Interpolatory Type Research Articles

Polynomial sequences generated by infinite Hessenberg matrices

Abstract We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

Numerical computation of integral of analytic functions in complex plane

A family of interpolatory type of quadrature rules of degree of precision at least seven has been constructed for the approximate evaluation of contour integrals of analytic functions along a directed line segment in the complex plane. The relative accuracies of rules have been studied in their respective classes and also numerically verified by integrating some standard test integrals.

A generalized Birkhoff–Young quadrature formula

A generalized (4n + 1)-point Birkhoff–Young quadrature of interpolatory type with the maximal degree of precision for numerical integration of analytic functions is derived. An explicit form of the node polynomial of such kind of quadratures is obtained. Special cases and an example are presented.

On the dense divergence of the product quadrature formulas of interpolatory type

This paper concerns divergent aspects regarding interpolatory product quadrature formulas generated by arbitrary projection operators associated to the Banach space C of all real continuous functions defined on the interval [−1,1] of R, in supremum norm. The main result is obtained by using principles of functional analysis and it highlights the phenomenon of double condensation of singularities for the considered quadrature formulas, meaning unbounded divergence on large subsets (in topological sense) belonging to C and to the Banach space of absolute integrable functions on [−1,1].

Radon-Nikodym Approximation in Application to Image Reconstruction

For an image, pixel information can be converted to the moments of some basis 𝑸ₖ, e.g. Fourier-Mellin, Zernike, monomials, etc. Given sufficient number of moments pixel information can be completely recovered, for insufficient number of moments only partial information can be recovered and the image reconstruction is, at best, of interpolatory type. Standard approach is to present interpolated value as a linear combination of basis functions, what is equivalent to least squares expansion. However, recent progress in numerical stability of moments estimation allows image information to be recovered from moments in a completely different manner, applying Radon-Nikodym type of expansion, what gives the result as a ratio of two quadratic forms of basis functions. In contrast with the least squares, the Radon-Nikodym approach has oscillations near the boundaries very much suppressed and does not diverge outside of basis support. While least squares theory operate with vectors 〈𝑓𝑸ₖ〉, Radon-Nikodym theory operates with matrices〈𝑓𝑸ⱼ𝑸ₖ〉, what make the approach much more suitable to image transforms and statistical propertyies estimation.

On generalized biparametric multipoint root finding methods with memory

A general family of biparametric n-point methods with memory for solving nonlinear equations is proposed using an original accelerating procedure with two parameters. This family is based on derivative free classes of n-point methods without memory of interpolatory type and Steffensen-like method with two free parameters. The convergence rate of the presented family is considerably increased by self-accelerating parameters which are calculated in each iteration using information from the current and previous iteration and Newton’s interpolating polynomials with divided differences. The improvement of convergence order is achieved without any additional function evaluations so that the proposed family has a high computational efficiency. Numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed methods.

Erratum to: On Gauss-type quadrature formulas with prescribed nodes anywhere on the real line

This is an erratum to our previous paper in this journal. In the paper we investigated among other things the existence of Gauss-type quadrature formulas with two prescribed nodes and a maximal domain of exactness. At some instances we claimed that the weights in the quadrature formulas are positive, while this only holds under more restrictive conditions than given. These are the corrections that have to be implemented. 1. In the discussion before Theorem 2.15, the case GL3c should be completed with the following additional information. T (b) = fn(b) iff a is a node of the quadrature and T (a) = fn(a) iff b is a node. If all the nodes are in (a, b) the corresponding weights will be positive. If a or b is one of the nodes, then the corresponding weight cannot be guaranteed to be positive. 2. The last sentence of Theorem 2.15 should be changed as follows. In all the cases where the Gauss-Lobatto quadrature rule exist, it must be of interpolatory type and the positivity of the weights for all the nodes inside (a, b) is guaranteed. The positivity of the weight corresponding to a node in one of the endpoints of the interval can not be guaranteed. 3. In Corollary 2.17, the sentence “The weights of the quadrature are positive” should be replaced by the following: The weights corresponding to the nodes different from a if μ(x) = (x− a)μ(x) (respectively different from b if μ(x) = (b− x)μ(x)) are positive, but the weight for a (respectively b) can not be guaranteed to be positive. More details about conditions that guarantee that all the weights are positive are provided in [1].

Some numerical integration methods based on interpolation polynomials

In this paper we have considered the asymptotic expressions for remainder term of quadrature formulas of the interpolator type. We derive some corrected versions of the quadrature formulas of interpolatory type, which provide a better approximation accuracy than the original rules. A method to improve the degree of exactness of the quadrature formulas is also considered. A numerical example of the proposed method is given.

A technique for the numerical solution of initial-value problems based on a class of Birkhoff-type interpolation method

In this paper a class of Birkhoff-type interpolation problems on arbitrary nodal points is studied. The explicit representation (characterization), the uniqueness and the error function are explicitly given. Furthermore, we apply the obtained Birkhoff-type interpolation method to find: (i) the numerical solution of high order initial-value problems (IVPs) and the corresponding error of this approximation, (ii) the approximation of some special functions with their explicit error functions, and (iii) new interpolatory type quadrature formulae of precision degree at least m+n−1 and m+kn−1(m,n,k∈N,n,k≥2). Numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results. The results reveal that the new method is effective, simple and accurate.

