Solving Large Systems of Linear Equations for Cubic Spline Polynomials

Finding cubic spline polynomials requires solving large systems of linear equations. Given N breakpoints, we need to determine N-1 cubic polynomials, each with 4 unknown coefficients. This leads to a total of 4*(N-1) unknowns that must be solved. Although this process appears complex, by expressing each cubic polynomial in a specific form and applying various constraints (such as continuity conditions, smoothness requirements, and boundary conditions), we can determine the cubic polynomials by solving N systems of equations with N unknowns each. In computational implementations, this typically involves constructing a tridiagonal matrix system where the continuity conditions at interior points and boundary conditions create a structured linear system. The Thomas algorithm (TDMA) is commonly employed for efficient solution of these tridiagonal systems. While this process may require substantial computation, it ensures we obtain precise results, enabling better understanding of data patterns and supporting more informed decisions. The cubic spline implementation often uses piecewise polynomial representation with coefficients stored in arrays, where each segment's polynomial is defined between consecutive breakpoints.