Dual Taylor Series, Spline Based Function and Integral Approximation and Applications
AbstractIn this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series approximation. It is shown that a spline based series approximation to an integral yields, in general, a higher accuracy for a set order of approximation than a dual Taylor series, a Taylor series and an antiderivative series. A spline based series for an integral has many applications and indicative examples are detailed. These include a series for the exponential function, which coincides with a Padé series, new series for the logarithm function as well as new series for integral defined functions such as the Fresnel Sine integral function. It is shown that these series are more accurate and have larger regions of convergence than corresponding Taylor series. The spline based series for an integral can be used to define algorithms for highly accurate approximations for the logarithm function, the exponential function, rational numbers to a fractional power and the inverse sine, inverse cosine and inverse tangent functions. These algorithms are used to establish highly accurate approximations for
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Howard, R.M. Dual Taylor Series, Spline Based Function and Integral Approximation and Applications. Math. Comput. Appl. 2019, 24, 35.
Howard RM. Dual Taylor Series, Spline Based Function and Integral Approximation and Applications. Mathematical and Computational Applications. 2019; 24(2):35.Chicago/Turabian Style
Howard, Roy M. 2019. "Dual Taylor Series, Spline Based Function and Integral Approximation and Applications." Math. Comput. Appl. 24, no. 2: 35.
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