an efficient two step iterative method for solving a class of complex symmetric linear systems abstract in this paper a new two step iterative method called the two step parameterized tsp iteration method for a class of complex symmetric linear systems is developed we investigate its convergence conditions and derive the quasi optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the tsp iteration method meanwhile some more practical ways to choose iteration parameters for the tsp iteration method are proposed furthermore comparisons of the tsp iteration method with some existing ones are given which show that the upper bound of the spectral radius of the tsp iteration method is smaller than those of the modified hermitian and skew hermitian splitting mhss the preconditioned mhss pmhss the combination method of real part and imaginary part cri and the parameterized variant of the fixed point iteration adding the asymmetric error pfpae iteration methods proposed recently inexact version of the tsp iteration itsp method and its convergence properties are also presented numerical experiments demonstrate that both tsp and itsp are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems

a practical asymptotical optimal sor method abstract this paper presents a practical asymptotical optimal successive over relaxation sor method for solving the large sparse linear system based on two optimization models asymptotically optimal relaxation factors are given which are computed by solving the low order polynomial equations in each iteration the coefficients of the two polynomials are determined by the residual vector and the coefficient matrix a of the real linear system the numerical examples show that the new methods are more feasible and effective than the classical sor method

on the order of uniprimitive permutation groups one of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group g of degree n where g x an we prove i g i exp 4v n log2 n for the case when g is not doubly transitive this result is best possible apart from an o log n factor in the exponent the best result previously known was i g i el h wielandt a similar estimate for the doubly transitive case follows in a subsequent paper for rank 3 groups with subdegree p n 2 we obtain i gi exp 4 n p log2 n in the proof we develop some new combinatorial properties of coherent configurations the results also have relevance to theoretical estimates on the computational complexity of graph isomorphism testing

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

is the optimal ω best for the sor iteration method abstract the successive overrelaxation sor iterative method for linear systems is well understood if the associated jacobi matrix b is consistently ordered and weakly cyclic of index 2 if in addition b 2 has only nonnegative eigenvalues and if b the spectral radius of b is strictly less than unity then by d m young s classical theorem the optimal relaxation parameter for the sor method is given by b 2 1 1 2 b young derived this result assuming that b 2 0 2 with b is the only information available about the spectrum b 2 of b 2 it is also well known that no polynomial acceleration can improve the asymptotic rate of convergence of the sor scheme if the optimal relaxation parameter has been selected the recent claim by j dancis that a smaller average spectral radius can be achieved by using a polynomial acceleration together with a suboptimal relaxation factor b therefore comes as a surprise a closer look however reveals that this improvement can only be achieved if more profound information on b 2 of the form b 2 0 2 2 with is at hand we show that no polynomial acceleration of the sor method for any real is asymptotically faster than the sor scheme with b under the assumption 08 thereby answering the question in the title of this paper in the affirmative as well as solving an old related conjecture of d m young we also carefully investigate the question of what can be gained from the additional information

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

a time integration algorithm for linear transient analysis based on the reproducing kernel method in this paper a new algorithm based on the reproducing kernel method is presented for linear transient analysis using very large time steps in the conventional time integration methods the required accuracy is always achieved by either decreasing the size of time steps or increasing the order of approximations the proposed method however does not require a restriction on the size of the time step the required approximating functions for integration of governing equations are determined using the concept of reproduction of the solution within each time step a recurrence relationship is obtained through solving the governing equation by the point collocation based reproducing kernel method within the time step in collocation points the values of external excitation can be obtained either through the approximation between the values at the beginning and the end of each time step or assigning the exact values of the external force hence one advantage of this new method is the independency of the sampling rates of external excitation from the size of the time step the proposed method is applied for solving first and second order differential equations and advantages of the method are illustrated through a number of numerical examples

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

quasi chebyshev accelerated iteration methods based on optimization for linear systems abstract in this paper we present a quasi chebyshev accelerated iteration method for solving a system of linear equations compared with the chebyshev semi iterative method the main difference is that the parameter is not obtained by a chebyshev polynomial but by optimization models we prove that the quasi chebyshev accelerated iteration method is unconditionally convergent if the original iteration method is convergent and also discuss the convergence rate finally three numerical examples indicate that our method is more efficient than the chebyshev semi iterative method

the norsett time integration methodology for finite element transient analysis this paper presents a set of methods for time integration of problems arising from finite element semidiscretizations the purpose is to obtain computationally efficient methods which possess higher order accuracy and controllable dissipation in the spurious high modes the methods are developed and analysed by a general collocation methodology which leads to the class of norsett approximants an algorithmic parameter is used to achieve an effective control over numerical dissipation moreover a simple and efficient implementation scheme is presented at each time step algorithms based on p order collocation polynomials require the solution of p sets of linear algebraic equations with the same coefficient matrix in this way a single factorization is needed and no transformations are required to recover the approximate solution at the end or within the time interval to demonstrate the performance of the proposed algorithms a wide experimental evaluation is carried out on typical test problems in finite element transient analysis

a practical formula for computing optimal parameters in the hss iteration methods in the hss iteration methods proposed by bai golub and ng z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 the determination of the optimal parameter is a tough task when solving a non hermitian positive definite linear system in this paper a new and simple strategy for obtaining the optimal parameter is proposed which computes the optimal parameter by solving a cubic polynomial equation the coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew symmetric parts of the coefficient matrix of the real linear system numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the hss iteration methods as it leads to fast convergence of the method

collocation time finite element procedures for integration of strongly nonlinear initial value problems by use of a collocation time finite element formulation a modification of the well known generalized midpoint method is derived the new family possesses an additional free parameter this allows for significant improvement of the numerical solution in strongly nonlinear situations numerical results show the accuracy of the proposed algorithm in comparison to the midpoint method

