a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

on adaptive choice of shifts in rational krylov subspace reduction of evolutionary problems we compute u t exp ta varphi using rational krylov subspace reduction for 0 leq t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n a priori optimization of the rational krylov subspaces for this problem was considered in v druskin l knizhnerman and m zaslavsky siam j sci comput 31 2009 pp 3760 3780 there was suggested an algorithm generating sequences of equidistributed shifts which are asymptotically optimal for the cases with uniform spectral distributions here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum the algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost the effectiveness of the developed approach is demonstrated in an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration we compare our approach with the one using the above mentioned equidistributed sequences of shifts numerical examples show that our algorithm is able to adapt to the spectral density of operator a for examples with near uniform spectral distributions both algorithms show the same convergence rates but the new algorithm produces superior convergence for cases with nonuniform spectra

a deflated version of the conjugate gradient algorithm we present a deflated version of the conjugate gradient algorithm for solving linear systems the new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin it can also be useful when solving linear systems with multiple right hand sides since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated

epidemic dynamics discrete time and cellular automaton models abstract we present a simple model of population dynamics in the presence of an infection the model is based on discrete time equations for sane and infected populations in interaction and correctly describes the dynamics of the epidemic we find that for some choices of the parameters the model can possess conserved quantities we also propose an ultra discrete cellular automaton version of the model which despite its extremely simple structure still captures the essence of the epidemic dynamics

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

block minimum perturbation algorithm based on block arnoldi process for nonsymmetric linear systems with multiple right hand sides abstract in many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right hand sides customarily one uses the residual error as a stopping condition but small residue does not imply accurate approximate solutions therefore minimizing quantities other than the residual norm may be more adequate in this context based on the above consideration a block minimum perturbation krylov subspace method bminpet for the solution xm satisfying a a x m b b at each step of nonsymmetric linear systems a x b is shown such that the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm a b f of the matrix a b and a stopping criterion is given for consistent system with multiple right hand sides to get a more robust way to stop the iterations this process is a generalization of minimum perturbation algorithm minpert by kasenally and simoncini 1997 analysis of a minimum perturbation algorithm for nonsymmetric linear systems siam j numer anal 34 48 66 to multiple right hand sides for the benefit of the amount of calculation we give lower and upper bounds about min a b f that are easier to compute as a by product we come up with a conception of function f g which is a generalization of function that is a symmetric gauge sg function the algorithm is used to analyze the performance of bgmres method by evaluating the proximity of their solutions to block joint backward perturbation optimality the relationships between bminpet method and relative methods are investigated numerical examples show us that bminpert shows its superiority compared with bfgmres s m pf gsgmres bl bicg rq bgmres and barnoldi in solving the large sparse ill conditioned problems

on restart and error estimation for krylov approximation of w f a v krylov algorithms are widely used for the solution of large scale linear systems a krylov approach is implemented also for the approximation of the vector w which results from operating with a function of a matrix on a vector v for example when solving a set of linear odes we have w f a v where f z rm exp tz the main drawback of the krylov algorithm lies in the need to store all the vectors spanning the approximation space when solving linear systems it is possible to overcome this drawback by restarting in this paper we show how to apply restarts in the general case of approximating w f a v this new algorithm allows a much more efficient approximation of the exponential operator than the standard krylov algorithm and it is especially useful in the case of functions which cannot be factored into a product of functions yet another interesting subject is the error estimate the scheme given here provides error estimates for free a trait which enables us to stop the algorithm whenever the desired accuracy is achieved

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

structural dynamics analysis using an unsymmetric block lanczos algorithm this paper describes the way in which an unsymmetric block lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and or repeated or closely spaced eigenvalues to reduce a square unsymmetric matrix which may have repeated eigenvalues to block tridiagonal form an unsymmetric block lanczos algorithm is developed right and left lanczos vectors generated by the reduction process are employed to define the reduced order model based on a discrete analytical model of general linear time invariant dynamic systems a special type of starting vector is also introduced which automatically includes the static displacement this reduced order model is employed to compute approximate eigenvalues vectors and dynamic responses of the original system to verify the proposed algorithm examples of an 8 dof beam rotor system are provided for eigensolutions and dynamic responses due to step and random external forces these examples showed promising results indicating that the proposed algorithm is worthy of further study

