the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

on the finite section method for computing exponentials of doubly infinite skew hermitian matrices abstract computing the exponential of large scale skew hermitian matrices or parts thereof is frequently required in applications in this work we consider the task of extracting finite diagonal blocks from a doubly infinite skew hermitian matrix these matrices usually have unbounded entries which impede the application of many classical techniques from approximation theory we analyze the decay property of matrix exponentials for several classes of banded skew hermitian matrices then finite section methods based on the decay property are presented we use several examples to demonstrate the effectiveness of these methods

on the time fractional schrodinger equation theoretical analysis and numerical solution by matrix mittag leffler functions abstract this paper presents a deep analysis of a time dependent schrodinger equation with fractional time derivative after the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations the main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations possibly with high frequency as it is typical with schrodinger equations and a persisting memory as a consequence of the fractional order moreover an accurate spatial discretization gives rise to systems of large to very large size involving a noteworthy computational complexity by means of a theoretical analysis the exact solution is split into two or three terms depending on the order of the fractional derivative thus to face the numerical computation by different and suitably selected methods direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step by step strategy in conjunction with matrix function for the oscillating term in both cases krylov subspace methods are employed for the computation of matrix functions and convergence results are presented

computation of the exponential of large sparse skew symmetric matrices in this paper we consider methods for evaluating both exp a and exp tau a q 1 where rm exp cdot is the exponential function a is a sparse skew symmetric matrix of large dimension q1 is a given vector and tau is a scaling factor the proposed method is based on two main steps a is factorized into its tridiagonal form h by the well known lanczos iterative process and then exp a is derived making use of an effective schur decomposition of h the procedure takes full advantage of the sparsity of a and of the decay behavior of exp h several applications and numerical tests are also reported

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

approximating the exponential from a lie algebra to a lie group consider a differential equation y a t y y y 0 y0 with y 0 g and a r g g where g is a lie algebra of the matricial lie group g every b g can be mapped to g by the matrix exponential map exp tb with t r most numerical methods for solving ordinary differential equations odes on lie groups are based on the idea of representing the approximation y n of the exact solution y t n t n r by means of exact exponentials of suitable elements of the lie algebra applied to the initial value y 0 this ensures that y n g when the exponential is difficult to compute exactly as is the case when the dimension is large an approximation of exp tb plays an important role in the numerical solution of odes on lie groups in some cases rational or polynomial approximants are unsuitable and we consider alternative techniques whereby exp tb is approximated by a product of simpler exponentials in this paper we present some ideas based on the use of the strang splitting for the approximation of matrix exponentials several cases of g and g are considered in tandem with general theory order conditions are discussed and a number of numerical experiments conclude the paper

nonlinear second order elliptic equations v the dirichlet problem for weingarten hypersurfaces on etudie le probleme de dirichlet pour une fonction u dans un domaine borne de r n a frontiere lisse strictement convexe en un point x de les courbures principales k k 1 k n du graphe x u x satisfont une relation f k 1 k n x 0 ou est une fonction positive lisse donnee sur de plus u doit satisfaire la condition aux limites de dirichlet u 0 sur sous certaines conditions on demontre qu il existe une solution u unique admissible lisse dans

krylov subspace exponential time domain solution of maxwell s equations in photonic crystal modeling the exponential time integration i e time integration which involves the matrix exponential is an attractive tool for time domain modeling involving maxwell s equations however its application in practice often requires a substantial knowledge of numerical linear algebra algorithms such as krylov subspace methods in this note we discuss exponential krylov subspace time integration methods and provide a simple guide on how to use these methods in practice while specifically aiming at nanophotonics applications we intentionally keep the presentation as general as possible and consider full vector maxwell s equations with damping i e with nonzero conductivity terms efficient techniques such as the krylov shift and invert method and residual based stopping criteria are discussed in detail numerical experiments are presented to demonstrate the efficiency of the discussed methods and their mesh independent convergence some of the algorithms described here are available as octave matlab codes from www math utwente nl botchevma expm

