gmres for the differentiation operator we investigate using the gmres method with the differentiation operator this operator is unbounded and thus does not fall into the framework of existing krylov subspace theory we establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane these conditions are used to determine when gmres converges this algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals even when the kernel of oscillations is unknown

projection methods isotone projection cones and the complementarity problem one studies some iterative methods for solving complementarity problems explicit or implicit in hilbert spaces the iterative methods presented are based on the fixed point theory on the coincidence equations on convex cones and on some properties of projection operators onto a special cone named the isotone projection cone the ordering defined by the cone is essential for the complementarity theory the idea is to use systematically the ordering in the iterative methods for solving complementarity problems in this sense one shows that the isotone projection cone is a good instrument

on uniform approximations to hypersingular finite part integrals abstract in this paper new uniform approximation schemes for computation of hypersingular finite part integrals are studied the methods are verified to be supremely qualified for oscillatory integrands in addition based on the results on chebyshev approximations the uniform convergence and computational error bounds are considered preliminary numerical results show the stability and efficiency of the schemes

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

filon clenshaw curtis rules for a class of highly oscillatory integrals with logarithmic singularities in this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities for these quadrature rules we derive error estimates in terms of n the number of nodes k the rate of oscillations and a sobolev like regularity of the function we prove that the method is not only robust but the error even decreases for fixed n as k increases practical issues about the implementation of the rule are also covered in this paper by a writing down ready to implement algorithms b analysing the numerical stability of the computations and c estimating the overall computational cost we finish by showing some numerical experiments which illustrate the theoretical results presented in this paper

finding solutions of implicit complementarity problems by isotonicity of the metric projection abstract isac and nemeth g isac and a b nemeth projection method isotone projection cones and the complementarity problem j math anal app 153 258 275 1990 proved that solving a coincidence point equation fixed point problem in turn solves the corresponding implicit complementarity problem nonlinear complementarity problem and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems nonlinear complementarity problems defined by these cones in this paper the notion of isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on isotone projection cones it is proved that if the sequence generated through the defined algorithm is convergent then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones the question of finding nonzero solutions of these problems is also studied

a well conditioned levin method for calculation of highly oscillatory integrals and its application abstract this paper is devoted to studying efficient calculation of generalized fourier transform 1 x f t e i g t d t for the general phase function g t we develop a modified levin method by the spectral coefficient approach a sparse and well conditioned linear system is constructed to help accelerate calculation of highly oscillatory integrals numerical examples are included to show the convergence properties of the new method with respect to both quantities of collocation points and the frequency furthermore we apply this approach to solving oscillatory volterra integral equations

entropy on abelian groups abstract we introduce the algebraic entropy for endomorphisms of arbitrary abelian groups appropriately modifying existing notions of entropy the basic properties of the algebraic entropy are given as well as various examples the main result of this paper is the addition theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups we give several applications of the addition theorem among them the uniqueness theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms furthermore we point out the delicate connection of the algebraic entropy with the mahler measure and lehmer problem in number theory

computation of integrals with oscillatory singular factors of algebraic and logarithmic type in this paper we present the clenshaw curtis filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration the methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in n 1 clenshaw curtis points then the required modified moments can be accurately and efficiently computed by constructing some recurrence relations moreover for these quadrature rules their absolute errors in inverse powers of the frequency are given the presented methods share the advantageous property that the accuracy improves greatly for fixed n as increases numerical examples show the accuracy and efficiency of the proposed methods

on the convergence of filon quadrature we analyze the convergence behavior of filon type quadrature rules by making explicit the dependence on both k the parameter that controls the oscillatory behavior of the integrand and n the number of function evaluations we provide explicit conditions on the domain of analyticity of the integrand to ensure convergence for n

clenshaw curtis type quadrature rule for hypersingular integrals with highly oscillatory kernels abstract the clenshaw curtis type quadrature rule is proposed for the numerical evaluation of the hypersingular integrals with highly oscillatory kernels and weak singularities at the end points for any smooth functions g x based on the fast hermite interpolation this paper provides a stable recurrence relation for these modified moments convergence rates with respect to the frequency k and the number of interpolation points n are considered these theoretical results and high accuracy of the presented algorithm are illustrated by some numerical examples

a new projection method for variational inequality problems we propose a new projection algorithm for solving the variational inequality problem where the underlying function is continuous and satisfies a certain generalized monotonicity assumption e g it can be pseudomonotone the method is simple and admits a nice geometric interpretation it consists of two steps first we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem this procedure requires a single projection onto the feasible set and employs an armijo type linesearch along a feasible direction then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set thus in contrast with most other projection type methods only two projection operations per iteration are needed the method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions preliminary computational experience is also reported

