A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon

  • Authors:

  • Jae-Hun Jung;Wolfgang Stefan

  • Affiliations:

  • Department of Mathematics, The State University of New York at Buffalo, Buffalo, USA 14260-2900;Department of Mathematics, The State University of New York at Buffalo, Buffalo, USA 14260-2900 and Department of Computational and Applied Mathematics, Rice University, Houston, USA 77251-1892

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2011

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Abstract

A polynomial interpolation based on a uniform grid yields the well-known Runge phenomenon, where maximum error is unbounded for functions with complex roots in the Runge zone. In this paper, we investigate the Runge phenomenon with the finite precision operation. We first show that the maximum error is bounded because of round-off errors inherent to the finite precision operation. Then we propose a simple truncation method based on the truncated singular value decomposition. The method consists of two stages: In the first stage, a new interpolating matrix is constructed using the assumption that the function is analytic. The new interpolating matrix is preconditioned using the statistical filter method. In the second stage, a truncation procedure is applied such that singular values of the new interpolating matrix are truncated if they are equal to or lower than a certain tolerance level. We generalize the method, by analyzing the singular vectors of both the original and new interpolation matrices based on the assumption in the first stage. We show that the structure of the singular vectors makes the first stage essential for an accurate reconstruction of the original function. Numerical examples show that exponential decay of the error can be achieved if an appropriate truncation is chosen.