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

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