Versatility of the Generalized Lagrange Interpolation Formula in the Matrix Form

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.