numerical approximations

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1: Annie A. M. Cuyt
Her main research interest is in the area of numerical approximation theory and its applications to a diversity of problems in scientific computing. …
2: 3.4 Differentiation
With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals $e^{k}$ at the dominant points $\theta=0,2\pi$, and in combination with the factor $k^{-k}$ in front of the integral sign this gives a rough approximation to $1/k!$. …
3: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(i) Numerical Use of Asymptotic Expansions
The rest of this section is devoted to general methods for increasing this accuracy. …
§2.11(vi) Direct Numerical Transformations
For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.
4: Bibliography W
• H. Werner, J. Stoer, and W. Bommas (1967) Rational Chebyshev approximation. Numer. Math. 10 (4), pp. 289–306.
• R. Wong (1995) Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.
• 5: 28.8 Asymptotic Expansions for Large $q$
Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). … Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). …
6: Bibliography G
• W. Gautschi (2004) Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation, Oxford University Press, New York.
• 7: Mathematical Introduction
All of the special function chapters contain sections that describe available methods for computing the main functions in the chapter, and most also provide references to numerical tables of, and approximations for, these functions. … In referring to the numerical tables and approximations we use notation typified by $x=0(.05)1$, 8D or 8S. …
For small values of $\|\mathbf{T}\|$ the zonal polynomial expansion given by (35.8.1) can be summed numerically. For large $\|\mathbf{T}\|$ the asymptotic approximations referred to in §35.7(iv) are available. Other methods include numerical quadrature applied to double and multiple integral representations. … Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …