Deferred Correction Methods for Initial Value Problems

In this paper we derive high order implicit difference methods for large systems of ODE. The methods are based on the deferred correction principle, yielding accuracy of order p by applying the trapezoidal rule p/2 times in each timestep. Numerical experiments demonstrate the efficiency of the method.This revised version was published online in October 2005 with corrections to the Cover Date.     

Higher-order Newton–Cotes rules with end corrections

We present higher-order quadrature rules with end corrections for general Newton–Cotes quadrature rules. The construction is based on the Euler–Maclaurin formula for the trapezoidal rule. We present examples with 6 well-known Newton–Cotes quadrature rules. We analyze modified end corrected quadrature rules, which consist on a simple modification of the Newton–Cotes quadratures with end corrections. Numerical tests and stability estimates show the superiority of the corrected rules based on the trapezoidal and the midpoint rules.     

Gregory type quadrature based on quadratic nodal spline interpolation

Summary. Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes, and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated, and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants for the Simpson rule. Received July 27, 1998/ Revised version received February 22, 1999 / Published online January 27, 2000     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

Error expansions for multidimensional trapezoidal rules with Sidi transformations

In 1993, Sidi introduced a set of trigonometric transformations x = ψ(t) that improve the effectiveness of the one-dimensional trapezoidal quadrature rule for a finite interval. In this paper, we extend Sidi's approach to product multidimensional quadrature over [0,1] ^N . We establish the Euler–Maclaurin expansion for this rule, both in the case of a regular integrand function f(x) and in the cases when f(x) has homogeneous singularities confined to vertices. This revised version was published online in August 2006 with corrections to the Cover Date.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

High-order quadratures for integral operators with singular kernels

A numerical integration method that has rapid convergence for integrands with known singularities is presented. Based on endpoint corrections to the trapezoidal rule, the quadratures are suited for the discretization of a variety of integral equations encountered in mathematical physics. The quadratures are based on a technique introduced by Rokhlin (1990). The present modification controls the growth of the quadrature weights and permits higher-order rules in practice. Several numerical examples are included.     

Hierarchical-matrix method for a class of diffusion-dominated partial integro-differential equations

A hierarchical matrix approach for solving diffusion-dominated partial integro-differential problems is presented. The corresponding diffusion-dominated differential operator is discretized by a second-order accurate finite-volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion-dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite-volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical-matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.     

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.     

Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule

Novel ideas in harmonic analysis are used to analyze the trapezoidal rule integration for two spheres. Sampling in spherical coordinates links three levels of harmonic analysis. Eigenfunctions of a nonstandard manifold Laplacian descend by restriction, first to a differential graph Laplacian, and then to difference operators. Trapezoidal rule integration with appropriate sampling is exact on eigenspaces of the manifold Laplacian, a fact which leads to trapezoidal rule error estimates on Sobolev-style spaces of functions. Singular functions with accurate trapezoidal rule integrals are identified, and a simplified analysis of smooth function numerical integration is provided.     

An Introduction to Lattice Rules and their Generator Matrices

For the one-dimensional quadrature of a naturally periodic functionover its period, the trapezoidal rule is an excellent choice,its efficiency being predicted theoretically and confirmed inpractice. However, for s-dimensional quadrature over a hypercube,the s-dimensional product trapezoidal rule is not generallycost effective even for naturally periodic functions. The searchfor more effective rules has led first to number theoretic rulesand then more recently to lattice rules. This survey outlinesthe motivation for and present results of this theory. It isparticularly designed to introduce the reader to lattice rules.     

A general approach to counterexamples in numerical analysis

Summary The purpose of this paper is to indicate a unified approach to quantitative negative results in numerical analysis. This is done via a rather general theorem which in fact subsumes our previous quantitative uniform boundedness principles. The proof is based upon a gliding hump method. The general theorem is exemplarily applied to discuss the sharpness of various direct and inverse approximation results, known for the compound trapezoidal rule and for the approximate solution of the heat equation. The treatment outlines a program which may also be worked out for other procedures.Supported by Deutsche Forschungsgemeinschaft Grant No. Ne 171/5-1     

An architecture of fuzzy neural networks for linguistic processing

Fuzzy sets have been used successfully in order to deal with imprecise data, linguistic terms or not well-defined concepts. Recently, considerable effort has been made in the direction of combining the neural network approach with fuzzy sets. In this paper a fuzzy feed-forward neural network, able to process trapezoidal fuzzy sets, has been investigated. Normalized trapezoidal fuzzy sets have been considered. The fuzzy generalized delta rule with different back-propagation algorithms is discussed. The more interesting and characteristic property of the proposed architecture is the ability of each node to process fuzzy sets or linguistic terms, preserving the simplicity of the back-propagation algorithm. Consequently, the resulting architecture is able to cope with problems in which the input parameters and the desired targets are described by linguistic terms. This methodology has the further interesting characteristic of being able to operate at the linguistic level rather than at the numerical level, that is it can work at a higher data abstraction level. An example in computerized electrocardiography will be illustrated in order to test the proposed approach.     

