The Carathéodory–Fejér Method for Real Rational Approximation
Lloyd N. Trefethen, Martin H. Gutknecht · SIAM Journal on Numerical Analysis · 1983
A “Carathéodory–Fejér method” is presented for near-best real rational approximation on intervals, based on the eigenvalue (or singular value) analysis of a Hankel matrix of Chebyshev coefficients. In approximation of a smooth function F, the CF approximant $R^{cf} $ frequently differs from the best approximation $R^ * $ by only one part in millions or billions. To account for this we show here under weak assumptions that if F is approximated on $[ - \varepsilon ,\varepsilon ]$, then as $\varepsilon \to 0$, $||F - R^ * || = O(\varepsilon ^{m + n + 1} )$ while $||R^{cf} - R^ * || = O(\varepsilon ^{3m + 2n + 3} )$. In contrast, the latter figure would be $O(\varepsilon ^{m + n + 2} )$ for the Chebyshev economization approximant of Maehly or the Chebyshev–Padé approximant of Gragg. It follows that as $\varepsilon \to 0$, best approximation error curves approach the real parts of $m + n + 1$-winding rational functions of constant modulus to within $O(\varepsilon ^{3m + 2n + 3} )$. Numerical examples are given, including applications to $e^x $ on $[ - 1,1]$ and $e^{ - x} $ on $[0,\infty )$. For the latter problem we conjecture that the errors in $(n,n)$ approximation decrease with each n by a ratio approaching a fixed constant $9.28903 \cdots $.