hessian and gradient estimates for three dimensional special lagrangian equations with large phase we obtain a priori interior hessian and gradient estimates for special lagrangian equations with phase larger than a critical value in dimension three gradient estimates are also derived for critical and super critical phases in general dimensions

sparse solution of the lyapunov equation for large scale interconnected systems we consider the problem of computing an approximate banded solution of the continuous time lyapunov equation a x x a t p where the coefficient matrices a and p are large symmetric banded matrices the sparsity pattern of a describes the interconnection structure of a large scale interconnected system recently it has been shown that the entries of the solution x are spatially localized or decaying away from a banded pattern we show that the decay of the entries of x is faster if the condition number of a is smaller by exploiting the decay of entries of x we develop two computationally efficient methods for approximating x by a banded matrix for a well conditioned and sparse banded a the computational and memory complexities of the methods scale linearly with the state dimension we perform extensive numerical experiments that confirm this and that demonstrate the effectiveness of the developed methods the methods proposed in this paper can be generalized to sparsity patterns of a and p that are more general than banded matrices the results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large scale riccati equation by a sparse matrix

existence of short time approximations of any polynomial order for the computation of density matrices by path integral methods in this paper i provide significant mathematical evidence in support of the existence of direct short time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials while for theorem 2 which is experimental i only provide a physicist s proof i believe the present development is mathematically sound as a verification i explicitly construct two short time approximations to the density matrix having convergence orders 3 and 4 respectively furthermore in appendix b i derive the convergence constant for the trapezoidal trotter path integral technique the convergence orders and constants are then verified by numerical simulations while the two short time approximations constructed are of sure interest to physicists and chemists involved in monte carlo path integral simulations the present paper is also aimed at the mathematical community who might find the results interesting and worth exploring i conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real time feynman path integral simulations

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

acceleration of contour integration techniques by rational krylov subspace methods we suggest a rational krylov subspace approximation for products of matrix functions and a vector appearing in exponential integrators we consider matrices with a field of values in a sector lying in the left complex half plane the choice of the poles for our method is suggested by a fixed rational approximation based on contour integration along a hyperbola around the sector compared to the fixed approximation our rational krylov subspace method exhibits an accelerated and more stable convergence of order o e c n

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

a residual based error estimate for leja interpolation of matrix functions abstract recent applications in science and engineering require a reliable implementation of the action of the matrix exponential and related functions for possibly large matrices an approximation up to a certain tolerance needs to be obtained in a robust and efficient way in this work we consider leja interpolation for performing this task our focus lies on a new a posteriori error estimate for the method we introduce the notion of a residual based estimate where the residual is obtained from differential equations defining the functions the properties of this new error estimate are investigated and compared to an existing one further a numerical investigation is performed based on test examples originating from spatial discretization of time dependent partial differential equations in two and three space dimensions the experiments show that this new approach is robust for various types of matrices and applications

on the exponential of semi infinite quasi toeplitz matrices let a z sum i in mathbb z a iz i be a complex valued function defined for z 1 such that sum i in mathbb z a i infty define t a t i j i j in mathbb z t i j a j i for i j in mathbb z the semi infinite toeplitz matrix associated with the symbol a z let e e i j i j in mathbb z be a compact operator in ell p with 1 le p le infty a semi infinite matrix of the kind a t a e is said quasi toeplitz qt the problem of the computation of exp a or exp a v with a quasi toeplitz and v a vector arises in many applications we prove that the exponential of a qt matrix a is qt that is exp a t exp a f where f is a compact operator in ell p this property allows the design of an algorithm for computing exp a and exp a v up to any precision the case of families of n times n matrices obtained by truncating infinite qt matrices to finite size is also considered numerical experiments show the effectiveness of this approach

levy ciesielski random series as a useful platform for monte carlo path integral sampling we demonstrate that the l evy ciesielski implementation of lie trotter products enjoys several properties that make it extremely suitable for path integral monte carlo simulations fast computation of paths fast monte carlo sampling and the ability to use different numbers of time slices for the different degrees of freedom commensurate with the quantum effects it is demonstrated that a monte carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all particle or all variable update sampler the sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent in the l evy ciesielski representation the path variables are grouped in a small number of layers with the variables from the same layer being statistically independent the superior performance of the fast sampling algorithm is shown to be a consequence of these observations both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler the l evy ciesielski implementation of path integrals and the fast sampling algorithm

