Practical Extrapolation Methods

Recent Progress on Univariate and Multivariate Polynomial and Spline Quasi-interpolants

“…[14] and [82]). Numerical experiments done by the author show that Richardson extrapolation is efficient while the use of variants of epsilon or ∆ 2 algorithms often introduce spurious poles in the interval of definition.…”

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

“…In developing methods for convergence acceleration, and numerical integration of singular integrands, the second author introduced some classes of polynomials of this type [15], [16], [17], [18]. These include polynomials P n that correspond to the following choices of exponents: (I) n;j = j + , 1 j n, some > 1; (II) 1 n;j n j=1 are the zeros of Sidi polynomials, which will be discussed in Example 1 of Section 2; (III) 1 n;j n j=1 are the zeros of Legendre polynomials scaled to (0; 1).…”

“…In much of the research of recent decades, the second sequence of polynomials is replaced by a sequence of functions that need not be polynomials at all. For example, the Sidi polynomials D (0;0) n are determined by the biorthogonality relation Their properties and generalizations have been studied in [9], [10], [11], [15], [18]. An elegant and general theory of biorthogonal polynomials was developed by Iserles and Norsett [5].…”

“…In developing methods for convergence acceleration, and numerical integration of singular integrands, the second author introduced [15], [18] the Sidi polynomials…”

A Generalized Technique in Numerical Integration

“…Traditional quadrature rules and summation techniques have failed to provide accurate approximations to such integrals. Numerous methods and techniques were developed for improving convergence of these challenging integrals [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and extremely efficient methods were introduced such as numerical steepest descent, Filon-type and Levin-type methods. Unfortunately, their application to complicated integrals is extremely challenging.…”