exponentially fitted runge kutta methods abstract exponentially fitted runge kutta methods with s stages are constructed which exactly integrate differential initial value problems whose solutions are linear combinations of functions of the form x j exp x x j exp x r or i r j 0 1 j max where 0 j max s 2 1 the lower bound being related to explicit methods the upper bound applicable for collocation methods explicit methods with s 2 3 4 belonging to that class are constructed for these methods a study of the local truncation error is made out of which follows a simple heuristic to estimate the value error and step length control is introduced based on richardson extrapolation ideas some numerical experiments show the efficiency of the introduced methods it is shown that the same techniques can be applied to construct implicit exponentially fitted runge kutta methods

parallel hybrid algebraic multilevel iterative methods abstract for the large scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self adjoint elliptic boundary value problem by making use of the blocked multilevel iteration idea we construct preconditioning matrices for the coefficient matrix and set up a class of parallel hybrid algebraic multilevel iterative methods for solving this kind of system of linear equations theoretical analysis shows that not only do these new methods lend themselves to parallel computation but also their convergence rates are independent of both the sizes and the level numbers of the grids and their computational work loads are also bounded by linear functions of the step sizes of the finest grids

the numerical solution of equality constrained quadratic programming problems this paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem the constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints among this class of convergent iterative schemes are generalizations of the relaxed jacobi gauss seidel and symmetric gauss seidel schemes

knot polynomials from q deformed algebras we present explicit polynomials of two knot invariants obtained from q deformed algebras braid group representations can be obtained from the r matrices which in turn arise in q deformed algebras a markov trace can be defined for r matrices based on representations of the q deformed algebras and hence knot polynomials can be defined in this paper the properties of coupling coefficients and r matrices based on each of the and representations for are used to calculate polynomials for knots of ten or fewer crossings we develop a new method to calculate the polynomials for the representation of there are five pairs of knots of ten or fewer crossings which have the same polynomial the exception is where n 2 in this case the polynomial is equivalent to the jones polynomial and has 14 pairs for knots of ten or fewer crossings the polynomial has four pairs for these knots each pair is different to the pairs thus the polynomial has slightly fewer pairs than the polynomial and is significantly better at predicting the amphichirality or non amphichirality of knots

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

racah wigner algebra for q deformed algebras the concepts of vector coupling coefficients and recoupling coefficients are generalized to q deformed algebras their properties under complex conjugation and permutation of irreps are derived relations between the coefficients for example the racah backcoupling rule are proved it is shown that these properties may be used to recursively calculate the coupling and recoupling coefficients for all q algebras the 3 jm and 6 j symbols for su 2 q are used to illustrate the building up method

a class of new hybrid algebraic multilevel preconditioning methods abstract a class of new hybrid algebraic multilevel preconditioning methods is presented for solving the large sparse systems of linear equations with symmetric positive definite coefficient matrices resulting from the discretization of many second order elliptic boundary value problems by the finite element method the new preconditioners are shown to be of optimal orders of complexities for two dimensional and three dimensional problem domains and their relative condition numbers are estimated to be bounded uniformly independent of the numbers of both the levels and the nodes

an optimized runge kutta method for the solution of orbital problems we present a new explicit runge kutta method with algebraic order four minimum error of the fifth algebraic order the limit of the error is zero when the step size tends to zero infinite order of dispersion and eighth order of dissipation the efficiency of the newly constructed method is shown through the numerical results of a wide range of methods when these are applied to well known periodic orbital problems

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

on the choice of the frequency in trigonometrically fitted methods for periodic problems in this paper the use of a trigonometrically fitted method to obtain the approximate solutions of some nonlinear periodic oscillators is presented a great number of different approaches have been considered to obtain analytical approximations for this kind of problems a generalized decomposition method gdm a linearized harmonic balance procedure lhb the homotopy perturbation method hpm the harmonic balanced method hbm the adomian decomposition method etc from those approaches analytical approximations to the frequency of oscillation and periodic solutions are obtained which are valid for a large range of amplitudes of oscillation however these techniques have been limited to obtain only one or two iterates because of the great amount of algebra involved we use a trigonometrically adapted method to obtain numerical approximations to the solutions yielding very acceptable results on the basis that the approximation of the frequency of the method is done with great accuracy there are a lot of trigonometrically fitted methods in the literature but there is not a definite way to obtain the optimal value of the frequency we present a strategy for the choice of the parameter value in the adapted method based on the minimization of the total energy some examples solved numerically confirm the good performance of the adopted strategy

