Algebraic Endpoint Singularities Research Articles

On the Convergence Rate of Clenshaw–Curtis Quadrature for Jacobi Weight Applied to Functions with Algebraic Endpoint Singularities

Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for this kind of function. In particular, in this case, the Clenshaw–Curtis quadrature for a new constructed Jacobi weight is exponentially convergent. Numerical examples illustrate the theoretical results.

On the convergence rate of Clenshaw–Curtis quadrature for integrals with algebraic endpoint singularities

In this paper, we are concerned with Clenshaw–Curtis quadrature for integrals with algebraic endpoint singularities. An asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the Chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of Chebyshev polynomials. Numerical examples are provided to confirm our analysis.

Gauss–Jacobi quadratures for weakly, strongly, hyper- and nearly-singular integrals in boundary integral equation methods for domains with sharp edges and corners

We present Gauss–Jacobi quadrature rules in terms of hypergeometric functions for the discretization of weakly singular, strongly singular, hypersingular, and nearly singular integrals that arise in integral equation formulations of potential problems for domains with sharp edges and corners. The rules are tailored to weight functions with algebraic endpoint singularities of a fairly general form, thus allowing one to easily incorporate a wide class of domains into the analysis. Numerical examples illustrate the accuracy and stability of the proposed algorithms; it is shown that the same level of high accuracy can be achieved for any choice of the external variable. The usefulness of the method is exemplified by application to the solution of a singular integral equation that arises in time-harmonic electromagnetic scattering by either closed or open perfectly conducting cylindrical objects with edges and corners, such as polygon cylinders and bent strips. Some practical aspects concerning the role of nearby singularities in achieving a highly accurate solution of singular integral equations are, also, discussed.

Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods

We present n-point Gauss–Gegenbauer quadrature rules for weakly singular, strongly singular, and hypersingular integrals that arise in integral equation formulations of potential problems in domains with edges and corners. The rules are tailored to weight functions with algebraic endpoint singularities related to the geometrical singularities of the domain. Each rule has two different expressions involving Legendre functions and hypergeometric functions, respectively. Numerical examples amply demonstrate the accuracy and stability of the proposed algorithms. Application to the solution of a singular integral equation is exemplified.

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| β , β being real. Depending on the value of β, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g?C ?[a,b], or g?C ?(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x j =a+jh and t?{x 1,?,x n?1}. These asymptotic expansions are based on some recent generalizations of the Euler---Maclaurin expansion due to the author (A. Sidi, Euler---Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which β=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler---Maclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) β , with g(x) as before and β=?1,?3,?5,?, from which suitable quadrature formulas can be obtained. We revisit the case of β=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h?0 ?μ>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.

Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities

In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the divergent finite-range integrals ∫ a b f ( x ) d x \int ^b_af(x)\,dx , where f ∈ C ∞ ( a , b ) f\in C^{\infty }(a,b) but can have arbitrary algebraic singularities at one or both endpoints. We assume that f ( x ) f(x) has asymptotic expansions of the general forms a m p ; f ( x ) ∼ K ( x − a ) − 1 + ∑ s = 0 ∞ c s ( x − a ) γ s as x → a + , a m p ; f ( x ) ∼ L ( b − x ) − 1 + ∑ s = 0 ∞ d s ( b − x ) δ s as x → b − , \begin{align*} &f(x)\sim K\,(x-a)^{-1}+\sum ^{\infty }_{s=0}c_s(x-a)^{\gamma _s} \quad \text {as}\ x\to a+,\\ &f(x)\sim L\,(b-x)^{-1}+\sum ^{\infty }_{s=0}d_s(b-x)^{\delta _s} \quad \text {as}\ x\to b-, \end{align*} where K , L K,L , and c s , d s c_s, d_s , s = 0 , 1 , … , s=0,1,\ldots , are some constants, | K | + | L | ≠ 0 , |K|+|L|\neq 0, and γ s \gamma _s and δ s \delta _s are distinct, arbitrary and, in general, complex, and different from − 1 -1 , and satisfy \[ ℜ γ 0 ≤ ℜ γ 1 ≤ ⋯ , lim s → ∞ ℜ γ s = + ∞ ; ℜ δ 0 ≤ ℜ δ 1 ≤ ⋯ , lim s → ∞ ℜ δ s = + ∞ . \Re \gamma _0\leq \Re \gamma _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \gamma _s=+\infty ;\quad \Re \delta _0\leq \Re \delta _1\leq \cdots , \ \ \lim _{s\to \infty }\Re \delta _s=+\infty . \] Hence the integral ∫ a b f ( x ) d x \int ^b_af(x)\,dx exists in the sense of Hadamard finite part. The results we obtain in this work extend some of the results in [A. Sidi, Numer. Math. 98 (2004), pp. 371–387] that pertain to the cases in which K = L = 0. K=L=0. They are expressed in very simple terms based only on the asymptotic expansions of f ( x ) f(x) as x → a + x\to a+ and x → b − x\to b- . With h = ( b − a ) / n h=(b-a)/n , where n n is a positive integer, one of these results reads h ∑ i = 1 n − 1 f ( a + i h ) ∼ I [ f ] a m p ; + K ( C − log ⁡ h ) + ∑ s = 0 γ s ∉ { 2 , 4 , … } ∞ c s ζ ( − γ s ) h γ s + 1 a m p ; + L ( C − log ⁡ h ) + ∑ s = 0 δ s ∉ { 2 , 4 , … } ∞ d s ζ ( − δ s ) h δ s + 1 as h → 0 , \begin{align*} h\sum ^{n-1}_{i=1}f(a+ih)\sim I[f]&+K\,(C -\log h) + \sum ^{\infty }_{\substack {s=0\\ \gamma _s\not \in \{2,4,\ldots \}}}c_s \zeta (-\gamma _s)\,h^{\gamma _s+1}\\ &+L\,(C -\log h) +\sum ^{\infty }_{\substack {s=0\\ \delta _s\not \in \{2,4,\ldots \}}}d_s\zeta (-\delta _s) h^{\delta _s+1}\quad \text {as $h\to 0$}, \end{align*} where I [ f ] I[f] is the Hadamard finite part of ∫ a b f ( x ) d x \int ^b_af(x)\,dx , C C is Euler’s constant and ζ ( z ) \zeta (z) is the Riemann Zeta function. We illustrate the results with an example.

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.

Whiff of a genome

In this paper, we are concerned with Clenshaw–Curtis quadrature for integrals with algebraic endpoint singularities. An asymptotic error expansion and convergence rate are derived by combining a delicate analysis of the Chebyshev coefficients of functions with algebraic endpoint singularities and the aliasing formula of Chebyshev polynomials. Numerical examples are provided to confirm our analysis.