Trapezoidal Rule Approximation Research Articles

Model-independent Reconstruction of the Cosmological Scale Factor as a Function of Lookback Time

Abstract We present a model-independent method of reconstructing scale factor against lookback time from the Observational Hubble parameter Data (OHD). The reconstruction method is independent of dynamical models and is only based on the Friedmann–Robertson–Walker metric. We also calculate the propagation of error in the reconstruction process. The reconstruction data errors mainly come from trapezoidal rule approximation and the uncertainty from OHD. Furthermore, the model discrimination ability of original OHD and reconstructed a–t data is discussed under a dimensionless standard method. The a–t data can present the differences between cosmology models more clearly than H–z data by comparing their coefficients of variations. Finally, we add 50 simulated H(z) data to estimate the influence of future observation. More Hubble measurements in the future will help constrain cosmological parameters more accurately.

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

AbstractA 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.

Computation of the Complex Error Function Using Modified Trapezoidal Rules

In this paper we propose a method for computing the Faddeeva function $w(z) := {e}^{-z^2}{erfc}(-{i}\,z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our s...

A unified direct approach for discretizing fractional-order differentiator in delta-domain

Fractional-order differentiator is a principal component of the fractional-order controller. Discretization of fractional-order differentiator is essential to implement the fractional-order controller digitally. Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the [Formula: see text]-domain as evident from the existing literature. In this paper, a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional [Formula: see text]-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency. The discretization of fractional-order differentiator is accomplished in two stages. In the first stage, the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage, the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method. The proposed method has been compared with two well-known direct discretization methods taken from the existing literature. Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.

Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one

In this paper, we study a fully discrete finite element method with second order accuracy in time for the equations of motion arising in the Oldroyd model of viscoelastic fluids. This method is based on a finite element approximation for the space discretization and the Crank-Nicolson/Adams-Bashforth scheme for the time discretization. The integral term is discretized by the trapezoidal rule to match with the second order accuracy in time. It leads to a linear system with a constant matrix and thus greatly increases the computational efficiency. Taking the nonnegativity of the quadrature rule and the technique of variable substitution for the trapezoidal rule approximation, we prove that this fully discrete finite element method is almost unconditionally stable and convergent. Furthermore, by the negative norm technique, we derive the $H^1$ and $L^2$-optimal error estimates of the velocity and the pressure.

The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities

This paper is aimed at deriving the error asymptotic expansion of trapezoidal rule approximation to integrals for the functions with fractional derivatives or algebraic singularities at some points and from which to design a modified Romberg extrapolation algorithm to effectively compute some singular integrals. Firstly, high-order local fractional derivatives are defined, then a general fractional Taylor's expansion is derived. Secondly, the error asymptotic expansion of trapezoidal rule for these integrals is obtained directly by using the formula of sums of non-integer powers and the general fractional Taylor's expansion. This method is different from the previous work. Thirdly, a modified Romberg extrapolation algorithm is designed to get numerical results efficiently. Finally, numerical examples are presented to verify the correctness of the theoretical analysis and the effectiveness of the extrapolation method.

Numerical Algorithms Based on Analytic Function Values at Roots of Unity

Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. Finally we highlight the power of rational in comparison with polynomial approximations for some of these problems.

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals \(I[f]=\int^{b}_{a}f(x)\,dx\), where f∈C ∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms Open image in new window where \(\widehat{P}(y),P_{s}(y)\) and \(\widehat{Q}(y),Q_{s}(y)\) are polynomials in y. The γ s and δ s are distinct, complex in general, and different from −1. They also satisfy Open image in new window The results we obtain in this work extend the results of a recent paper [A. Sidi, Numer. Math. 98:371–387, 2004], which pertain to the cases in which \(\widehat{P}(y)\equiv0\) and \(\widehat{Q}(y)\equiv0\). They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let \(D_{\omega}=\frac{d}{d\omega}\), h=(b−a)/n, where n is a positive integer, and define \(\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)\). Then with \(\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}\) and \(\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}\), one of these results reads Open image in new window where ζ(z) is the Riemann Zeta function and σ i are Stieltjes constants defined via \(\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]\), i=0,1,… .

