Abstract
General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by Tan and Fang. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method.
Highlights
Newton interpolation and Thiele-type continued fractions interpolation may be the favored linear interpolation and nonlinear interpolation [1]
If f0(x) = x, fi(x) = 1/x, 1 ≤ i ≤ N − 1, fi,i(x) = x, fi,i+1(x) = fi+1,i(x) = 1/x, fi,j(x) = fj,i(x) = x, j ≥ i + 2, gi(x) = x − x2i, hi(y) = y − y2i, ai,i+s(x) = a2i,2i+s, 1 ≤ s ≤ n − i, ai+t,i(x) = a2i+t,2i, 1 ≤ t ≤ m − i, i = 0, 1, . . . , n, j = 0, 1, . . . , m, Q(x, y) is bivariate continued fraction interpolation over orthotriples studied by Wang and Qian [11]
It could be used to deal with the interpolation problems where inverse differences are nonexistent or unattainable points occur via choosing fi(x), fi,j(x) appropriately [17]
Summary
Received 6 February 2014; Revised 14 April 2014; Accepted 7 May 2014; Published 25 June 2014. General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by Tan and Fang. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method
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