methodes de projections pour les systemes lineaires et non lineaires habilitation thesis university of lille1 lille france 1994 h sadok cmrh a new method for solving nonsymmetric linear systems based on the hessenberg reduction algorithm numer algorithms 20 1999 303 321 is an algorithm for solving nonsymmetric linear systems in which the arnoldi component of gmres is replaced by the hessenberg process which generates krylov basis vectors which are orthogonal to standard unit basis vectors rather than mutually orthogonal the iterate is formed from these vectors by solving a small least squares problem involving a hessenberg matrix like gmres this method requires one matrix vector product per iteration however it can be implemented to require half as much arithmetic work and less storage moreover numerical experiments show that this method performs accurately and reduces the residual about as fast as gmres with this new implementation we show that the cmrh method is the only method with long term recurrence which requires not storing at the same time the entire krylov vectors basis and the original matrix as in the gmres algorithm a comparison with gaussian elimination is provided

a new iterative method for solving complex symmetric linear systems based on implementation of the quasi minimal residual qmr and biconjugate a orthogonal residual bicor method a new krylov subspace method is presented for solving complex symmetric linear systems the new method can be combined with arbitrary symmetric preconditioners the preconditioned modified hermitian and skew hermitian splitting pmhss preconditioner is used to accelerate the convergence rate of this method numerical experiments indicate that the proposed method and its preconditioned version are efficient and robust in comparison with other krylov subspace methods

the methods of vorobyev and lanczos abstract this paper deals with the method of moments of vorobyev which is a particular case of galerkin s method this method is generalized in different ways and some of its properties are studied in detail as an application it is showed that the lanczos method for solving systems of linear equations can be derived from it and that some generalizations could be obtained

quadrature rule based bounds for functions of adjacency matrices bounds for entries of matrix functions based on gauss type quadrature rules are applied to adjacency matrices associated with graphs this technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities estrada index subgraph centrality communicability that describe properties of networks

local in time solvability and space analyticity for the navier stokes equations with bmo type initial data it is proved that there exists a local in time solution u in c 0 t bmo mathbb r d d of the navier stokes equations such that every u t has an analytic extension on a complex domain whose size only depends on t and increases with t and the external force f assuming only that the initial velocity u 0 is a local bmo function our method for proving this is a combination and refinement of the work by grujic and kukavica j funct anal 152 2 447 466 1998 https doi org 10 1006 jfan 1997 3167 guberovic discrete contin dyn syst 27 1 231 236 2010 https doi org 10 3934 dcds 2010 27 231 and kozono et al kyushu j math 57 2 303 324 2003 https doi org 10 2206 kyushujm 57 303 one challenging step is the estimation of the heat and stokes semigroups from bmo type spaces to l infty a result that is itself of independent interest we also apply the idea to the analyticity of vorticity with the assistance of calderon zygmund theory

a krylov subspace projection method for simultaneous solution of helmholtz problems at multiple frequencies abstract a krylov subspace projection method which provides simultaneous solutions of the helmholtz equation at multiple frequencies in one solution step is presented the projector is obtained with an unsymmetric block lanczos algorithm applied to a transfer function derived from a finite element discretization this approach is equivalent to a matrix valued pade approximation of the transfer function the proposed method is an extension of the formulation presented in j comput acoust 8 2000 223 to unsymmetric systems and allows the treatment of a much wider range of practical problems including near field and fluid structure interaction computations

rational interpolation via orthogonal polynomials abstract relations between rational interpolants and hankel matrices are investigated a modification of a jacobi like algorithm for rational interpolation proposed in 1 is developed the proposed algorithm eliminates all the auxiliary assumptions required to implement the algorithm of 1 in this way it can be used to construct all existing rational interpolants with the cost of o n 2 arithmetic operations where n 1 is the number of data points

a partial pade via lanczos method for reduced order modeling abstract the classical lanczos process can be used to efficiently generate pade approximants of the transfer function of a given single input single output time invariant linear dynamical system unfortunately in general the resulting reduced order models based on pade approximation do not preserve the stability and possibly passivity of the original linear dynamical system in this paper we describe the use of partial pade approximation for reduced order modeling partial pade approximants have a number of prescribed poles and zeros while the remaining degrees of freedom are used to match the taylor expansion of the original transfer function in as many leading coefficients as possible we present an algorithm for computing partial pade approximants via suitable rank 1 updates of the tridiagonal matrices generated by the lanczos process numerical results for two circuit examples are reported

