Adaptive Padé-Chebyshev Type Approximation to Piecewise Smooth Functions
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The aim of this article is to study the role of piecewise implementation of Padé-Chebyshev type approximation in minimising Gibbs phenomena in approximating piecewise smooth functions. A piecewise Padé-Chebyshev type (PiPCT) algorithm is proposed and an L^1-error estimate for at most continuous functions is obtained using a decay property of the Chebyshev coefficients. An advantage of the PiPCT approximation is that we do not need to have an a prior knowledge of the positions and the types of singularities present in the function. Further, an adaptive piecewise Padé-Chebyshev type (APiPCT) algorithm is proposed in order to get the essential accuracy with a relatively lower computational cost. Numerical experiments are performed to validate the algorithms. The numerical results are also found to be well in agreement with the theoretical results. Comparison results of the PiPCT approximation with the singular Padé-Chebyshev and the robust Padé-Chebyshev methods are also presented.
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