acoustic propagation in a refracting ocean waveguide with an irregular interface abstract a method is derived for computing the acoustic field in a cylindrically symmetric ocean with an irregular water sediment interface and a general depth dependent sound speed profile the approach is based on a stepwise coupled mode solution which was originally developed for a homogeneous water column the extension to a general sound speed profile is carried out with the aid of galerkin s method numerical examples are given for a shallow water waveguide with an upward refracting sound speed profile the first example shows that propagation loss is significantly greater in a rough bottomed waveguide than in a smooth bottomed waveguide in the second example it is shown that upward refraction reduces the effect of bottom roughness on sound propagation by reducing the insonification of the rough water sediment interface

a new method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation abstract we present a fast method for computing the closed loop eigenvalues of a discrete time algebraic riccati equation the method is close to the symplectic method for finding all the eigenvalues of a hamiltonian matrix and is based on a g orthogonal transformation which preserves structure and has desirable numerical properties the algorithm requires about one fourth the number of floating point operations and one half the storage of the qz algorithm

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

solution of lyapunov equations by alternating direction implicit iteration abstract in this report a new procedure is presented for solving the lyapunov matrix equation first the system is reduced to tridiagonal form with gaussian similarity transformations then the resulting system is solved with alternating direction implicit adi iteration a matrix commutation property essential for model problem convergence of adi iteration applied to elliptic difference equations is not needed for this application all stable lyapunov matrix equations are model adi problems

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

on matrix differential equations and abstract fg algorithm abstract there is well known relationship between both the isospectral flows and the unitarily self equivalent flows and the abstract fg eigenvalue algorithm namely the values of the isospectral and self equivalent flows at integer times coincide with the iterates of fg algorithm we show that such a relation is characteristic of a whole class of matrix differential equations all these equations are built by means of various representations of an arbitrary lie algebra on k n n

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

the constrained newton method on a lie group and the symmetric eigenvalue problem the newton method for estimating a critical point of a real function is formulated in a coordinate free manner on an arbitrary lie group convergence proofs for the numerical method are given an application of the general approach to computing the eigenvalues of a symmetric matrix is given and the resultant algorithm is compared with the classical shifted qr algorithm properties of the method described suggest that it is of interest for certain computations in online and adaptive environments

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

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved

markov chains with memory tensor formulation and the dynamics of power iteration a markov chain with memory is no different from the conventional markov chain on the product state space such a markovianization however increases the dimensionality exponentially instead markov chain with memory can naturally be represented as a tensor whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products in this context the progression of a markov chain can be interpreted as variants of power like iterations moving toward the limiting probability distributions what is not clear is the makeup of the second dominant eigenvalue that affects the convergence rate of the iteration if the method converges at all casting the power method as a fixed point iteration this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence as an application it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the z eigenvector type power iteration fails to converge

the lr cholesky algorithm for symmetric hierarchical matrices abstract we investigate the application of the lr cholesky algorithm to symmetric hierarchical matrices symmetric simple structured hierarchical matrices and symmetric hierarchically semiseparable hss matrices the data sparsity of these matrices make the otherwise expensive lr cholesky algorithm applicable as long as the data sparsity is preserved we will see in an example that the ranks of the low rank blocks grow and the data sparsity gets lost we will explain this behavior by applying a theorem on the structure preservation of diagonal plus semiseparable matrices under lr cholesky transformations therefore we have to give a new more constructive proof for the theorem we will show that the structure of h l matrices is almost preserved and so the lr cholesky algorithm is of almost quadratic complexity for h l matrices

effects of strategy migration direction and noise in the evolutionary spatial prisoner s dilemma spatial games are crucial for understanding patterns of cooperation in nature and to some extent society they are known to be more sensitive to local symmetries than e g spin models this paper concerns the evolution of the prisoner s dilemma game on regular lattices with three different types of neighborhoods char22 the von neumann moore and kagom e types we investigate two kinds of dynamics for the players to update their strategies that can be unconditional cooperator or defector depending on the payoff difference an individual can adopt the strategy of a random neighbor a voter model like dynamics vmld or impose its strategy on a random neighbor i e invasion process like dynamics ipld in particular we focus on the effects of noise in combination with the strategy dynamics on the evolution of cooperation we find that vmld compared to ipld better supports the spreading and sustaining of cooperation we see that noise has nontrivial effects on the evolution of cooperation maximum cooperation density can be realized either at a medium noise level in the limit of zero noise or in both these regions the temptation to defect and the local interaction structure determine the outcome especially in the low noise limit the local interaction plays a crucial role in determining the fate of cooperators we elucidate these both by numerical simulations and mean field cluster approximation methods

