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Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell'Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691-1707, ) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell'Accio, Appl Numer Math 52:339-361, ). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77-90, ) and drawn on by Agarwal et al. () and Agarwal and Wong (J Comput Appl Math 234:2543-2561, ). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided. [ABSTRACT FROM AUTHOR]

A class of polynomials Lk(x), k=1, 2, ... of degree 2k satisfying the differential equation L''k(x)=Lk-1(x) and the initial condition L'k(0)=0 is introduced. A relationship with a particular sequence of Appell polynomials of even degree is given. For any choice of one more boundary condition, a special sequence of polynomials is obtained. Special sequences of these polynomials are the basis of an interpolatory problem related to a fixed linear functional. Some of these polynomials are well known in literature. Finally, the problem of the polynomial expansion in these basis of real C∞ functions is posed. Some examples are given. [ABSTRACT FROM AUTHOR]

The class of polynomials Lk(x), k=1, 2, ..., of degree 2k+1 satisfying the differential equation L''k(x)=Lk-1(x) and the initial condition Lk(0)=0 is introduced. For any choice of one more condition, we have a definite and special sequence of polynomials. The sequences of polynomials related to a linear functional F on C2k+1 are interesting, since they are the basis of a particular non-classic interpolatory problem. [ABSTRACT FROM AUTHOR]

For the numerical solution of the second order nonlinear two-point boundary value problems a family of polynomial global methods is derived. Numerical examples provide favorable comparisons with other existing methods. [ABSTRACT FROM AUTHOR]

It is well-known that Muller’s method for the computation of the zeros of continuous functions has order ≈ 1.84 [10], and does not have the character of global convergence. Muller’s method is based on the interpolating polynomial built on the last three points of the iterative sequence. In this paper the authors take as nodes of the interpolating polynomial the last two points of the sequence and the middle point between them. The resulting method has order p=2 for regular functions. This method leads to a globally convergent algorithm because it uses dichotomic techniques. Many numerical examples are given to show how the proposed code improves on Muller’s method. [ABSTRACT FROM AUTHOR]

Two new general classes of polynomial sequences called respectively odd and even Lidstone-type polynomials are considered. These classes include classic Lidstone polynomials of first and second kind. Some characterizations of the two classes are given, including matrix form, conjugate sequences, generating function, recurrence relations, and determinant forms. Some examples are presented and some applications are sketched. [ABSTRACT FROM AUTHOR]