Adaptive Polynomial Interpolation on Evenly Spaced Meshes

The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches that the oscillations may be contained and that the resulting polynomials are data bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each sub-interval. Computational results for a number of challenging functions including a number of problems similar to Runge’s function with as many as 511 points per interval are shown. keywords Adaptive polynomial interpolation, data bounded polynomials, Runge’s function This report is to appear in SIAM Review 2007. ADAPTIVE POLYNOMIAL INTERPOLATION ON EVENLY SPACED MESHES M. BERZINS ‡ Abstract. The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches that the oscillations may be contained and that the resulting polynomials are data bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each sub-interval. Computational results for a number of challenging functions including a number of problems similar to Runge’s function with as many as 511 points per interval are shown. The problem of oscillatory polynomial interpolants arising from equally spaced mesh points is considered. It is shown that by making use of adaptive approaches that the oscillations may be contained and that the resulting polynomials are data bounded and monotone on each interval. This is achieved at the cost of using a different polynomial on each sub-interval. Computational results for a number of challenging functions including a number of problems similar to Runge’s function with as many as 511 points per interval are shown.