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

ml n bicgstabt a ml n bicgstab variant with a transpose the 1980 idr method wesseling and sonneveld 1980 12 plays an important role in the history of krylov subspace methods it started the research of transpose free krylov subspace methods the ml n bicgstab method yeung 2012 is one of such methods in this paper we present a new ml n bicgstab variant that involves a transpose in its implementation comparison of this new algorithm with the existing ml n bicgstab algorithms and some other krylov subspace algorithms will be presented

network properties revealed through matrix functions the emerging field of network science deals with the tasks of modeling comparing and summarizing large data sets that describe complex interactions because pairwise affinity data can be stored in a two dimensional array graph theory and applied linear algebra provide extremely useful tools here we focus on the general concepts of centrality communicability and betweenness each of which quantifies important features in a network some recent work in the mathematical physics literature has shown that the exponential of a network s adjacency matrix can be used as the basis for defining and computing specific versions of these measures we introduce here a general class of measures based on matrix functions and show that a particular case involving a matrix resolvent arises naturally from graph theoretic arguments we also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering we finish with computational examples showing the new matrix resolvent version applied to real networks

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 implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

embedding calderon multiplicative preconditioners in multilevel fast multipole algorithms calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation efie for perfect electric conductors at lower frequencies previous authors have shown that by using a dense matrix preconditioner based on the calderon identities the low frequency instability is removed while still maintaining the inherent accuracy of the efie it was also demonstrated that the spectral properties of the caldero n preconditioner are conserved during discretization if the efie operator is discretized with rao wilton glisson expansion functions and the preconditioner with buffa christiansen expansion functions in this article we will show how the calderon multiplicative preconditioner cmp can be combined with fast multipole methods to accelerate the numerical solution leading to an overall complexity of o n logn for the entire iterative solution at low frequencies where the cmp is most useful the traditional multilevel fast multipole algorithm mlfma is unstable and we apply the nondirectional stable plane wave mlfma nspwmlfma that resolves the low frequency breakdown of the mlfma the combined algorithm will be called the cmp nspwmlfma applying the cmp nspwmlfma at open surfaces or very low frequencies leads to certain problems which will be discussed in this article

preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation the dc operating point of a circuit may be computed by tracking the zero curve of an associated artificial parameter homotopy homotopy algorithms exist that are globally convergent with probability one for the dc operating point problem these algorithms require computing the one dimensional kernel of the jacobian matrix of the homotopy mapping at each step along the zero curve and hence the solution of a linear system of equations at each step these linear systems are typically large highly sparse non symmetric and indefinite several iterative methods which are applicable to such problems including craig s method gmres k bicgstab qmr kacz and lsqr are applied to a suite of test problems derived from simulations of actual bipolar circuits preconditioning techniques considered include incomplete lu factorization ilu sparse submatrix ilu and ilu allowing restricted fill in bands or blocks timings and convergence statistics are given for each iterative method and preconditioner

boltzmann s dilemma an introduction to statistical mechanics via the kac ring the process of coarse graining here in particular of passing from a deterministic simple and time reversible dynamics at the microscale to a typically irreversible description in terms of averaged quantities at the macroscale is of fundamental importance in science and engineering at the same time it is often difficult to grasp and if not interpreted correctly implies seemingly paradoxical results the kinetic theory of gases historically the first and arguably most significant example occupied physicists for the better part of the 19th century and continues to pose mathematical challenges to this day in this paper we describe the so called kac ring model suggested by mark kac in 1956 which illustrates coarse graining in a setting so simple that all aspects can be exposed both through elementary explicit computation and through easy numerical simulation in this setting we explain a boltzmannian stoszahlansatz ensemble averages the difference between ensemble averaged and typical system behavior and the notion of entropy

the cal d q r algorithm basic theory convergence and conditional stability we describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on d orthogonal reduction techniques where d mbox diag pm1 is computed from the given matrix a it operates on a similarity reduction of a real matrix a nn to general tridiagonal form t and computes all of a s eigenvalues in frac 4 3 n 3 o n 2 operations where the o n 2 part of the operations is possibly performed over bbb c instead of the 7 8 n 3 real flops required by the qr eigenvalue algorithm potential breakdo wn of the dqr algorithm can occur in the reduction to tridiagonal form and in the d orthogonal dqr reductions both however can be monitored during the computations the former occurs rather rarely for dimensions n leq 300 and can essentially be bypassed while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the dqr steps we prove an implicit dq theorem which allows implicit shifts give a convergence proof for the dqr algorithm and show that dqr is conditionally stable for general balanced tridiagonal matrices t rm b

