Approximating a function from its values f(x_(i))at a set of evenly spaced points x_(i)through(N+1)-point polynomial interpolation often fails because of divergence near the endpoints,the“Runge Phenomenon”.Here we b...Approximating a function from its values f(x_(i))at a set of evenly spaced points x_(i)through(N+1)-point polynomial interpolation often fails because of divergence near the endpoints,the“Runge Phenomenon”.Here we briefly describe seven strategies,each employing a single polynomial over the entire interval,to wholly or partially defeat the Runge Phenomenon such that the error decreases exponentially fast with N.Each is successful in obtaining high accuracy for Runge’s original example.Unfortunately,each of these single-interval strategies also has liabilities including,depending on the method,various permutations of inefficiency,ill-conditioning and a lack of theory.Even so,the Fourier Extension and Gaussian RBF methods are worthy of further development.展开更多
基金This work was supported by NSF grants OCE0451951 and ATM0723440by the Un-dergraduate Research Opportunities Program(UROP)of the University of Michigan.We thank Martin Berzins for sending us a preprint and Rodrigo Platte for sharing his un-published work.
文摘Approximating a function from its values f(x_(i))at a set of evenly spaced points x_(i)through(N+1)-point polynomial interpolation often fails because of divergence near the endpoints,the“Runge Phenomenon”.Here we briefly describe seven strategies,each employing a single polynomial over the entire interval,to wholly or partially defeat the Runge Phenomenon such that the error decreases exponentially fast with N.Each is successful in obtaining high accuracy for Runge’s original example.Unfortunately,each of these single-interval strategies also has liabilities including,depending on the method,various permutations of inefficiency,ill-conditioning and a lack of theory.Even so,the Fourier Extension and Gaussian RBF methods are worthy of further development.