partial retrieval of cad models based on the gradient flows in lie group based on the gradient flows in lie group a partial retrieval approach for cad models is presented in this paper first a representation of the face attributed relational graph arg for a cad model is created from its b rep model and thus partial retrieval is converted to a subgraph matching problem then an optimization method is adopted to solve the matching problem where the optimization variable is the vertex mapping and the objective function is the measurement of compatibility between the mapped vertices and between the mapped edges different from most previously proposed methods a homogeneous transformation matrix is introduced to represent the vertex mapping in subgraph matching whose translational sub matrix gives the vertex selection in the larger graph and whose orthogonal sub matrix presents the vertex permutation for the same sized mapping from the selected vertices to the smaller graph s vertices finally a gradient flow method is developed to search for optimal matching matrix in special euclidean group se n here a penalty approach is used to handle the constraints on the elements of the matching matrix which leads its orthogonal part to be a permutation matrix and its translational part to have different integer elements experimental results show that it is a promising method to support the partial retrieval of cad models

accurate singular values and differential qd algorithms in september 1991 j w demmel and w m kahan were awarded the second siam prize in numerical linear algebra for their paper accurate singular values of bidiagonal matrices 1 referred to as dk hereafter among several valuable results was the observation that the standard bidiagonal qr algorithm used in linpack 2 and in many other svd programs can be simplified when the shift is zero and of greater importance no subtractions occur the last feature permits very small singular values to be found with almost all the accuracy permitted by the data and at no extra cost

the first initial boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the hessian abstract for a class of elliptic hessian operators raised by caffarelli nirenberg spruck the corresponding parabolic monge ampere equation was studied the existence and uniqueness of the admissible solution to the first initial boundary value problem for the equation were established which extended a result of ivochkina ladyzhenskaya

numerical methods for ordinary differential equations on matrix manifolds in recent years differential systems whose solutions evolve on manifolds of matrices have acquired a certain relevance in numerical analysis a classical example of such a differential system is the well known toda flow this paper is a partial survey of numerical methods recently proposed for approximating the solutions of ordinary differential systems evolving on matrix manifolds in particular some results recently obtained by the author jointly with his co workers will be presented we will discuss numerical techniques for isospectral and isodynamical flows where the eigenvalues of the solutions are preserved during the evolution and numerical methods for odes on the orthogonal group or evolving on a more general quadratic group like the symplectic or lorentz group we mention some results for systems evolving on the stiefel manifold and also review results for the numerical solution of odes evolving on the general linear group of matrices

computationally enhanced projection methods for symmetric sylvester and lyapunov matrix equations abstract in the numerical treatment of large scale sylvester and lyapunov equations projection methods require solving a reduced problem to check convergence as the approximation space expands this solution takes an increasing portion of the overall computational effort when data are symmetric we show that the frobenius norm of the residual matrix can be computed at significantly lower cost than with available methods without explicitly solving the reduced problem for certain classes of problems the new residual norm expression combined with a memory reducing device make classical krylov strategies competitive with respect to more recent projection methods numerical experiments illustrate the effectiveness of the new implementation for standard and extended krylov subspace methods

interior gradient estimates for mean curvature equations the interior gradient estimate for the prescribed mean curvature equation has been extensively studied see 9 and the references therein for high order mean curvature equations it has also been obtained in 11 18 in most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion from the view point of partial differential equations such estimate should hold for equations with similar structural conditions in 13 6 the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions different proofs for the gradient estimate for mean curvature equations have been given in 1 3 in this note we show that for curvature and hessian equations the interior gradient estimate can be obtained very easily our proof which is based on suitable choice of auxiliary functions is elementary and avoids geometric computations on the graph of solutions the technique in this note has actually been widely used in literature see e g 9

the exponential map for the conformal group o 2 4 we present a general method to obtain a closed finite formula for the exponential map from the lie algebra to the lie group for the defining representation of orthogonal groups our method is based on the hamilton cayley theorem and some special properties of the generators of the orthogonal group and is also independent of the metric we present an explicit formula for the exponential of generators of the so p q groups with p q 6 in particular dealing with the conformal group so 2 4 which is homomorphic to the su 2 2 group this result is needed in the generalization of u 1 gauge transformations to spin gauge transformations where the exponential plays an essential role we also present some new expressions for the coefficients of the secular equation of a matrix