On the problem of mutual deviation of certain quadrature sums of interpolation type

We obtained efficient computational formulae for quadrature sums that are optimal with respect to coefficients for arbitrary distribution of knots for certain classes of differentiable functions. Based on this, we found exact values of mutual deviation of interpolatory type quadrature sums.

A generalized Birkhoff–Young–Chebyshev quadrature formula for analytic functions

A generalized N -point Birkhoff–Young quadrature of interpolatory type, with the Chebyshev weight, for numerical integration of analytic functions is considered. The nodes of such a quadrature are characterized by an orthogonality relation. Some special cases of this quadrature formula are derived.

Explicit forms of weighted quadrature rules with geometric nodes

A weighted quadrature rule of interpolatory type is represented as ∫abf(x)w(x)dx=∑k=0nwkf(xk)+Rn+1[f], where w(x) is a weight function, {xk}k=0n are integration nodes, {wk}k=0n are the corresponding weight coefficients, and Rn+1[f] denotes the error term. During the past decades, various kinds of formulae of the above type have been developed. In this paper, we introduce a type of interpolatory quadrature, whose nodes are geometrically distributed as xk=aqk, k=0,1,…,n, and obtain the explicit expressions of the coefficients {wk}k=0n using the q-binomial theorem. We give an error analysis for the introduced formula and finally we illustrate its application with a few numerical examples.

Uniform approximations to Cauchy principal value integrals of oscillatory functions

This paper presents an interpolatory type integration rule for the numerical evaluation of Cauchy principal value integrals of oscillatory integrands ⨍ - 1 1 e i ω x f ( x ) x - τ dx , where - 1 < τ < 1 , for a given smooth function f ( x ) . The proposed method is constructed by interpolating f ( x ) at practical Chebyshev points and subtracting out the singularity. A numerically stable procedure is obtained and the corresponding algorithm can be implemented by fast Fourier transform. The validity of the method has been demonstrated by several numerical experiments.

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.

Construction of aggregation operators for automated decision making via optimal interpolation and global optimization

This paper examines methods of point wise construction of aggregation operators via optimal interpolation. It is shown that several types of application-specific requirements lead to interpolatory type constraints on the aggregation function. These constraints are translated into global optimization problems, which are the focus of this paper. We present several methods of reduction of the number of variables, and formulate suitable numerical algorithms based on Lipschitz optimization.

Accelerating the convergence of the iterative methods of interpolatory type

In this paper we deal with iterative methods of interpolatory type, for solving nonlinear equations in Banach spaces. We show that the convergence order of the iterations may considerably grow if the nodes are properly controlled.

A Generalized Bezoutian Matrix With Respect to a Polynomial Sequence of Interpolatory Type

In this note, a generalized Bezoutian matrix with respect to a polynomial sequence of interpolatory type is introduced. The operator representation relative to a pair of dual bases and the generalized Barnett-type factorization formula are derived. An intertwining relation and a Bezoutian reduction to a block diagonal form by congruence via a generalized Vandermonde matrix are presented. Fujiwara-Hermite and Routh-Hurwitz criteria in terms of this generalized Bezout matrix are obtained.

Two trigonometric quadrature formulae for evaluating hypersingular integrals

AbstractTwo trigonometric quadrature formulae, one of non‐interpolatory type and one of interpolatory type for computing the hypersingular integral ${\int\hskip-0.33cm=}_{-1}^{1} w(\tau)g(\tau)/(\tau-t)^{2} \,{\rm d}\tau$ are developed on the basis of trigonometric quadrature formulae for Cauchy principal value integrals. The formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae. Fast three‐term recurrence relations for evaluating the quadrature weights are derived. Numerical tests are carried out using the current formula. As applications, two simple crack problems are considered. One is a semi‐infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions. It is found that the present method generally gives superior results. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circle

A method is proposed for the computation of the Riesz-Herglotz transform. Numerical experiments show the effectiveness of this method. We study its application to the computation of integrals over the unit circle in the complex plane of analytic functions. This approach leads us to the integration by Taylor polynomials. On the other hand, with the goal of minimizing the quadrature error bound for analytic functions, in the set of quadrature formulas of Hermite interpolatory type, we found that this minimum is attained by the quadrature formula based on the integration of the Taylor polynomial. These two different approaches suggest the effectiveness of this formula. Numerical experiments comparing with other quadrature methods with the same domain of validity, or even greater such as Szego formulas, (traditionally considered as the counterpart of the Gauss formulas for integrals on the unit circle) confirm the superiority of the numerical estimations.

Whitney theorem of interpolatory type for k-monotone functions

One of the main results of this paper is the following Whitney theorem of interpolatory type for k-monotone functions (i.e., functions f such that divided differences f[x 0,…, x k ] are nonnegative for all choices of (k + 1) distinct points x 0,…, x k .