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The accurate solution of certain continuous problems using only single precision arithmetic (1985)

Jankowski, M. ; Woźniakowski, H.

Springer

BIT 25 (1985), S. 635-651

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ISSN: 1572-9125

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Abstract A typical approach for finding the approximate solution of a continuous problem is through discretization with meshsizeh such that the truncation error goes to zero withh. The discretization problem is solved in floating point arithmetic. Rounding-errors spoil the theoretical convergence and the error may even tend to infinity. In this paper we present algorithms of moderate cost which use only single precision and which compute the approximate solution of the integration and elliptic equation problems with full accuracy. These algorithms are based on the modified Gill-Møller algorithm for summation of very many terms, iterative refinement of a linear system with a special algorithm for the computation of residuals in single precision and on a property of floating point subtraction of nearby numbers.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01936142

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Digitale Medien

Is Gauss quadrature optimal for analytic functions? (1985)

Kowalski, M. A. ; Werschulz, A. G. ; Woźniakowski, H.

Springer

Numerische Mathematik 47 (1985), S. 89-98

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ISSN: 0945-3245

Schlagwort(e): AMS(MOS): 65D30 ; CR: G1.4

Quelle: Springer Online Journal Archives 1860-2000

Thema: Mathematik

Notizen: Summary We consider the problem of optimal quadratures for integrandsf: [−1,1]→ℝ which have an analytic extension $$\bar f$$ to an open diskD r of radiusr about the origin such that $$\left| {\bar f} \right|$$ ≦1 on $$\bar D_r $$ . Ifr=1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degreen rather than at optimal points, tends to infinity withn. In particular there is an “infinite” penalty for using Gauss quadrature. On the other hand, ifr〉1, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1007/BF01389877

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