A lightweight micropayment scheme based on Lagrange interpolation formula

An efficient quadrature formula was developed for evaluating numerically certain singular Fredholm integral equations of the first kind with oscillatory trigonometric kernels.  The method is based on the Lagrange interpolation formula and the orthogonal polynomial considered are the Legendre polynomials whose zeros served as interpolation nodes. A test example was provided for the verification and validation of the rule developed. The results showed the convergence of the solution and can be improved by increasing n.

The purpose of this paper is to derive lagrange interpolation formula for a single variable and two independent variables. Firstly, single variable interpolation is derived and then two independent variable interpolation derived. Some practical problems are computed by virtue of these interpolation formula.

The purpose of this paper is to derive lagrange interpolation formula for a single variable and two independent variables in triangular form. Firstly, single variable interpolation is derived and then two independent variable interpolation derived in triangular form. In this paper, we derived the formula of three points of fit approximation, six points of fit approximation and ten points of fit approximation with their examples respectively. And also derived the approximation using a general triangular form of points with examples.

The problem of the best recovery in the sense of Sard of a linear functional Lf on the basis of information T(f) = {Ljf,j = 1, 2,... N} is studied. It is shown that in the class of bivariate functions with restricted (n,m) -derivative, known on the (n,m)-grid lines, the problem of the best recovery of a linear functional leads to the best approximation of L(KnKm) in the space S = span Lj(Kn_Km), j=1; 2,...N}, where Kn(x,t) = K(x,t)- Lxn(K(.,t):x) is the difference between the truncated power kernel K(x,t) = (x-t)n-1+ =(n-1)! and its Lagrange interpolation formula. In particular, the best recovery of a bivariate function is considered, if scattered data points and blending grid are given. An algorithm is designed and realized using the software product MATLAB.