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

interior gradient estimates for solutions of certain fully nonlinear elliptic equations korevaar has given a new proof for interior gradients estimates for solutions of prescribed mean curvature equations in n j korevaar proc sympos pure math 45 part 2 1986 interior gradient estimates for elliptic weingarten equations are also obtained in n j korevaar ann inst h poincare 43 1987 1 17 in this paper we apply the normal perturbation method introduced in n j korevaar proc sympos pure math 45 part 2 1986 n j korevaar ann inst h poincare 43 1987 1 17 to generalize the result in n j korevaar ann inst h poincare 43 1987 1 17 to more general fully nonlinear elliptic equations

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

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

some existence results for fully nonlinear elliptic equations of monge ampere type we study the existence and uniqueness of solutions of monge ampere type equations this type of equations has been studied extensively by caffarelli nirenberg spruck and many others see 5 through 8 and the references therein we present some existence and uniqueness results for this type of equations on compact riemannian manifolds with non negative sectional curvature we have also generalized some results in 7

efficient computation of the matrix exponential by generalized polar decompositions in this paper we explore the computation of the matrix exponential in a manner that is consistent with lie group structure our point of departure is the method of generalized polar decompositions which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation we develop techniques valid for a range of lie groups the orthogonal group the symplectic group lorentz isotropy and scaling groups however the gpd approach is equally promising in a more general context even when lie group structure is not at issue our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential

gradient estimates for semi linear elliptic equations with prescribed contact angle problem abstract in this paper we find a new auxiliary function and use the maximum principle to give a new proof for the gradient of a solution for semi linear elliptic equations with capillarity type problem

the interior gradient estimate of hessian quotient equations abstract in this paper we establish the interior gradient estimate of k admissible solutions of hessian quotient equations k d 2 u l d 2 u f x u d u with 0 l k n which generalizes the result of 1 7

computing f a b via least squares polynomial approximations given a certain function f various methods have been proposed in the past for addressing the important problem of computing the matrix vector product f a b without explicitly computing the matrix f a such methods were typically developed for a specific function f a common case being that of the exponential this paper discusses a procedure based on least squares polynomials that can in principle be applied to any continuous function f the idea is to start by approximating the function by a spline of a desired accuracy then a particular definition of the function inner product is invoked that facilitates the computation of the least squares polynomial to this spline function since the function is approximated by a polynomial the matrix a is referenced only through a matrix vector multiplication in addition the choice of the inner product makes it possible to avoid numerical integration as an important application we consider the case when f t sqrt t and a is a sparse symmetric positive definite matrix which arises in sampling from a gaussian process distribution the covariance matrix of the distribution is defined by using a covariance function that has a compact support at a very large number of sites that are on a regular or irregular grid we derive error bounds and show extensive numerical results to illustrate the effectiveness of the proposed technique

random series and discrete path integral methods the levy ciesielski implementation we perform a thorough analysis of the relationship between discrete and series representation path integral methods which are the main numerical techniques used in connection with the feynman kac formula first an interpretation of the so called standard discrete path integral methods is derived by direct discretization of the feynman kac formula second we consider a particular random series technique based upon the levy ciesielski representation of the brownian bridge and analyze its main implementations namely the primitive the partial averaging and the reweighted versions it is shown that the n 2 k 1 subsequence of each of these methods can also be interpreted as a discrete path integral method with appropriate short time approximations we therefore establish a direct connection between the discrete and the random series approaches in the end we give sharp estimates on the rates of convergence of the partial averaging and the reweighted levy ciesielski random series approach for sufficiently smooth potentials the asymptotic rates of convergence are found to be o 1 n 2 in agreement with the rates of convergence of the best standard discrete path integral techniques

