1,720,980 research outputs found
Computation of infinite integrals involving Bessel functions of arbitrary order by the D-transformation
No full textAbstractThe D-transformation due to the author is an effective extrapolation method for computing infinite oscillatory integrals of various kinds. In this work two new variants of this transformation are designed for computing integrals of the form ∫a∞g(t)Cv(t)dt, where g(x) is a nonoscillatory function and Cv(x) may be an arbitrary linear combination of the Bessel functions of the first and second kinds Jv(x) and Yv(x), of arbitrary real order v. When applied to such integrals, the D-transformation and its new variants are observed to produce very accurate results. It is also seen that their performance is very similar to that of the modified W-transformation due to the author, as extended in a recent work by Lucas and Stone with Cv(x) = Jv(x). The present paper is concluded by stating the relevant convergence and stability results and by appending a numerical example
A new approach to vector-valued rational interpolation
No full textAbstractIn this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where F:C→CN, and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived
A de Montessus type convergence study of a least-squares vector-valued rational interpolation procedure
No full textAbstractIn a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where F:C→CN, were proposed, and some of their algebraic properties were studied. One of these procedures, denoted IMPE, was defined via the solution of a linear least-squares problem. In the present work, we concentrate on IMPE, and study its convergence properties when it is applied to meromorphic functions with simple poles and orthogonal vector residues. We prove de Montessus and Koenig type theorems when the points of interpolation are chosen appropriately
Interpolation by a sum of exponential functions when some exponents are preassigned
No full textAbstractIn a recent paper (A. Sidi, J. Approx. Theory 34 (1982), 194–210) the author has given the solutions to the problems of interpolation at equidistant points and confluent interpolation by a sum of exponential functions when none of the exponents is known. In the present work we generalize these results by specifying some of the exponents. Necessary and sufficient conditions for existence and uniqueness of solutions are given, and the solutions are provided in closed form. The connection of these problems with Padé approximants is exploited to prove a limit result
Extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability
No full textAbstractIn a recent work by the author an extrapolation method, the W-transformation, was developed, by which a large class of oscillatory infinite integrals can be computed very efficiently. The results of this work are extended to a class of divergent oscillatory infinite integrals in the present paper. It is shown in particular that these divergent integrals exist in the sense of Abel summability and that the W-transformation can be applied to them without any modifications. Convergence results are stated and numerical examples given
Further extension of a class of periodizing variable transformations for numerical integration
No full textAbstractClass Sm 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 sinm-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 sinm-transformation was generalized by introducing two integers p and q, instead of the single integer m; we denote this generalization as the sinp,q-transformation here. When p=q=m, the sinp,q-transformation becomes the sinm-transformation. Unlike the sinm-transformation which is symmetric, the sinp,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 Sm by introducing a new class of nonsymmetric variable transformations, which we denote as Sp,q, where p and q can assume arbitrary noninteger values, such that the sinp,q-transformation is a representative of this class and Sm⊂Sm,m. We provide a detailed analysis of the trapezoidal rule approximation following a variable transformation from the class Sp,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 sinp,q-transformation. Finally, we discuss the computation of surface integrals in R3 containing point singularities with the help of class Sp,q transformations
Extension and Completion of Wynn's Theory on Convergence of Columns of the Epsilon Table
No full textAbstractLet {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
Development of iterative techniques and extrapolation methods for Drazin inverse solution of consistent or inconsistent singular linear systems
No full textAbstractConsider the linear system of equations Bx=ƒ, where B is an NxN singular matrix. In an earlier work by the author it was shown that iterative techniques coupled with standard vector extrapolation methods can be used to obtain or approximate a solution of this system when it is consistent. In the present work we expand on that approach to treat the case in which this system is in general inconsistent. Starting with Richardson's iterative method, we develop a family of new iterative techniques and vector extrapolation methods that enable us to obtain or approximate the Drazin inverse solution of this system whether the index of B is 1 or greater than 1. We show that the Drazin inverse solution can be constructed from a finite number of iterations, this number being at most N+2. We also provide detailed convergence analyses of the new iterative techniques and vector extrapolation methods and give their precise rates of convergence
Application of class Sm variable transformations to numerical integration over surfaces of spheres
No full textAbstractClass Sm variable transformations with integer m for finite-range integrals were introduced by the author (Numerical Integration IV, International series of Numerical Mathematics, Basel, 1993, pp. 359–373) about a decade ago. These transformations “periodize” the integrand functions in a way that enables the trapezoidal rule to achieve very high accuracy, especially with even m. In a recent work by the author (Math. Comp. (2005)), these transformations were extended to arbitrary m, and their role in improving the convergence of the trapezoidal rule for different classes of integrands was studied in detail. It was shown that, with m chosen appropriately, exceptionally high accuracy can be achieved by the trapezoidal rule. In the present work, we make use of these transformations in the computation of integrals on surfaces of spheres in conjunction with the product trapezoidal rule. We treat integrands that have point singularities of the single-layer and double-layer types. We propose different approaches and provide full analyses of the errors incurred in each. We show that surprisingly high accuracies can be achieved with suitable values of m. We also illustrate the theoretical results with numerical examples. Finally, we also recall analogous procedures developed in another work by the author (Appl. Math. Comput. (2005)) for regular integrands
DGMRES: A GMRES-type algorithm for Drazin-inverse solution of singular non-symmetric linear systems
No full textAbstractIn a recent work by the author [Linear Algebra Appl. 298 (1999) 99] Krylov subspace methods were derived for Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b, where A∈CN×N is a singular and in general non-hermitian matrix that has arbitrary index. One of these methods, modeled after the Generalized Conjugate Residual method (GCR) and denoted DGCR, is considered in the present work again. It is shown that all of the approximations produced by DGCR exist, and a GMRES like algorithm, denoted DGMRES, for its implementation is derived. Like GMRES, DGMRES too is economical computationally and storagewise
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