Variable transformations in numerical integration

AbstractWe discuss some recent developments in the use of class Sm variable transformations for improving the accuracy of the trapezoidal rule approximations in numerical integration via the trapezoidal rule. These concern optimal choices of m. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Further extension of a class of periodizing variable transformations for numerical integration

Class S m variable transformations with integer m , for accurate numerical computation of finite-range integrals via the trapezoidal rule, were introduced and studied by the author. A representative of this class is the sin m -transformation. In a recent work of the author, this class was extended to arbitrary noninteger values of m , and it was shown that exceptionally high accuracies are achieved by the trapezoidal rule in different circumstances with suitable values of m . In another recent work by Monegato and Scuderi, the sin m -transformation was generalized by introducing two integers p and q , instead of the single integer m ; we denote this generalization as the sin p , q -transformation here. When p = q = m , the sin p , q -transformation becomes the sin m -transformation. Unlike the sin m -transformation which is symmetric, the sin p , q -transformation is not symmetric when p ≠ q , and this offers an advantage when the behavior of the integrand at one endpoint is quite different from that at the other endpoint. In view of the developments above, in the present work, we generalize the class S m by introducing a new class of nonsymmetric variable transformations, which we denote as S p , q , where p and q can assume arbitrary noninteger values, such that the sin p , q -transformation is a representative of this class and S m ⊂ S m , m . We provide a detailed analysis of the trapezoidal rule approximation following a variable transformation from the class S p , q , and show that, with suitable and not necessarily integer p and q , it achieves an unusually high accuracy when the integrand has algebraic endpoint singularities. We also illustrate our results with numerical examples via the sin p , q -transformation. Finally, we discuss the computation of surface integrals in R 3 containing point singularities with the help of class S p , q transformations.

Extension of a class of periodizing variable transformations for numerical Integration

ClassSm{\mathcal S}_mvariable transformations with integermm, for numerical computation of finite-range integrals, were introduced and studied by the author in the paper [A. Sidi, A new variable transformation for numerical integration,Numerical Integration IV,1993 (H. Brass and G. Hämmerlin, eds.), pp. 359–373.] A representative of this class is thesinm\sin ^m-transformation that has been used with lattice rules for multidimensional integration. These transformations “periodize” the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with evenmm. In the present work, we extend these transformations toarbitraryvalues ofmm, and give a detailed analysis of the resulting transformed trapezoidal rule approximations. We show that, with suitablemm, they can be very useful in different situations. We prove, for example, that if the integrand function is smooth on the interval of integration and vanishes at the endpoints, then results of especially high accuracy are obtained by taking2m2mto be an odd integer. Such a situation can be realized in general by subtracting from the integrand the linear interpolant at the endpoints of the interval of integration. We also illustrate some of the results with numerical examples via the extendedsinm\sin ^m-transformation.

Euler–Maclaurin expansions for integrals with endpoint singularities: a new perspective

In this note, we provide a new perspective on Euler–Maclaurin expansions of (offset) trapezoidal rule approximations of the finite-range integrals I[f]=∫baf(x),dx, where f ∈ C∞(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms * where Ps(y) and Qs(y) are some polynomials in y. Here the γs and δs are complex in general and different from −1,−2,... . The results we obtain in this work generalize, and include as special cases, those pertaining to the known special cases in which f(x)=(x−a)γ[ log (x−a)]pga(x)=(b−x)δ[log (b−x)]qgb(x), where p and q are nonnegative integers and ga ∈ C∞[a,b) and gb ∈ C∞(a,b]. In addition, they have the pleasant feature that they are expressed in very simple terms based only on the asymptotic expansions of f(x) as x→a+ and x→b−. With h=(b−a)/n, where n is a positive integer, and with * one of these results reads, as h→0,* where ζ(z) is the Riemann Zeta function.

Error Analysis of a Derivative-Free Algorithm for Computing Zeros of Holomorphic Functions

We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69-91, 1999) for computing all the zeros of a holomorphic function that lie inside the unit circle. The algorithm uses only the function values and no (first or higher order) derivatives. Information about the location of the zeros is obtained from certain integrals along the unit circle. In numerical computations these are replaced by their trapezoidal rule approximations. We investigate the resulting quadrature error. Our error analysis shows that the zeros located inside the unit circle do not affect the accuracy of the computed approximations whereas the quadrature error related to the zeros located outside the unit circle tends to zero exponentially as the number of quadrature points tends to infinity.