differential equations and the qr algorithm in this paper we consider a variety of isospectral flows on the set of n times n matrices these flows arise from lax pairs and can all be interpreted in terms of the qr decomposition for nonsingular matrices the asymptotics of these differential equations are considered in detail and for symmetric matrices these asymptotics provide a new method of solving the eigenvalue problem

evolutionary prisoner s dilemma game on newman watts networks maintenance of cooperation was studied for a two strategy evolutionary prisoner s dilemma game where the players are located on a one dimensional chain and their payoff comes from games with the nearest and next nearest neighbor interactions the applied host geometry makes it possible to study the impacts of two conflicting topological features the evolutionary rule involves some noise affecting the strategy adoptions between the interacting players using monte carlo simulations and the extended versions of dynamical mean field theory we determined the phase diagram as a function of noise level and a payoff parameter the peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by newman and watts

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

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

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

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

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

the evolutionary public goods game on scale free networks with heterogeneous investment abstract the public goods game pgg is generally considered as a suitable paradigm to explain ubiquitous cooperative behavior in this study we investigated the evolutionary pgg on scale free networks and studied the effect of individual heterogeneity by setting the cooperator x an investment value correlated to its degree as i x n k x j k j where k x is the degree of x j runs over all players and is a tunable parameter it is shown that the cooperation level is remarkably promoted by negative values of whereas it is highly depressed by positive values of moreover the effect of environmental noise has also been investigated our result may sharpen the understanding of cooperation induced by the individual diversity

on hamiltonian and symplectic lanczos processes abstract large sparse hamiltonian eigenvalue problems arise in a variety of contexts these problems can be attacked directly or they can first be transformed to problems having some related structure such as symplectic or skew hamiltonian in the interest of efficiency stability and accuracy such problems should be solved by methods that preserve the structure whether it be hamiltonian skew hamiltonian or symplectic the present work outlines krylov subspace methods for computing partial eigensystems of skew hamiltonian hamiltonian and symplectic matrices and records some of the relationships between them the ordinary unsymmetric lanczos process is a structure preserving method for skew hamiltonian matrices the hamiltonian and symplectic lanczos processes developed here are condensed versions of the processes that have been published previously the condensed hamiltonian lanczos process applied to h is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian h 2 but costs half as much to execute the condensed symplectic lanczos process applied to s is equivalent to the unsymmetric lanczos process applied to the skew hamiltonian matrix s s 1 but also costs half as much to execute implicit restarts of the hamiltonian and symplectic lanczos processes can be effected by the sr algorithm because of the known relationship between the sr and hr algorithms the much simpler hr algorithm can be used to restart the condensed symplectic and hamiltonian lanczos processes each of these restart procedures is equivalent to restarting the unsymmetric lanczos process using the hr algorithm the hr algorithm is most effectively implemented as an implicit bulge chasing hz algorithm on a symmetric tridiagonal diagonal pencil t d

evolutionary prisoner s dilemma game with dynamic preferential selection we study a modified prisoner s dilemma game taking place on two dimensional disordered square lattices the players are pure strategists and can either cooperate or defect with their immediate neighbors in the generations each player updates its strategy by following one of the neighboring strategies with a probability dependent on the payoff difference the neighbor selection obeys a dynamic preferential rule i e the more frequently a neighbor s strategy was adopted by the focal player in the previous rounds the larger probability it will be chosen to refer to in the subsequent rounds it is found that cooperation is substantially promoted due to this simple selection mechanism corresponding analysis is provided by the investigation of the distribution of the players impact weights persistence and correlation function

on the sensitivity of the sr decomposition abstract first order componentwise and normwise perturbation bounds for the sr decomposition are presented the new normwise bounds are at least as good as previously known results in particular for the r factor the normwise bound can be significantly tighter than the previous result

the pseudosymmetric tridiagonalization of an arbitrary real matrix this paper presents an algorithm for similarly reducing an arbitrary real matrix to a pseudosymmetric tridiagonal one discusses the possibility of overcoming breakdowns in the tridiagonalization process and gives some methods it also discusses the perturbation bounds of eigenvalues of a pseudosymmetric tridiagonal matrix some numerical examples are given

