Abstract
Fifteen years ago Meinardus made a conjecture on the degree of the rational approximation of the function ex on the interval [−1,+1]. The conjecture was recently proved via the approximation on the circle |z|=½ in the complex plane. The same method is now applied to the approximation of the square root function. Here we have a gap between the upper and the lower bound, the amount of which depends on the location of the branch point. To close the gap some folklore about Heron's method is collected and completed.
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References
P. Borwein, On a method of Newman and a theorem of Bernstein. J. Approximation Theory 34, 37–41 (1982).
D. Braess, On the conjecture of Meinardus on rational approximation of ex. J. Approximation Theory 36, 317–320 (1982).
G. Meinardus, Approximation of Functions: Theory and Numerical Methods. Springer, New-York-Heidelberg 1967.
D.J. Newman, Approximation with Rational Functions. Regional Conference Series, No.41. Amer.Math.Soc.Providence,Rhode Island 1979.
I. Ninomiya, Best Rational Starting Approximations and Improved Newton Iteration for the Square Root. Math.Comp. 24, 391–407 (1970).
O. Perron, Die Lehre von den Kettenbrüchen II. Teubner, Suttgart 1957.
H. Rutishauser, Betrachtungen zur Quadratwurzeliteration. Monatshefte Math. 67, 452–464 (1963).
E. Saff, On the degree of best rational approximation to the exponential function. J. Approximation Theory 9,97–101 (1973).
L.N. Trefethen, Near circularity of the error curve in complex Chebyshev approximation. J. Approximation Theory 31, 344–367 (1981).
L.N. Trefethen, The asymptotic accuracy of rational best approximation to ez on a disk. J. Approximation Theory 40, 380–383 (1984).
K. Zeller, Newton-Čebyšev-Approximation. In "Iterationsverfahren, Numerische Mathematik, Approximationstheorie" (L. Collatz et al, Eds.) pp. 101–104, Birkhäuser, Basel 1970.
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Braess, D. (1984). On rational approximation of the exponential and the square root function. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072401
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DOI: https://doi.org/10.1007/BFb0072401
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