Computation of dynamic adsorption with adaptive integral, finite difference, and finite element methods

Analysis of diffusion-controlled adsorption and surface tension in one-dimensional planar coordinates with a finite diffusion length and a nonlinear isotherm, such as the Langmuir or Frumkin isotherm, requires numerical solution of the governing equations. This paper presents three numerical methods for solving this problem. First, the often-used integral (I) method with the trapezoidal rule approximation is improved by implementing a technique for error estimation and choosing time-step sizes adaptively. Next, an improved finite difference (FD) method and a new finite element (FE) method are developed. Both methods incorporate (a) an algorithm for generating spatially stretched grids and (b) a predictor–corrector method with adaptive time integration. The analytical solution of the problem for a linear dynamic isotherm (Henry isotherm) is used to validate the numerical solutions. Solutions for the Langmuir and Frumkin isotherms obtained using the I, FD, and FE methods are compared with regard to accuracy and efficiency. The results show that to attain the same accuracy, the FE method is the most efficient of the three methods used.

An implementation of a Fourier series method for the numerical inversion of the Laplace transform

Our method is based on the numerical evaluation of the integral which occurs in the Riemann Inversion formula. The trapezoidal rule approximation to this integral reduces to a Fourier series. We analyze the corresponding discretization error and demonstrate how this expression can be used in the development of an automatic routine , one in which the user needs to specify only the required accuracy.

Study on radiation in 3-D irregular systems using the trapezoidal rule approximation on the transport equation

In this study, the integral of the radiative source function appearing in the apparent solution of the Radiative Transfer Equation (RTE) is approximated by the two-point trapezoidal rule which is different from the Taylor series expansion approximation used in the Finite Volume Method (FVM) or the discrete ordinates interpolation method. The resulting equation derived from the trapezoidal rule approximation has much simpler form than that obtained from the existing Taylor series approximation. The approximate equation of transfer by the trapezoidal rule is applied for the discrete ordinates interpolation method using nonorthogonal grid systems to predict the radiative heat transfer in 3-D enclosures filled with a gray, absorbing, emitting and scattering medium. The upstream intensity and the source function required for analyses are determined by a linear interpolation on a diagonally placed triangular plane that is reported to be very simple for radiative transfer analyses in three dimensional irregular systems. Numerical results indicate that good accuracy is obtained by using the trapezoidal rule which showed fairly good agreement with the results from the zone method or the original discrete ordinates interpolation method both of which are considered to be more accurate as compared to the conventional S-N discrete ordinate method and the FVM. The trapezoidal rule proposed in this study is successful for nonorthogonal grid systems and it can be used for analyses of the radiative transfer in three dimensional irregular enclosures.

A discrete collocation method for Symm's integral equation on curves with corners

This paper is devoted to the approximate solution of the classical first-kind boundary integral equation with logarithmic kernel (Symm's equation) on a closed polygonal boundary in ℝ 2. We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions. The convergence results are illustrated with some numerical examples.

Extension and Completion of Wynn's Theory on Convergence of Columns of the Epsilon Table

Let {Sn}∞n=0be such thatSn∽S+∑∞j=1ajλnjasn→∞, with 1>|λ1|>|λ2|>…, such that limj→∞λj=0. A well-known result by Wynn states that when the Shanks transformation or its equivalentε-algorithm is applied to {Sn}∞n=0, thenε(n)2k−S∽ak+1[∏ki=1(λk+1−λi)/(1−λi)]2λnk+1asn→∞. In the present work we extend this result (i) by allowing some of theλjto have the same modulus and (ii) by replacing the constantsajby some polynomialsPj(n) inn. Sequences {Sn}∞n=0with these characteristics arise frequently, e.g., in fixed point iterative solution of linear systems and in trapezoidal rule approximation of finite range integrals with logarithmic endpoint singularities and their multidimensional analogues. The results of this work are obtained by exploiting the connection between the Shanks transformation and Padé approximants and by using some recent results of the author on Padé approximants for meromorphic functions.

Further results on convergence and stability of a generalization of the Richardson extrapolation process

In an earlier paper by the author a detailed convergence and stability analysis of a generalization of the Richardson extrapolation process was given under certain conditions. In the present work these conditions are modified and relaxed considerably, and results are obtained on convergence and stability under the new conditions. As the previous ones, these new results are asymptotic in nature, and contain the former. The conditions of the present paper are naturally satisfied, e.g., by the trapezoidal rule approximations of finite range integrals of functions having algebraic and logarithmic end point singularities.

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.