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

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

dynamical systems and non hermitian iterative eigensolvers simple preconditioned iterations can provide an efficient alternative to more elaborate eigenvalue algorithms we observe that these simple methods can be viewed as forward euler discretizations of well known autonomous differential equations that enjoy appealing geometric properties this connection facilitates novel results describing convergence of a class of preconditioned eigensolvers to the leftmost eigenvalue provides insight into the role of orthogonality and biorthogonality and suggests the development of new methods and analyses based on more sophisticated discretizations these results also highlight the effect of preconditioning on the convergence and stability of the continuous time system and its discretization

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

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

systems of hess appel rot type we construct higher dimensional generalizations of the classical hess appel rot rigid body system we give a lax pair with a spectral parameter leading to an algebro geometric integration of this new class of systems which is closely related to the integration of the lagrange bitop performed by us recently and uses mumford relation for theta divisors of double unramified coverings based on the basic properties satisfied by such a class of systems related to bi poisson structure quasi homogeneity and conditions on the kowalevski exponents we suggest an axiomatic approach leading to what we call the class of systems of hess appel rot type

evolutionary prisoner s dilemma game on a square lattice a simplified prisoner s game is studied on a square lattice when the players interacting with their neighbors can follow two strategies to cooperate c or to defect d unconditionally the players updated in random sequence have a chance to adopt one of the neighboring strategies with a probability depending on the payoff difference using monte carlo simulations and dynamical cluster techniques we study the density c of cooperators in the stationary state this system exhibits a continuous transition between the two absorbing states when varying the value of temptation to defect in the limits stackrel ensuremath rightarrow c 0 and 1 we have observed critical transitions belonging to the universality class of directed percolation

a grassmann rayleigh quotient iteration for computing invariant subspaces the classical rayleigh quotient iteration rqi allows one to compute a one dimensional invariant subspace of a symmetric matrix a here we propose a generalization of the rqi which computes a p dimensional invariant subspace of a cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature

the dynamics of rayleigh quotient iteration this paper analyzes the rayleigh quotient iteration algorithm for symmetric matrices dynamical systems techniques are employed to characterize the sets of points for which the algorithm will converge to an eigenvector it is shown that these sets have full measure and that their geometric nature is related to the spacing of the eigenvalues

asymptotic analysis of toda lattice on diagonalizable matrices soit x x 0 g x x 0 x 0 x est une matrice a valeur complexe et g x une integrale de contour a valeur matricielle on etudie la dynamique de x t sous les hypotheses suivantes 1 x 0 x 0 r n n est diagonalisable 2 g z z 3 x 0 est une matrice de hessenberg superieure irreductible

on computing the eigenvalues of a symplectic pencil this paper presents an algorithm for computing the eigenvalues of a symplectic pencil that arises in one of the commonly used approaches for solving the discrete time algebraic riccati equation the algorithm is numerically efficient and reliable in that it employs only orthogonal transformations and makes use of the structure of the symplectic pencil it requires about one fourth the number of floating point operations that the qz algorithm uses to compute the eigenvalues of the pencil directly the proposed method can be regarded as being analogous for the case of symplectic pencils to the method developed by van loan for computing the eigenvalues of hamiltonian matrices

rational qr iteration without inversion in this manuscript a new method will be presented for performing a qr iteration with a i a i 1 qr without explicit inversion of the factor a i 1 a qr method driven by a rational function is attractive since convergence can occur at both sides of the matrix each step of this new iteration consists of two substeps in the explicit version first an rq factorization of the initial matrix a i rq will be computed followed by a qr factorization of the matrix a i q h the factorization of a i q h can be computed in an intelligent manner exploiting properties of the already known rq factorization of a i the similarity transformation yielding the qr step is defined by the unitary factor q in the qr factorization of the transformed matrix a i q h examples will be given illustrating how to efficiently compute the factorization for some specific classes of matrices the novelties of this approach with respect to these matrix classes will be discussed