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

exponentially fitted two step hybrid methods for y f x y it is the purpose of this paper to derive two step hybrid methods for y f x y with oscillatory or periodic solutions specially tuned to the behaviour of the solution through the usage of the exponential fitting technique the construction of two step exponentially fitted hybrid methods is shown and their properties are discussed some numerical experiments confirming the theoretical expectations are provided

variable stepsize chebyshev type methods for the integration of second order i v p s panovsky and richardson a family of implicit chebyshev methods for the numerical integration of second order differential equations j comput appl math 23 1988 35 51 presented a method based on chebyshev approximations for numerically solving the problem y f x y being the steplength constant coleman and booth analysis of a family of chebyshev methods for y f x y j comput appl math 44 1992 95 114 made an analysis of the above method and suggested the convenience to design a variable steplength implementation as far as we know this goal has not been achieved until now later on we extended the above method this journal 2003 and obtained a scheme for numerically solving the equation y 2gy g 2 w 2 f x y the question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper

dissipative chebyshev exponential fitted methods for numerical solution of second order differential equations a family of implicit methods based on intra step chebyshev interpolation has been developed to integrate oscillatory second order initial value problems of the form y t 2g y t g2 w2 y t f t y t the procedure integrates the homogeneous part exactly in the absence of round off errors the chebyshev approach uses stepsizes that are considerably larger than those typically used in runge kutta or multistep methods computational overheads are comparable to those incurred by high order conventional procedures chebyshev interpolation coupled with the exponential fitted nature of the method substantially reduces local errors global error propagation rates are also reduced making these procedures good candidates to be used in long term simulations of perturbed oscillatory systems with a dissipative term

exponentially fitted explicit runge kutta nystrom methods exponentially fitted runge kutta nystrom efrkn methods for the numerical integration of second order ivps with oscillatory solutions are derived these methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions exp t exp t c or equivalently sin t cos t when i r explicit efrkn methods with two and three stages and algebraic orders 3 and 4 are constructed in addition a 4 3 embedded pair of explicit efrkn methods based on the fsal technique is obtained which permits to introduce an error and step length control with a small cost added some numerical experiments show the efficiency of our explicit efrkn methods when they are compared with other exponential fitting type codes proposed in the scientific literature

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

a 3d rectangular mixed finite element method to solve the stationary semiconductor equations a mixed finite element method using rectangular elements is presented to solve the drift diffusion model of semiconductors the method which is on the one hand a classical primal mixed finite element method generalizes the one dimensional scharfetter gummel scheme to three dimensions in a natural way the major contribution to a successful implementation of the method is the development of a fast and exact numerical evaluation of exponential fitted integrals to which considerable attention has been devoted the essentials of the efficient computation of these integrals are described with full details to be presented elsewhere in order to use computer resources parsimoniously with the help of lagrange multipliers admissible bases on irregular rectangular meshes are constructed which allow a local element refinement to validate the method the modeling of a three dimensional magneto transistor is presented

scheifele two step methods for perturbed oscillators two step methods specially adapted to the numerical integration of perturbed oscillators are obtained the formulation of the methods is based on a refinement of classical taylor expansions due to scheifele g scheifele on the numerical integration of perturbed linear oscillating systems z angew math phys 22 1971 186 210 the key property is that those algorithms are able to integrate exactly harmonic oscillators with frequency w the methods depend on a parameter email protected where h is the stepsize based on the b2 series theory of coleman j p coleman order conditions for a class of two step methods for y f x y ima j numer anal 23 2003 197 220 we derive the order conditions of this new type of method the linear stability and phase properties are examined the theory is illustrated with some fourth and fifth order explicit schemes numerical results carried out on an assortment of test problems such as the integration of the orbital motion of earth satellites show the relevance of the theory

relations for clebsch gordan and racah coefficients in suq 2 and yang baxter equations relations are exploited among clebsch gordan cg and racah coefficients in the algebra suq 2 known as a deformation of su 2 these are used to show that the yang baxter yb relation for the irf interaction round a face model results from one of the symmetry relations for the 9 j symbol specific to suq 2 and that in an asymptotic limit this yb relation becomes the yb relation for the two dimensional vertex model the racah coefficient which has a particularly simple dependence on q is efficiently used such that an asymptotic limit of the racah coefficient is the cg coefficient and another limit gives the factorized s matrix of the vertex model