space analyticity for the navier stokes and related equations with initial data inlp abstract we introduce a method of estimating the space analyticity radius of solutions for the navier stokes and related equations in terms of l p and l norms of the initial data the method enables us to express the space analyticity radius for 3d navier stokes equations in terms of the reynolds number of the flow also for the kuramoto sivashinsky equation we give a partial answer to a conjecture that the radius of space analyticity on the attractor is independent of the spatial period

new look ahead lanczos type algorithms for linear systems a breakdown due to a division by zero can arise in the algorithms for implementing lanczos method because of the non existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate such a breakdown can be avoided by jumping over the polynomials involved this strategy was already used in some algorithms such as the mrz and its variants in this paper we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors independent of the length of the jump these new algorithms are based on horner s rule and on a different way for computing the coefficients of the recurrence relationships moreover these new algorithms seem to be more stable than the old ones and they provide better numerical results numerical examples and comparisons with other algorithms will be given

computation of gauss type quadrature formulas gaussian formulas for a linear functional l such as a weighted integral are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to l in gauss type formulas one or more extraneous conditions such as pre assigning certain nodes replace some of the equations expressing exactness when applied to high order polynomials these extraneous conditions may be applied by modifying the same number of recursion coefficients we survey the methods of computing formulas from recursion coefficients methods of obtaining recursion coefficients and modifying them for gauss type formulas and questions of existence and numerical accuracy associated with those computations

a review of formal orthogonality in lanczos based methods krylov subspace methods and their variants are presently the favorite iterative methods for solving a system of linear equations although it is a purely linear algebra problem it can be tackled by the theory of formal orthogonal polynomials this theory helps to understand the origin of the algorithms for the implementation of krylov subspace methods and moreover the use of formal orthogonal polynomials brings a major simplification in the treatment of some numerical problems related to these algorithms this paper reviews this approach in the case of lanczos method and its variants the novelty being the introduction of a preconditioner

computation of matrix valued formally orthogonal polynomials and applications we present a computational procedure for generating formally orthogonal polynomials associated with a given bilinear hankel form with rectangular matrix valued moments our approach covers the most general case of moments of any size and is not restricted to square moments moreover our algorithm has a built in deflation procedure to handle linearly dependent or almost linearly dependent columns and rows of the block hankel matrix associated with the bilinear form possible singular or close to singular leading principal submatrices of the deflated block hankel matrix are avoided by means of look ahead techniques applications of the computational procedure to eigenvalue computations reduced order modeling the solution of multiple linear systems and the fast solution of block hankel systems are also briefly described

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

adaptive control of a class of slowly time varying plants abstract an adaptive control system of the type considered earlier by the author which is designed to be stable under the assumption that the unknown plant parameters are constant is shown to maintain stability when the plant parameters are slowly time varying

the role of the number of degrees of freedom and chaos in macroscopic irreversibility abstract this article aims at revisiting with the aid of simple and neat numerical examples some of the basic features of macroscopic irreversibility and thus of the mechanical foundation of the second principle of thermodynamics as drawn by boltzmann emphasis will be put on the fact that in systems characterized by a very large number of degrees of freedom irreversibility is already manifested at a single trajectory level for the vast majority of the far from equilibrium initial conditions a property often referred to as typicality we also discuss the importance of the interaction among the microscopic constituents of the system and the irrelevance of chaos to irreversibility showing that the same irreversible behaviors can be observed both in chaotic and non chaotic systems

local analyticity radii of solutions to the 3d navier stokes equations with locally analytic forcing abstract we introduce a new method for establishing local analyticity and estimating the local analyticity radius of a solutions to the 3d navier stokes equations at interior points the approach is based on rephrasing the problem in terms of second order parabolic systems which are then estimated using the mild solution approach the estimates agree with the global analyticity radius from 16 up to a logarithm

irreversibility and typicality a simple analytical result for the ehrenfest model abstract with the aid of simple analytical computations for the ehrenfest model we clarify some basic features of macroscopic irreversibility the stochastic character of the model allows us to give a non ambiguous interpretation of the general idea that irreversibility is a typical property for the vast majority of the realizations of the stochastic process a single trajectory of a macroscopic observable behaves irreversibly remaining very close to the deterministic evolution of its ensemble average which can be computed using probability theory the validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit

