1,720,983 research outputs found
Computation of infinite integrals involving Bessel functions of arbitrary order by the D -transformation
No full textAbstract The /9-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 f,,~ ,q(t)cC(t)dt, where g(x) i
Analysis of convergence of the 𝑇-transformation for power series
No full textRecently the present author has given some convergence theorems of general nature for Levin’s nonlinear sequence transformations. In this work these theorems are extended and sharpened to cover the case of power series, both inside and on their circle of convergence. It is shown that one of the two limiting processes considered in the previous work can be used for analytic continuation and a realistic estimate of its rate of convergence is given. Three illustrative examples are also appended.</p
Convergence analysis for a generalized Richardson extrapolation process with an application to the 𝑑⁽¹⁾-transformation on convergent and divergent logarithmic sequences
No full textIn an earlier work by the author the Generalized Richardson Extrapolation Process (GREP) was introduced and some of its convergence and stability properties were discussed. In a more recent work by the author a special case of GREP, which we now call
GREP
(
1
)
{\text {GREP}^{(1)}}
, was considered and its properties were reviewed with emphasis on oscillatory sequences. In the first part of the present work we give a detailed convergence and stability analysis of
GREP
(
1
)
{\text {GREP}^{(1)}}
as it applies to a large class of logarithmic sequences, both convergent and divergent. In particular, we prove several theorems concerning the columns and the diagonals of the corresponding extrapolation table. These theorems are very realistic in the sense that they explain the remarkable efficiency of
GREP
(
1
)
{\text {GREP}^{(1)}}
in a very precise manner. In the second part we apply this analysis to the Levin-Sidi
d
(
1
)
{d^{(1)}}
-transformation, as the latter is used with a new strategy to accelerate the convergence of infinite series that converge logarithmically, or to sum the divergent extensions of such series. This is made possible by the observation that, when the proper analogy is drawn, the
d
(
1
)
{d^{(1)}}
-transformation is, in fact, a
GREP
(
1
)
{\text {GREP}^{(1)}}
. We append numerical examples that demonstrate the theory.</p
The numerical evaluation of very oscillatory infinite integrals by extrapolation
No full textRecently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.</p
A user-friendly extrapolation method for oscillatory infinite integrals
No full textIn a recent publication [4] the author developed an extrapolation method, the W-transformation, for the accurate computation of convergent oscillatory infinite integrals. In yet another publication [6] this method was shown to be applicable to divergent oscillatory infinite integrals that are defined in the sense of summability. The application of the W-transformation involves some asymptotic analysis of the integrand as the variable of integration tends to infinity. In the present work the W-transformation is modified so as to keep this asymptotic analysis to a minimum, involving only the phase of oscillations. This modified version, which turns out to be as efficient as the original W-transformation, can also be applied to convergent or divergent oscillatory infinite integrals other than those dealt with in [4] and [6]. The convergence properties of the modified transformation are analyzed in detail for the integrals of [4] and [6], and numerical examples are provided.</p
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
Numerical quadrature rules for some infinite range integrals
No full textRecently the present author has given a new approach to numerical quadrature and derived new numerical quadrature formulas for finite range integrals with algebraic and/or logarithmic endpoint singularities. In the present work this approach is used to derive new numerical quadrature formulas for integrals of the form
∫
0
∞
x
α
e
−
x
f
(
x
)
d
x
\smallint _0^\infty {x^\alpha }{e^{ - x}}f(x)\,dx
and
∫
0
∞
x
α
E
p
(
x
)
f
(
x
)
d
x
\smallint _0^\infty {x^\alpha }{E_p}(x)f(x)\,dx
, where
E
p
(
x
)
{E_p}(x)
is the exponential integral. It turns out the new rules are of interpolatory type, their abscissas are distinct and lie in the interval of integration and their weights, at least numerically, are positive. For fixed
α
\alpha
the new integration rules have the same set of abscissas for all p. Finally, the new rules seem to be at least as efficient as the corresponding Gaussian quadrature formulas. As an extension of the above, numerical quadrature formulas for integrals of the form
∫
−
∞
+
∞
|
x
|
β
e
−
x
2
f
(
x
)
d
x
\smallint _{ - \infty }^{ + \infty }|x{|^\beta }{e^{ - {x^2}}}f(x)\,dx
too are considered.</p
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
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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