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

square conservative schemes for a class of evolution equations using lie group methods a new method for constructing square conservative schemes for a class of evolution equations using lie group methods is presented the basic idea is as follows first we discretize the space variable appropriately so that the resulting semidiscrete system of equations can be cast into a system of ordinary differential equations evolving on a sphere second we apply lie group methods to the semidiscrete system and then square conservative schemes can be constructed since the obtained numerical solution evolves on the same sphere both exponential and cayley coordinates are used numerical experiments are also reported

an exact formulation of the time ordered exponential using path sums we present the path sum formulation for the time ordered exponential of a time dependent matrix the path sum formulation gives the time ordered exponential as a branched continued fraction of finite depth and breadth the terms of the path sum have an elementary interpretation as self avoiding walks and self avoiding polygons on a graph our result is based on a representation of the time ordered exponential as the inverse of an operator the mapping of this inverse to sums of walks on a graphs and the algebraic structure of sets of walks we give examples demonstrating our approach we establish a super exponential decay bound for the magnitude of the entries of the time ordered exponential of sparse matrices we give explicit results for matrices with commonly encountered sparse structures

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

error estimates for krylov subspace approximations of matrix exponentials the paper develops a better upper estimate of the error for the krylov subspace approximations to the exponential of a large matrix than that given by gallopoulos and saad 1992 this approach is based on chebyshev series approximation

stopping criteria for rational matrix functions of hermitian and symmetric matrices building upon earlier work by golub meurant strakos and tich we derive new a posteriori error bounds for krylov subspace approximations to f a b the action of a function f of a real symmetric or complex hermitian matrix a on a vector b to this purpose we assume that a rational function in partial fraction expansion form is used to approximate f and the krylov subspace approximations are obtained as linear combinations of galerkin approximations to the individual terms in the partial fraction expansion our error estimates come at very low computational cost in certain important special situations they can be shown to actually be lower bounds of the error our numerical results include experiments with the matrix exponential as used in exponential integrators and with the matrix sign function as used in lattice quantum chromodynamics simulations and demonstrate the accuracy of the estimates the use of our error estimates within acceleration procedures is also discussed

the first initial boundary value problem for hessian equations of parabolic type on riemannian manifolds abstract in this paper we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on riemannian manifolds as usual the establishment of the a priori c 2 estimates is our main part based on these estimates the existence of classical solutions is proved under conditions which are nearly optimal

closed forms for the exponential mapping on matrix lie groups based on putzer s method we present closed forms for the exponential of some infinitesimal generators of lie groups which play an important role in physics and engineering applications these explicit forms are based on the putzer s method we also compare this methodology and results with related work by other authors

computation of matrix functions with deflated restarting a deflated restarting krylov subspace method for approximating a function of a matrix times a vector is proposed in contrast to other krylov subspace methods the performance of the method in this paper is better we further show that the deflating algorithm inherits the superlinear convergence property of its unrestarted counterpart for the entire function and present the results of numerical experiments

interior and boundary gradient estimates for solutions to hessian equations satisfying neumann boundary conditions abstract in this paper we study the a priori gradient estimate for admissible solutions to the neumann problem of fully nonlinear hessian equations on riemannian manifolds we derive the interior gradient estimate for admissible solutions and then we establish the boundary gradient estimate by reducing it to the interior gradient estimate

upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices abstract we give easily computable upper and lower bounds for the inverse elements of finite tridiagonal diagonally dominant matrices and we improve the well known upper bounds due to ostrowski the results are extended to infinite systems the theory is also applied to some earlier results and to the evaluation of bessel functions and mathieu functions by using their recurrence relations

a note on the numerical integration of the kdv equation via isospectral deformations the main purpose of this paper is to test the performance of lie group integrators as applied to a semi discrete version of the kdv equation

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

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