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

the lanczos algorithm and hankel matrix factorization abstract we explore the connections between the lanczos algorithm for matrix tridiagonalization and two fast algorithms for hankel matrix factorization we show how the asymmetric lanczos process is related to the berlekamp massey algorithm and how the symmetrized lanczos process is related to the phillips algorithm we also discuss conditions under which the analysis of phillips applies

a breakdown free lanczos type algorithm for solving linear systems lanczos type algorithms for solving systems of linear equations have their foundations in the theory of formal orthogonal polynomials and the method of moments which leads to a determinantal formula for their iterates the various lanczos type algorithms mainly differ by the way of computing the coefficients entering into the recurrence formulae if the denominator in the formula for one of these coefficients is zero then a breakdown occurs in the algorithm and it must be stopped such a breakdown is in fact due to the non existence of some orthogonal polynomial in this paper we show how to jump over such a singularity by computing the next existing orthogonal polynomial by the block bordering method the resulting algorithm called mrz is equivalent to the nongeneric biodir algorithm which is a look ahead lanczos type algorithm but our derivation is much simpler

a quasi minimal residual variant of the bi cgstab algorithm for nonsymmetric systems motivated by a recent method of freund siam j sci comput 14 1993 pp 470 482 who introduced a quasi minimal residual qmr version of the conjugate gradients squared cgs algorithm a qmr variant of the biconjugate gradient stabilized bi cgstab algorithm of van der vorst that is called qmrcgstab is proposed for solving nonsymmetric linear systems the motivation for both qmr variants is to obtain smoother convergence behavior of the underlying method the authors illustrate this by numerical experiments that also show that for problems on which bi cgstab performs better than cgs the same advantage carries over to qmrcgstab

block krylov subspace methods for the computation of structural response to turbulent wind abstract in this paper the numerical computation of the dynamic response to turbulent wind excitations of slender structures is addressed a numerical procedure capable to effectively estimate the three dimensional structural behavior is proposed based on a direct frequency domain approach a probabilistic description of the wind velocity field accounting for the correlation between the turbulence components is combined to a linearized fluid structure interaction model under the quasi steady hypothesis we propose robust implementations of multiple right hand side and multiple shift krylov subspace methods with deflation of basis vectors which allow us to efficiently analyze the dynamic response for a wide range of frequency values and wind time histories numerical experiments are reported with data stemming from real structure modeling

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

level sets of the vorticity and the stream function for the 2d periodic navier stokes equations with potential forces abstract by a result of foias and saut the quotient of the enstrophy and the energy of a solution of the navier stokes equations with potential forces converges to the square root of an eigenvalue k of the stokes operator we study the hausdorff length h 1 of the level sets of the vorticity and the stream function of solutions we provide upper bounds for these quantities and for large t we express them in terms of the corresponding eigenvalue k

an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem abstract an implicitly restarted symplectic lanczos method for the hamiltonian eigenvalue problem is presented the lanczos vectors are constructed to form a symplectic basis the inherent numerical difficulties of the symplectic lanczos method are addressed by inexpensive implicit restarts the method is used to compute eigenvalues eigenvectors and invariant subspaces of large and sparse hamiltonian matrices and low rank approximations to the solution of continuous time algebraic riccati equations with large and sparse coefficient matrices

stable row recurrences for the pade table and generically superfast lookahead solvers for non hermitian toeplitz systems abstract we present general recurrences for the pade table that allow us to skip ill conditioned pade approximants while we proceed along a row of the table in conjunction with a certain inversion formula for toeplitz matrices these recurrences form the basis for fast algorithms for solving non hermitian toeplitz systems under the assumption that the lookahead step size i e the number of successive skipped approximants remains bounded we give both o n 2 and o n log 2 n algorithms which are presumably weakly stable with little additional work still in o n 2 operations we can also obtain a decomposition of the toeplitz matrix t according to tr ld where r is upper triangular l is unit lower triangular and d is block diagonal the relation to continued fractions is also discussed

a transpose free quasi minimal residual variant of the cors method for solving non hermitian linear systems recently the conjugate a orthogonal residual squared cors method is often competitive and superior to other krylov subspace methods for solving complex non hermitian linear systems from many realistic problems however like the conjugate gradient squared cgs method the cors method often exhibits irregular convergence behavior with wild oscillations in the residual norm to try and remedy this problem we develop a transpose free quasi minimal residual tfqmr variant of the cors method which leads to smooth convergence curves with a convergence rate similar to the cors method we also give convergence results for this new method finally numerical examples are reported to illustrate the effectiveness of the proposed method