eigenvalues of ax λbx for real symmetric matrices a and b computed by reduction to a pseudosymmetric form and the hr process abstract the paper presents a method for solving the eigenvalue problem ax bx where a and b are real symmetric but not necessarily positive definite matrices and b is nonsingular the method reduces the general case into a form cz z where c is a pseudosymmetric matrix a further reduction of c produces a tridiagonal pseudosymmetric form to which the iterative hr process is applied the tridiagonal pseudosymmetric form is invariant under the hr transformations the amount of computation is significantly less than in treating the problem by a general method

preferential selection promotes cooperation in a spatial public goods game abstract we introduce a preferential selection mechanism into a spatial public goods game where players are located on a square lattice each individual chooses one of its neighbors as a reference with a probability proportional to exp p y a where p y is the neighbor s payoff and a 0 is a tunable parameter it is shown that the introduction of such a preferential selection can remarkably promote the emergence of cooperation over a wide range of the multiplication factor we find that the mean payoffs of cooperators along the boundary are higher than that of defectors and cooperators form larger clusters as a increases the extinction thresholds of cooperators and defectors for different values of noise are also investigated

the lagrange bitop on so 4 so 4 and geometry of the prym varieties a four dimensional integrable rigid body system is considered and it is shown that it represents two twisted three dimensional lagrange tops a polynomial lax representation which doesn t fit neither in dubrovin s nor in adler van moerbeke s picture is presented the algebro geometric integration procedure is based on deep facts from the geometry of the prym varieties of double coverings of hyperelliptic curves mumford s relation and mumford dalalyan theory the correspondence between all such coverings with prym varieties splitted as a sum of two varieties of the same dimension and the integrable hierarchy associated to the initial system is established

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

two connections between the sr and hr eigenvalue algorithms abstract the sr and hr algorithms are members of the family of gr algorithms for calculating eigenvalues and invariant subspaces of matrices this paper makes two connections between the sr and hr algorithms 1 an iteration of the sr algorithm on a 2 n 2 n symplectic butterfly matrix using shifts i 1 i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts i 1 i i 1 k 2 an iteration of the sr algorithm on a 2 n 2 n j tridagonal hamiltonian matrix using shifts i i i 1 k is equivalent to an iteration of the hr algorithm on an n n tridiagonal sign symmetric matrix using shifts 2 i i 1 k

impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies cooperation and defection may be considered to be two extreme responses to a social dilemma yet the reality is much less clear cut between the two extremes lies an interval of ambivalent choices which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool if strategies are chosen from the unit interval where 0 corresponds to pure defection and 1 corresponds to the maximal contribution the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal however as the cost to benefit ratio of cooperation increases the characteristic threshold decreases while the corresponding steepness increases our observations remain valid if more complex sigmoid functions are used thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

qr like algorithms for eigenvalue problems in the year 2000 the dominant method for solving matrix eigenvalue problems is still the qr algorithm this paper discusses the family of gr algorithms with emphasis on the qr algorithm included are historical remarks an outline of what gr algorithms are and why they work and descriptions of the latest highly parallelizable versions of the qr algorithm now that we know how to parallelize it the qr algorithm seems likely to retain its dominance for many years to come

product eigenvalue problems many eigenvalue problems are most naturally viewed as product eigenvalue problems the eigenvalues of a matrix a are wanted but a is not given explicitly instead it is presented as a product of several factors a a k a k 1 cdots a 1 usually more accurate results are obtained by working with the factors rather than forming a explicitly for example if we want eigenvalues vectors of b t b it is better to work directly with b and not compute the product the intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems for all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic gr algorithm applied to a related cyclic matrix

self similar flows abstract large classes of self similar isospectral flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms in particular there exist families of flows associated with the qr lr and cholesky eigenvalue algorithms this paper uses lie theory to develop a general theory of self similar flows which includes the qr lr and cholesky flows as special cases also included are new families of flows associated with the sr and hr eigenvalue algorithms the basic theory produces analogues of unshifted single step eigenvalue algorithms but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple step eigenvalue algorithms

two sided grassmann rayleigh quotient iteration the two sided rayleigh quotient iteration proposed by ostrowski computes a pair of corresponding left right eigenvectors of a matrix c we propose a grassmannian version of this iteration i e its iterates are pairs of p dimensional subspaces instead of one dimensional subspaces in the classical case the new iteration generically converges locally cubically to the pairs of left right p dimensional invariant subspaces of c moreover grassmannian versions of the rayleigh quotient iteration are given for the generalized hermitian eigenproblem the hamiltonian eigenproblem and the skew hamiltonian eigenproblem

a symplectic method for approximating all the eigenvalues of a hamiltonian matrix a fast method for computing all the eigenvalues of a hamiltonian matrix m is given the method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties the algorithm is about four times faster than the standard q r algorithm the computed eigenvalues are shown to be the exact eigenvalues of a matrix m e where vert e vert depends on the square root of the machine precision the accuracy of a computed eigenvalue depends on its condition and its magnitude larger eigenvalues typically being more accurate

