On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon

In this note, we use the Gaussian elimination for the inversion of the transformation matrix W between Fourier and Legendre basis functions, Eq. (6). We show that the exponential growth of the L∞ errors when the round-off errors become dominant with the inverse polynomial reconstruction method is due to the ill-posedness of the transformation matrix W and consequently due to the numerical calculation of W-1. We show that the algebraically decaying Fourier coefficients over(f, ̂)k are mapped to hk by the Gaussian upper triangularization procedure which exhibit an exponential decay rate. The hk are mapped into over(g, ̃)l by the inversion of the upper triangular matrix U and do not maintain the desired exponential decay rate due to the ill-conditioned matrix U. Based on this observation, a simple truncation method has been proposed with which we use the truncation of h with a certain tolerance level ε{lunate}t and show that the L∞ errors decay exponentially with N without any growth. It is also shown that the proposed truncation method yields the same performance with various Gegenbauer polynomials of different λ and that the inverse reconstruction provides accurate results even with large λ. Additional numerical examples are provided for the filtered IPRM and the IPRM reconstruction of the Runge function. From the numerical results, it is confirmed that the truncated IPRM yields the best results for both examples. Numerical examples used in our previous works are used to confirm the resolution of the L∞ error growth due to the ill-posedness of the IPRM with the proposed truncation method of h. © 2007 Elsevier Inc. All rights reserved.