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Extra resources for Approximation Theory: Proceedings of an International Colloquium Held at Bonn, Germany, June 8–11, 1976

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Schoenberg,Academic Press, New York, 1976, 163-201 15. J. Schoenberg, Cardinal interpolation and spline functions, J. Approximation Theory ~ (1969) 167-206 16. , J°Approximation Theory ~ (1972) 404-420 17. Ju. N. Mat. 78 (1965) 2442; Engl. Transl. Soc. (1967) 23-42 Mathematics Research Center 610 Walnut Street Madison, WI. USA 53706 Zur numerischen Stabilitgt des Newton-Verfa~'ens bei der nichtlinearen Tschebyscheff-Approximation Dietrich Braess When the nonlinear approximation problem is treated by using Newton's method, at each iteration step the solution of a linear approximation problem is required.

In einem gro~en Prozentsatz hat man eine extrem langsame Konvergenz, so da~ man vom praktischen Standpunkt nicht mehr von Konvergenz sprechen kann. Die Schwierigkeiten scheinen ~hnlich gelagert zu sein, wie sie Krabs yon dem in [8] geschilderten Verfahren berichtete. Dort wird einpassen- der Tangentenvektor h aus 8berlegungen gewonnen, die sich an das Kolmogorov- Kriterium anlehnen. Die in der lh'axis auftretenden Komplikationen zeigen, da~ das Verfahren numerisch instabil ist. Woran liegt diese Instabilit~t?

20 (1976) 7. " 1-1~ , A bound on the ~ -norm of ~2-approximation by splines in terms of a global mesh ratio, MRC TSR 1597, Aug. 1975; ~ t h . Comp. 30 (1976) 8. J. Schoenberg, Cardinal interpolation and spline funct- ions VIII. The Budan-Fourier theorem for splines and applications, in "Spline functions, Karlsruhe 1975", Springer Lecture Notes in Mathematics 501, Springer, Berlin, 1976, 1-79 9. B. J. Schoenberg, On P61ya frequency functions IV. d'Analyse Math. 1_~7 (1966) 71-i07 10. S. Demko, Inverses of band matrices and local convergence of spline projections, Feb.

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