Numerical calculation of incomplete gamma functions by the trapezoidal rule

Summary The trapezoidal rule is applied to the numerical calculation of a known integral representation of the complementary incomplete gamma function (a,x) in the regiona<–1 andx>0. Since this application of the rule is not standard, a careful investigation of the remainder terms using the Euler-Maclaurin formula is carried out. The outcome is a simple numerical procedure for obtaining values of incomplete gamma functions with surprising accuracy in the stated region.This work has been supported by the Ministero della Pubblica Istruzione and the Consiglio Nazionale delle Ricerche     

Comments on “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””

In a recently published paper “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””, Khalil and Hassan pointed out that assertions (3) and (4) of Theorem 3.2 in our previous paper “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets” are not true [2]. Furthermore, they introduced the notions of a generalized trapezoidal interval type-2 fuzzy soft subset and a generalized trapezoidal interval type-2 fuzzy soft equal and used these two notions to correct the flaw in assertions (3) and (4) of Theorem 3.2 in our previous paper. In this paper, we show by a counterexample that Khalil and Hassan's correction is incorrect and provide the modified versions of assertions (3) and (4) of Theorem 3.2, along with a strict proof. In addition, Khalil and Hassan pointed out by a counterexample that assertions (3)–(6) of Theorem 3.5 in our paper are not true and proposed the corrections of those assertions. In this paper, we show that Khalil and Hassan's counterexample and corrections are incorrect and provide a new example to verify the inaccuracies of assertions (3) and (5) of Theorem 3.5 in our paper. Moreover, we offer the modified versions of assertions (3) and (5) of Theorem 3.5 and prove them. Finally, Khalil and Hassan's statement that assertions (4) and (6) of Theorem 3.5 in our previous paper are not true is proven to be incorrect, i.e. assertions (4) and (6) of Theorem 3.5 in our previous paper are true.     

The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals

The composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels. However, this quadrature rule has been less employed for Hadamard finite part integrals due to the fact that its global convergence rate for Hadamard finite part integrals with (p+1)-order singularity is p-order lower than that for the Riemann integrals in general. In this paper, we study the superconvergence of the composite trapezoidal rule for Hadamard finite part integrals with the second-order and the third-order singularity, respectively. We obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points. Numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for Hadamard finite part integrals by noting the superconvergence phenomenon. The work of this author was partially supported by the National Natural Science Foundation of China(No.10271019), a grant from the Research Grants Council of the Hong Kong Special Administractive Region, China (Project No. City 102204) and a grant from the Laboratory of Computational Physics The work of this author was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102204).     

DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods

In this paper, the theoretical convergence rate of the trapezoidal rule combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. It is well known that the DE transformation enables the rule to achieve a much higher rate of convergence than the SE transformation, and the convergence rate has been analyzed and justified theoretically under a proper assumption. Here, it should be emphasized that the assumption is more severe than the one for the SE transformation, and there actually exist some examples such that the trapezoidal rule with the SE transformation achieves its usual rate, whereas the rule with DE does not. Such cases have been observed numerically, but no theoretical analysis has been given thus far. This paper reveals the theoretical rate of convergence in such cases, and it turns out that the DE’s rate is almost the same as, but slightly lower than that of the SE. By using the analysis technique developed here, the theoretical convergence rate of the Sinc approximation with the DE transformation is also given for a class of functions for which the SE transformation is suitable. The result is quite similar to above; the convergence rate in the DE case is slightly lower than in the SE case. Numerical examples which support those two theoretical results are also given.     

Characterization of the speed of convergence of the trapezoidal rule

Summary Our aim is to determine the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity. Excluding or controlling odd functions in some way it is possible to establish a correspondence between the speed of convergence and regularity properties of the function to be integrated. In this way we characterize Sobolev spaces, certain spaces of infinitely differentiable functions, of functions holomorphic in a strip, of entire functions of order greater than 1 and of entire functions of exponential type by the speed of convergence.Dedicated to Professor G. Hämmerlin on the occasion of his 60th birthday     

A numerical approach for solving an extended Fisher-Kolomogrov-Petrovskii-Piskunov equation

In the present paper a numerical method, based on finite differences and spline collocation, is presented for the numerical solution of a generalized Fisher integro-differential equation. A composite weighted trapezoidal rule is manipulated to handle the numerical integrations which results in a closed-form difference scheme. A number of test examples are solved to assess the accuracy of the method. The numerical solutions obtained, indicate that the approach is reliable and yields results compatible with the exact solutions and consistent with other existing numerical methods. Convergence and stability of the scheme have also been discussed.     

Same families of geometric aggregation operators with intuitionistic trapezoidal fuzzy numbers

The aim of this work is to present some cases of aggregation operators with intuitionistic trapezoidal fuzzy numbers and study their desirable properties. First, some operational laws of intuitionistic trapezoidal fuzzy numbers are introduced. Next, based on these operational laws, we develop some geometric aggregation operators for aggregating intuitionistic trapezoidal fuzzy numbers. In particular, we present the intuitionistic trapezoidal fuzzy weighted geometric (ITFWG) operator, the intuitionistic trapezoidal fuzzy ordered weighted geometric (ITFOWG) operator, the induced intuitionistic trapezoidal fuzzy ordered weighted geometric (I-ITFOWG) operator and the intuitionistic trapezoidal fuzzy hybrid geometric (ITFHG) operator. It is worth noting that the aggregated value by using these operators is also an intuitionistic trapezoidal fuzzy value. Then, an approach to multiple attribute group decision making (MAGDM) problems with intuitionistic trapezoidal fuzzy information is developed based on the ITFWG and the ITFHG operators. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.