Mathematics – Numerical Analysis
Scientific paper
2009-10-09
Found. Comput. Math. 11 (2011) 1-63
Mathematics
Numerical Analysis
Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Fo
Scientific paper
10.1007/s10208-010-9075-z
High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.
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