using b spline frames to represent solutions of acoustics scattering problems abstract although frames which are a generalization of bases are important tools used in signal processing their potential in other fields of engineering and applied mathematics e g acoustics has not been fully explored yet gabor frames that are specific type of frames are very well adapted to oscillating functions and therefore have a great potential to efficiently represent functions in connection with the helmholtz operator in this paper representations of the solution of a scattering problem in 2d using gabor frames based on b splines as building blocks are investigated practical issues concerning the implementation of frames like the restriction of the frame elements to a finite interval and methods to determine the unknown coefficients for the representation i e by using dual frames or by solving a least squares problem are discussed numerical experiments are made comparing the different ways to determine the unknown expansion coefficients and the representations are compared in terms of their efficiency i e the number of used frame coefficients versus the accuracy of the approximation in all cases the coefficients calculated with a slightly modified orthogonal matching pursuit algorithm provide the best accuracy versus sparsity ratio

filon clenshaw curtis rules for highly oscillatory integrals with algebraic singularities and stationary points in this paper we propose and analyze composite filon clenshaw curtis quadrature rules for integrals of the form i k a b f g int a b f x exp mathrm i kg x mathrm d x where k geq 0 f may have integrable singularities and g may have stationary points our composite rule is defined on a mesh with m subintervals and requires mn 1 evaluations of f it satisfies an error estimate of the form c n k r m n 1 r where r is determined by the strength of any singularity in f and the order of any stationary points in g and c n is a constant which is independent of k and m but depends on n the regularity requirements on f and g are explicit in the error estimates for fixed k the rate of convergence of the rule as m rightarrow infty is the same as would be obtained if f was smooth moreover the quadrature error decays at least as fast as k rightarrow infty as does the original integral i k a b f g for the case of nonlinear oscill

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

efficient methods for highly oscillatory integrals with weakly singular and hypersingular kernels abstract in this paper we present a special clenshaw curtis filon ccf type scheme for approximation of highly oscillatory integrals with weak and hypersingular kernels the non oscillatory and nonsingular part of the integrand is replaced by a special hermite interpolation polynomial error bounds with respect to the frequency k and the number of the clenshaw curtis points n are considered the overall computational complexity for the scheme is o nlog n numerical experiments support the theoretical analysis

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

an efficient quadrature for 2 5d boundary element calculations abstract in recent years the boundary element method has become a widely used tool for calculating the mitigation effects of noise barriers however since for large structures calculations in 3d become very inefficient most of the standard implementations are only in 2d this means that the noise source is implicitly assumed to be given by a coherent line source which is not realistic in most cases by using a fourier transform with respect to a spatial coordinate along the length of the structure it is possible to reduce the 3d problem to several 2d problems with distinct wavenumbers which allows the simulation of more realistic noise sources and which is typically referred to as 2 5d bem to that end it is necessary to numerically calculate a fourier like integral over all the 2d solutions in this work an efficient way to calculate this integral is given building on existing approaches using clenshaw curtis filon quadrature and demodulation combined with an adaptive order selection scheme as bem calculations are costly the main focus of the method introduced lies on avoiding too many of these calculations the efficiency of the method is illustrated using two different examples a reflecting cylinder and an l shaped noise barrier

efficient filon type methods for int_a bf x rm e rm i omega g x rm d x based on the transformation y g x some new efficient filon type methods for integration of highly oscillatory function int a bf x rm e rm i omega g x rm d x with an irregular oscillator are presented one is a moment free filon type method for the case that g x has no stationary points in a b the others are based on the filon type method or the asymptotic method together with filon type method for the case that g x has stationary points the effectiveness and accuracy are tested by numerical examples

oscillation free solutions to volterra integral and integro differential equations with periodic force terms volterra equations governed by oscillatory functions arise frequently in applied fields to construct efficient algorithms for solving these equations oscillatory properties of their solutions should be studied in this paper oscillatory orders of solutions to a class of highly oscillatory volterra integral equations are presented then some modified solutions are given with the help of asymptotic expansions of highly oscillatory integrals finally theoretical analysis verifies these modified solutions enjoy less oscillation than the original one and extensions to highly oscillatory volterra integro differential equations are considered

regularizing the abstract convex program abstract characterizations of optimality for the abstract convex program inf p x g x s x p where s is an arbitrary convex cone in a finite dimensional space is a convex set and p and g are respectively convex and s convex on were given in 10 these characterizations hold without any constraint qualification they use the minimal cone s f of p and the cone of directions of constancy d g s f in the faithfully convex case these cones can be used to regularize p i e transform p into an equivalent program p r for which slater s condition holds we present an algorithm that finds both s f and d g s f the main step of the algorithm consists in solving a particular complementarity problem we also present a characterization of optimality for p in terms of the cone of directions of constancy of a convex functional d g rather than d g s f

iterative methods for nonlinear complementarity problems on isotone projection cones in this paper we present a recursion related to a nonlinear complementarity problem defined by a closed convex cone in a hilbert space and a continuous mapping defined on the cone if the recursion is convergent then its limit is a solution of the nonlinear complementarity problem in the case of isotone projection cones sufficient conditions are given for the mapping so that the recursion to be convergent