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

the qr algorithm revisited the qr algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices most discussions of the qr algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used this paper outlines a pedagogical path that leads directly to the implicit multishift qr algorithms that are used in practice bypassing the basic qr algorithm completely

an l a pair for the hess apel rot system and a new integrable case for the euler poisson equations on so 4 so 4 we present an l a pair for the hess apel rot case of a heavy rigid three dimensional body using it we give an algebro geometric integration procedure generalizing this l a pair we obtain a new completely integrable case of the euler poisson equations in dimension four explicit formulae for integrals that are in involution are given this system is a counterexample to one of ratiu s theorems a corrected version of this classification theorem is proved

ordinary differential equations and the symmetric eigenvalue problem in this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations new algorithms are suggested and old algorithms including qr are interpreted

the pfaff lattice on symplectic matrices the pfaff lattice is an integrable system arising from the sr group factorization in an analogous way to how the toda lattice arises from the qr group factorization in our earlier paper kodama and pierce 2007 int math res not arxiv 0705 0510 we studied the pfaff lattice hierarchy for the case where the lax matrix is defined to be a lower hessenberg matrix in this paper we deal with the case of a symplectic lower hessenberg lax matrix this forces the lax matrix to take a 2 2 block tridiagonal shape we then show that the odd members of the pfaff lattice hierarchy are trivial while the even members are equivalent to the indefinite toda lattice hierarchy defined in kodama and ye 1996 physica d 91 321 39 this is analogous to the case of the toda lattice hierarchy in relation to the kac van moerbeke system in the case with the initial matrix having only real or imaginary eigenvalues the fixed points of the even flows are given by 2 2 block diagonal matrices with zero diagonals we also consider a family of skew orthogonal polynomials with a symplectic recursion relation related to the pfaff lattice and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite toda lattice

a quadratically convergent qr like method without shifts for the hermitian eigenvalue problem abstract we propose a new qr like algorithm symmetric squared qr ssqr method that can be readily parallelized using commonly available parallel computational primitives such as matrix matrix multiplication and qr decomposition the algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts we provide a rigorous convergence analysis of ssqr and derive structures for several of the important quantities generated by the algorithm we also discuss various practical implementation issues such as stopping criteria and deflation techniques we demonstrate the convergence behavior of ssqr using several numerical examples

effects of inhomogeneous activity of players and noise on cooperation in spatial public goods games we study the public goods game in the noisy case by considering the players with inhomogeneous activity of teaching on a square lattice it is shown that the introduction of the inhomogeneous activity of teaching the players can remarkably promote cooperation by investigating the effects of noise on cooperative behavior in detail we find that the variation of cooperator density c with the noise parameter displays several different behaviors c monotonically increases decreases with c first increases decreases with and then it decreases increases monotonically after reaching its maximum minimum value which depends on the amount of the multiplication factor r on whether the system is homogeneous or inhomogeneous and on whether the adopted updating is synchronous or asynchronous these results imply that the noise plays an important and nontrivial role in the evolution of cooperation

diversity optimized cooperation on complex networks we propose a strategy for achieving maximum cooperation in evolutionary games on complex networks each individual is assigned a weight that is proportional to the power of its degree where the exponent is an adjustable parameter that controls the level of diversity among individuals in the network during the evolution every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor and updates its strategy depending on their payoff difference it is found that there exists an optimal value of for which the level of cooperation reaches maximum this phenomenon indicates that although high degree individuals play a prominent role in maintaining the cooperation too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation other pertinent quantities such as the payoff the cooperator density as a function of the degree and the payoff distribution are also investigated computationally and theoretically our results suggest that in order to achieve strong cooperation on a complex network individuals should learn more frequently from neighbors with higher degrees but only to a certain extent