generalized block anti gauss quadrature rules golub and meurant describe how pairs of gauss and gauss radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real valued matrix functionals defined by a symmetric matrix however there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds in this situation it may be possible to determine upper and lower bounds by evaluating pairs of gauss and anti gauss rules unfortunately it is difficult to ascertain whether the values determined by gauss and anti gauss rules bracket the value of the given real valued matrix functional therefore generalizations of anti gauss rules have recently been described such that pairs of gauss and generalized anti gauss rules may determine upper and lower bounds for real valued matrix functionals also when pairs of gauss and standard anti gauss rules do not the available generalization requires the matrix that defines the functional to be real and symmetric the present paper reviews available anti gauss and generalized anti gauss rules and extends them in several ways that allow applications in new situations in particular the genarlized anti gauss rules for a real valued non negative measure described in pranic and reichel j comput appl math 284 235 243 2015 are extended to allow the estimation of the error in matrix functionals defined by a non symmetric matrix as well as to matrix valued matrix functions modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described

further optimized look ahead recurrences for adjacent rows in the pade table and toeplitz matrix factorizations abstract in a recent paper we introduced a new look ahead algorithm for recursively computing pade approximants this algorithm generates a subsequence of the pade approximants on two adjacent rows defined by fixed numerator degree of the pade table its two basic versions reduce to the classical levinson and schur algorithms if no look ahead is required in this paper we show that the computational overhead of the look ahead steps in the o n2 versions of the look ahead levinson and the look ahead schur type algorithm can be further reduced if the algorithms are used to solve toeplitz systems of equations tx b then the corresponding block ldu decompositions of t 1 or t respectively can be found with less computational effort than with any other look ahead algorithm available today

generalized biorthogonal bases and tridiagonalisation of matrices in connection with the breakdown problem of the lanczos algorithm a theory of generalized biorthogonal bases is developed the connection between the generalized biorthogonal bases of krylov chains and look ahead lanczos recursions is worked out in detail it is shown how generalized biorthogonal bases with antidiagonal blocks can be constructed with small computational effort finally a special look ahead lanczos algorithm is derived which requires minimal computational effort and storage

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

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

a quasi minimal residual method for simultaneous primal dual solutions and superconvergent functional estimates the adjoint solution has found many uses in computational simulations where the quantities of interest are the functionals of the solution including design optimization error estimation and control in those applications where both the solution and the adjoint are desired the conventional approach is to apply iterative methods to solve the primal and dual problems separately however we show that there is an advantage associated with iterating the primal and dual problem simultaneously since this enables the construction of iterative methods where both the primal and the dual iterates may be chosen so that they provide functional estimates that are superconvergent in that the error converges at twice the order of the optimal global solution error norm in particular we show that the structure of the lanczos process allows for this superconvergence property and propose a modified qmr method which uses the same lanczos process to simultaneously solve the primal and dual problems thus both the primal and the dual systems are solved at essentially the same computational cost as the conventional qmr method applied to the primal problem alone numerical experiments show that our proposed method does indeed exhibit superconvergence behavior

a block qmr algorithm for non hermitian linear systems with multiple right hand sides abstract many applications require the solution of multiple linear systems that have the same coefficient matrix but differ in their right hand sides instead of applying an iterative method to each of these systems individually it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously in this paper we propose a block version of freund and nachtigal s quasi minimal residual qmr method for the iterative solution of non hermitian linear systems the block qmr method uses a novel lanczos type process for multiple starting vectors which was recently developed by aliaga boley freund and hernandez to compute suitable basis vectors for the underlying block krylov subspaces we describe the basic block qmr method and also give important implementation details in particular we show how to incorporate deflation to drop converged linear systems and to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences numerical results are reported that illustrate typical features of the block qmr method

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

space analyticity for the nonlinear heat equation in a bounded domain 1 introductionin fot foias and temam introduced a method for estimating thespace analyticity radius of solutions of the navier stokes equation withperiodic boundary conditions for the numerous applications seee g cees crt dth ft gr k lo in our previous paper gk we did not use the fourier series and were thus able to treat the navier stokes and other semilinear parabolic equations with singular l