residual restarting and richardson iteration for the matrix exponential a well known problem in computing some matrix functions iteratively is the lack of a clear commonly accepted residual notion an important matrix function for which this is the case is the matrix exponential suppose the matrix exponential of a given matrix times a given vector has to be computed we develop the approach of druskin greenbaum and knizhnerman siam j sci comput 19 1998 pp 38 54 and interpret the sought after vector as the value of a vector function satisfying the linear system of ordinary differential equations odes whose coefficients form the given matrix the residual is then defined with respect to the initial value problem for this ode system the residual introduced in this way can be seen as a backward error we show how the residual can be computed efficiently within several iterative methods for the matrix exponential this resolves the question of reliable stopping criteria for these methods further we show that the residual concept can be used to construct new residual based iterative methods in particular a variant of the richardson method for the new residual appears to provide an efficient way to restart krylov subspace methods for evaluating the matrix exponential

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

exact solutions for discrete and ultradiscrete modified kdv equations and their relation to box ball systems a new class of solutions is proposed for discrete and ultradiscrete modified kdv equations these are directly related to solutions of the box and ball system with a carrier moreover an extended box and ball system and its exact solutions are discussed

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

block krylov subspace methods for approximating the linear combination of ź functions arising in exponential integrators in this paper we present numerical methods for computing the matrix functions arising in exponential integrators the matrix functions are the linear combination of the form j 0 p j a b j where j is related to exponential function by reducing the corresponding systems of linear odes with polynomial inhomogeneity based on block krylov subspace methods two effective exponentials of low dimension matrix are derived to approximate the objective functions some useful a posteriori error estimates are established the algorithms can be combined with time stepping strategy to satisfy the accuracy requirements under a moderate dimension of the block krylov subspace several numerical experiments are presented to demonstrate the efficiency of the methods

on analysis of projection methods for rational function approximation to the matrix exponential we revisit the analysis of projection methods for rational function approximation to the matrix exponential and present a new result for establishing convergence of the pade plus krylov approximation when a 1

computing the action of the matrix exponential with an application to exponential integrators a new algorithm is developed for computing e ta b where a is an n times n matrix and b is n times n 0 with n 0 ll n the algorithm works for any a its computational cost is dominated by the formation of products of a with n times n 0 matrices and the only input parameter is a backward error tolerance the algorithm can return a single matrix e ta b or a sequence e t ka b on an equally spaced grid of points t k it uses the scaling part of the scaling and squaring method together with a truncated taylor series approximation to the exponential it determines the amount of scaling and the taylor degree using the recent analysis of al mohy and higham siam j matrix anal appl 31 2009 pp 970 989 which provides sharp truncation error bounds expressed in terms of the quantities a k 1 k for a few values of k where the norms are estimated using a matrix norm estimator shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems and analysis of rounding errors and of the conditioning of the problem provides theoretical support experimental comparisons with matlab codes based on krylov subspace chebyshev polynomial and laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy an important application of the algorithm is to exponential integrators for ordinary differential equations it is shown that the sums of the form sum k 0 p varphi k a u k that arise in exponential integrators where the varphi k are related to the exponential function can be expressed in terms of a single exponential of a matrix of dimension n p built by augmenting a with additional rows and columns and the algorithm of this paper can therefore be employed

analysis of projection methods for rational function approximation to the matrix exponential krylov subspace methods for approximating the action of the matrix exponential exp a on a vector v are analyzed with a hermitian and negative semidefinite our approach is based on approximating the exponential with the commonly employed diagonal pade and chebyshev rational functions which yield a system of equations with a polynomial coefficient matrix we derive optimality properties and error bounds for the convergence of a galerkin type approximation and of a computationally feasible and extensively used alternative as complementary results we theoretically justify the use of a popular a posteriori error estimate and we provide upper bounds for the components of the solution vector our theoretical and numerical results show that this methodology may provide an appropriate framework to devise new strategies such as more powerful acceleration schemes

ultradiscretization without positivity we present a new ultradiscretization approach which can be applied to discrete systems the solutions of which are not positive definite this was made possible thanks to an ansatz involving the hyperbolic sine function we apply this new procedure to simple mappings for the linear and homographic mappings we obtain ultradiscrete forms and explicitly construct their solutions two discrete painleve ii equations are also analysed and ultradiscretized we show how to construct the ultradiscrete analogues of their rational and airy type solutions