fast management of permutation groups i we present new algorithms for permutation group manipulation our methods result in an improvement of nearly an order of magnitude in the worst case analysis for the fundamental problems of finding strong generating sets and testing membership the normal structure of the group is brought into play even for such elementary issues an essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group the analysis of the algorithm depends on the classification of finite simple groups in a sequel to this paper using an enhancement of the present method we shall achieve a further order of magnitude improvement

a time stepping procedure for structural dynamics using gauss point collocation when a cubic function is interpolated between the prescribed initial displacement and velocity and the exact displacement and velocity at the end of a time step for a single degree of freedom system the error or residual in the governing equation is zero at a number of times it is shown that for a general undamped system in the limit as the time step approaches zero these times correspond to gauss points this observation is verified by considering a general collocation procedure in which the displacement in any time step is approximated as a cubic function of time with two coefficients chosen to satisfy the displacement and velocity at the beginning of the time step with the other two coefficients being chosen to satisfy the governing differential equation at any two times it is shown that optimum accuracy is obtained if these points are the gauss points detailed expressions are then presented for this particular case and stability of the algorithm is investigated showing that the procedure is conditionally stable for time steps which are a small proportion of the least period of vibration of the structure the algorithm is considered to be the most accurate possible procedure based on cubic approximation of the displacement

unconditionally stable collocation algorithms for second order initial value problems abstract in this paper unconditionally stable higher order accurate time step integration algorithms suitable for second order initial value problems in collocation form are presented the second order equations are manipulated directly if the approximate solution is expressed as a polynomial of degree n 1 there are n unknowns to be determined after taking into account the two given initial conditions it is well known that by suppressing the residuals of the governing equations at n distinct collocation points only the resultant algorithms are only conditionally stable in this paper linear combinations of the residuals at n 1 distinct collocation points are used to solve for the n unknowns the collocation points and the relative weights between the residuals are derived from the weighted residual method the weighting functions are arbitrary polynomials of degree not exceeding n 1 to control the accuracy and stability properties of the resultant algorithms the reduced integration technique is used to evaluate the integrals in the formulation once the reduced integration rules are decided the equivalent collocation form can be derived it is found that the resultant algorithms cast in the collocation form are easy to implement and can be used to tackle non linear problems directly numerical examples are given to illustrate the validity of the present formulation

unconditionally stable higher order accurate collocation time step integration algorithms for first order equations abstract in this paper unconditionally stable higher order accurate time step integration algorithms for linear first order differential equations based on the collocation method are presented the amplification factor at the end of the spectrum is a controllable algorithmic parameter the collocation parameters for unconditionally stable higher order accurate algorithms are found to be given by the roots of a polynomial in terms of the ultimate amplification factor in general when the numerical solution is approximated by a polynomial of degree n this approximation is at least n th order accurate however by using the above collocation parameters the order of accuracy can be improved to 2 n 1 or 2 n the approximate solutions are found to be equivalent to the generalized pade approximations furthermore it is shown that the accuracy of the particular solution due to excitation given by the present method is compatible with the homogeneous solutions no modification of the collocation parameters is required

complex time step newmark methods with controllable numerical dissipation in this paper unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation are presented the algorithms are based on the newmark method with complex time steps the ultimate spectral radius the sub step locations j and the weighting factors j are the algorithmic parameters for an algorithm that is 2n 1 th order accurate the sub step locations which may be complex are shown to be the roots of an nth degree polynomial the polynomial is given explicitly in terms of n and the weighting factors are then obtained by solving a system of n simultaneous equations it is further shown that the order of accuracy is increased by one for the non dissipative algorithms with 1 the stability properties of the present algorithms are studied it is shown that if the ultimate spectral radius is set between 1 and 1 the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity the algorithms are therefore unconditionally c stable when the ultimate spectral radius is set to 0 or 1 the algorithms are found to be equivalent to the first sub diagonal and diagonal pade approximations respectively the present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers the accuracy of the excitation responses is found to be enhanced by the present complex time step procedure to maintain high order accuracy the excitation may need some modifications 1998 john wiley sons ltd

successive overrelaxation sor and related methods abstract covering the last half of the 20th century we present some of the basic and well known results for the sor theory and related methods as well as some that are not as well known most of the earlier results can be found in the excellent books by varga matrix iterative analysis prentice hall englewood cliffs nj 1962 young iterative solution of large linear systems academic press new york 1971 and berman and plemmons nonnegative matrices in the mathematical sciences siam philadelphia pa 1994 while some of the most recent ones are given in the bibliography of this paper in this survey both the point and the block sor methods are considered for the solution of a linear system of the form ax b where a c n n and b c n 0 some general results concerning the sor and related methods are given and also some more specific ones in cases where a happens to possess some further property e g positive definiteness l m h matrix property p cyclic consistently ordered property etc

a transient response analysis in the state space applying the average velocity concept an implicit direct time integration method for obtaining transient responses of general dynamic systems is described the conventional newmark method cannot be directly applied to state space first order differential equations which contain no explicit acceleration terms the method proposed here is the state space newmark method that incorporates the average velocity concept and can be applied to an analysis of general dynamic systems that are expressed by state space first order differential equations it is also readily coded into a program stability and accuracy analyses indicate that the method is numerically unconditionally stable like the conventional newmark method and has a period error of second order accuracy for small damping and fourth order for large damping and an amplitude error of second order regardless of damping in addition its utility and validity are confirmed by two application examples the results suggest that the proposed state space newmark method based on average velocity be generally applied to the analysis of transient responses of general dynamic systems with a high degree of reliability with respect to stability and accuracy