Numerical integration over the square in the presence of algebraic/logarithmic singularities with an application to aerodynamics

Tensor-product formulae based on one-dimensional Gaussian quadratures are developed for evaluating double integrals of the type indicated in the title. If the singularities occur only along the diagonal and the regular part of the integrand is a polynomial of total degree d, the formulae can be made exact by choosing the number of quadrature points larger than, or equal to, 1?+?d/2. Numerical examples are given as well as an application to a problem in aerodynamics.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

Gauss-Green cubature and moment computation over arbitrary geometries

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n−1 using N∼cmn² nodes, 1≤c≤p, m being the total number of points given on the boundary. It does not need any decomposition of the domain, but relies directly on univariate Gauss-Legendre quadrature via Green’s integral formula. Several numerical tests are presented, including computation of standard as well as orthogonal moments over a nonstandard planar region.     

Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal <Emphasis Type="Italic">B</Emphasis>-spline

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ _m as the weight functions on [0, m] (m ∈ ?) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.     

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→∞.     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.     

An analytical formula for pricing m-th to default swaps

The computational effort of pricing an m-th to default swap depends highly on the size n of the underlying basket. Usually, n different default times are modeled, but in fact the valuation only depends on the m-th smallest default time of this tuple. In this paper we attain an analytical formula for the distribution of this m-th default time. With the help of this distribution we simplify the valuation problem from an n-dimensional quadrature to a one-dimensional quadrature and break the curse of dimensionality. Applications of this modification are efficient pricing of m-th to default swaps, estimation of sensitivities and pricing of European max/min options.     

Gauss-Hermite quadratures for functions from Hilbert spaces with Gaussian reproducing kernels

We study univariate integration with the Gaussian weight for a positive variance α. This is done for the reproducing kernel Hilbert space with the Gaussian kernel for a positive shape parameter γ. We study Gauss-Hermite quadratures, although this choice of quadratures may be questionable since polynomials do not belong to this space of functions. Nevertheless, we provide the explicit formula for the error of the Gauss-Hermite quadrature using n function values. In particular, for 2αγ ²<1 we have an exponential rate of convergence, and for 2αγ ²=1 we have no convergence, whereas for 2αγ ²>1 we have an exponential divergence.     

Further properties of inner product quadrature formulas

An inner product quadrature formula is of the form $$\int_{ - 1}^1 {w(x)f(x)g(x)dx \cong \sum\limits_{i = 0}^m {\sum\limits_{j = 0}^n {f(x_i )a_{ij} g(y_j ) = f^T Ag.} } } $$ Conditions are established for which these quadrature formulas are exact whenf andg are polynomials of degree not greater thanm+k andn?l (also, analogouslym?k andn+l) respectively. The structure and properties of the matrixA are also considered.     

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szeg?-Radau and Szeg?-Lobatto quadrature formulas on the unit circle. These are (n+m)-point formulas for which m nodes are fixed in advance, with m=1 and m=2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. This means that the free parameters (free nodes and positive weights) are chosen such that the quadrature formula is exact for all powers z^j, −p≤j≤p, with p=p(n,m) as large as possible.     

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Given an ensemble of N×N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N→∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a “dial” we can “turn” from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f _m show a visually stunning convergence to the semi-circle as m→∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f _m is the product of a Gaussian and a certain even polynomial of degree 2m?2; the formula is the same as that for the m×m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending not only on the frequency at which each element appears, but also on the way the elements are arranged.     

Gaussian quadrature of Chebyshev polynomials

We investigate the behaviour of the maximum error in applying Gaussian quadrature to the Chebyshev polynomials T_m. This quantity has applications in determining error bounds for Gaussian quadrature of analytic functions.     

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m ² nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m ? 1.     

Best quadrature formulas for evaluating line integrals of the first kind for certain classes of functions and curves determined by moduli of continuity

The extremal problem of minimizing the error of approximate evaluation of a line integral of the first kind is considered for certain classes of functions and spatial curves determined by moduli of continuity.It is proved that if the endpoints of the interval [0, L] (where L is the length of the curve along which the integration is performed) are not included in the set of nodes of a quadrature formula for evaluating the line integral of the first kind, then the best quadrature formula for the classes m^(p) _ρ of functions and \({H^{{\omega _1}, \ldots ,{\omega _m}}}\) of curves is the midpoint rectangle formula. If the extreme points x = 0 and x = L of the interval are included in the set of nodes of a quadrature formula for approximately evaluating the line integral (such formulas are said to be Markov-type), then, for these classes, the best formula is the trapezoidal rule. Sharp error estimates for all considered classed of functions and curves are calculated and a generalization to more general classes is given.     

On a conjecture of D. B. Hunter

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare.     

Product type quadrature formulas

This paper is concerned with the numerical approximation of integrals of the form _a ^b f(x)g(x)dx by means of a product type quadrature formula. In such a formula the functionf (x) is sampled at a set ofn+1 distinct points and the functiong(x) at a (possibly different) set ofm+1 distinct points. These formulas are a generalization of the classical (regular) numerical integration rules. A number of basic results for such formulas are stated and proved. The concept of a symmetric quadrature formula is defined and the connection between such rules and regular quadrature formulas is discussed. Expressions for the error term are developed. These are applied to a specific example.The work of the first author was supported in part by NIH Grant No. FRO 7129-01 and that of the second author in part by U.S. Army Ballistic Research Laboratories Contract DA-18-001-AMC-876 X.     

Product type multiple integration formulas

An integration formula of the type $$\int_a^b {f(x)g(x)dx \cong \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {a_{ij} f(xi)g(y_j ),} } } $$ referred to as a product quadrature, was first considered by R. Boland and C. Duris. In this paper, the author extends the concept of a product formula to multiple integrals. The definitions and some of the results for interpolatory, compound, and symmetric product quadratures have an analog for product cubatures and these are given.     

Quadrature formulae on half-infinite intervals

We develop two classes of quadrature rules for integrals extended over the positive real axis, assuming given algebraic behavior of the integrand at the origin and at infinity. Both rules are expressible in terms of Gauss-Jacobi quadratures. Numerical examples are given comparing these rules among themselves and with recently developed quadrature formulae based on Bernstein-type operators.Work supported, in part, by the National Science Foundation under grant CCR-8704404.     

Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space $L_{2}^{(m)}(0,1)$

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P _m?1(x), where P _m?1(x) is a polynomial of degree m?1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.