a new approach on the construction of trigonometrically fitted two step hybrid methods the construction of trigonometrically fitted two step hybrid methods for the numerical solution of second order initial value problems is considered these methods are suitable for the numerical integration of problems with periodic or oscillatory behavior of the solution and have variable coefficients depending on the frequency of each problem the modification of classical two step hybrid methods is done by inserting extra parameters at each stage we derive the coefficients of the modified methods for the general case of s stages as examples we present the modifications of three methods of algebraic orders five six and seven

trigonometrically fitted multi step rkn methods for second order oscillatory initial value problems abstract trigonometrically fitted multi step runge kutta nystrom tfmsrkn methods for solving numerically oscillatory special second order initial value problems are introduced tfmsrkn methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set exp i w t exp i w t or equivalently the set cos wt sin wt where w represents an approximation of the main frequency of the problem the corresponding order conditions are given and two explicit four stage tfmsrkn methods with order five are constructed stability of the new methods is examined and the regions of stability are depicted numerical results show that our new methods are more efficient in comparison with other well known high quality methods proposed in the scientific literature

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

the block preconditioned lsqr and gl lsqr algorithms for the block partitioned matrices in this paper a new block preconditioner is proposed for the block partitioned matrices this preconditioner is based on the block c orthogonalization where c is a symmetric positive definite matrix the block preconditioned least squares bpls and block preconditioned global least squares bpgls algorithms are presented to solve the linear system of equations with block partioned coefficient matrix and multiple linear system of equations respectively the bpls algorithm is applied to solve the complex linear system of equations and also bpls and classical preconditioned least squares pls algorithms are compared finally some numerical experiments are given to show the efficiency of the new block preconditioner

on generalized parameterized inexact uzawa method for a block two by two linear system recently chen and jiang f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 presented a parameterized inexact uzawa piu algorithm for solving symmetric saddle point problems where the 1 2 and the 2 1 blocks are the transpose of each other in this paper we extend the piu method to the block two by two linear system by allowing the 1 2 block to be not equal to the transpose of the 2 1 block and the 2 2 block may not be zero we prove that the iteration method is convergent under certain conditions with different choices of the parameter matrices we obtain several new algorithms for solving the block two by two linear system numerical experiments confirm our theoretical results and show that our method is feasible and effective

analysis of iterative methods for saddle point problems a unified approach in this paper two classes of iterative methods for saddle point problems are considered inexact uzawa algorithms and a class of methods with symmetric preconditioners in both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems the obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature a few numerical tests are given which confirm the sharpness of the estimates

diagonally implicit runge kutta nystro m methods for oscillatory problems implicit runge kutta nystrom rkn methods are constructed for the integration of second order differential equations possessing an oscillatory solution based on a linear homogeneous test model we analyse the phase errors or dispersion introduced by these methods and derive so called dispersion relations diagonally implicit rkn methods of relatively low algebraic order are constructed which have a high order of dispersion up to 10 application of these methods to a number of test examples linear as well as nonlinear yields a greatly reduced phase error when compared with conventional dirkn methods

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

quantum group invariants and link polynomials a general method is developed for constructing quantum group invariants and determining their eigenvalues applied to the universal matrix this method leads to the construction of a closed formula for link polynomials to illustrate the application of this formula the quantum groups uq es uq so 2m 1 and uq gl m are considered as examples and corresponding link polynomials are obtained

recursive calculation of the r matrices of q deformed algebras the authors show that the pentagonal equation relating r matrices and vector coupling coefficients can be used to calculate recursively general r matrices from those containing the primitive irrep all r matrices for su 2 q are calculated by this method as an illustration of the more general case the primitive vector coupling coefficients for su 3 q are obtained and used to derive formulae for the r matrices with a certain restriction on the representations

the nonlinear inexact uzawa hybrid algorithms based on one step newton method for solving nonlinear saddle point problems this paper mainly consider the construction of efficient uzawa type algorithms for solving the nonlinear saddle point problem nspp two nonlinear inexact uzawa hybrid algorithms based on one step newton method are proposed for solving the nspps by using the so called energy norm the convergence of the proposed algorithms is verified under some practical conditions in addition some numerical results are reported to illustrate the behavior of the considered algorithm