Mahesh D. Pandey

In this paper, a generalized Lagrange interpolation formula expressed in matrix form is developed to systematically expand a sampled function with enhanced flexibility and computational rigor. The proposed formulation employs appropriate coordinate functions that not only satisfy prescribed boundary conditions but also exploit the symmetry or anti-symmetry inherent in the function under consideration. When such conditions are absent, the coordinate functions naturally degenerate into polynomial bases, thereby reproducing the classical Lagrange interpolation as a special case. The expansion coefficients are efficiently obtained through the collocation method, ensuring numerical simplicity and stability. The matrix-based generalized Lagrange interpolation exhibits substantial versatility beyond traditional interpolation tasks. It can be readily applied to numerical differentiation and integration under both uniform and non-uniform sampling schemes. Moreover, the approach proves useful in solving ordinary differential equations with specified boundary constraints, as well as in problems involving root-finding and extremum detection of functions. Numerical experiments demonstrate the accuracy and robustness of the proposed method, revealing a marked reduction in the Runge phenomenon even when the number of sampling points is limited. The results further indicate that computational efficiency and precision improve progressively as the number of samples increases. Overall, the generalized interpolation framework developed herein provides a unified and reliable computational tool for interpolation, differentiation, integration, and boundary-value problems, thereby offering broad potential for applications in numerical analysis and scientific computing.

Numerical differentiation has been widely applied in engineering practice due to its remarkable simplicity in the approximation of derivatives. Existing formulas rely on only three-point interpolation to compute derivatives when dealing with irregular sampling intervals. However, it is widely recognized that employing five-point interpolation yields a more accurate estimation compared to the three-point method. Thus, the objective of this study is to develop formulas for numerical differentiation using more than three sample points, particularly when the intervals are irregular. Based on Lagrange interpolation in matrix form, formulas for numerical differentiation are developed, which are applicable to both regular and irregular intervals and can use any desired number of points. The method can also be extended for numerical integration and for finding the extremum of a function from its samples. Moreover, in the proposed formulas, the target point does not need to be at a sampling point, as long as it is within the sampling domain. Numerical examples are presented to illustrate the accuracy of the proposed method and its engineering applications. It is demonstrated that the proposed method is versatile, easy to implements, efficient, and accurate in performing numerical differentiation and integration, as well as the determination of extremum of a function.