Exponentially-convergent strategies for defeating the Runge Phenomenon for the approximation of non-periodic functions, part two: Multi-interval polynomial schemes and multidomain Chebyshev interpolation

• Authors: John P. Boyd;Jun Rong Ong
• Venue: Applied Numerical Mathematics
• Year: 2011

Approximating a smooth function from its values f(x"i) at a set of evenly spaced points x"i through P-point polynomial interpolation often fails because of divergence near the endpoints, the ''Runge Phenomenon''. This report shows how to achieve an error that decreases exponentially fast with P by means of polynomial interpolation on N"s subdomains where N"s increases with P. We rigorously prove that in the limit both N"s and M, the degree on each subdomain, increase simultaneously, the approximation error converges proportionally to exp(-constantPlog(P)). Thus, division into ever-shrinking, ever-more-numerous subdomains is guaranteed to defeat the Runge Phenomenon in infinite precision arithmetic. (Numerical ill-conditioning is also discussed, but is not a great difficulty in practice, though not insignificant in theory.) Although a Chebyshev grid on each subdomain is well known to be immune to the Runge Phenomenon, it is still interesting, and the same methodology can be applied as to a uniform grid. When a Chebyshev grid is used on each subdomain, there are two regimes. If c is the distance from the middle of the interval [-1,1] to the nearest singularity of f(x) in the complex plane, then when cN"s@?1, the error is proportional to exp(-cP), independent of the number of subdomains. When cN"s@?1, the rate of convergence slows to exp(-constantPlog(P)), the same as for equispaced interpolation. However, the Chebyshev multidomain error is always smaller than the equispaced multidomain error.