Solving Large Systems of Linear Equations for Cubic Spline Polynomials

Finding cubic spline polynomials requires solving extensive systems of linear equations. Given N breakpoints, we need to determine N-1 cubic polynomials, each containing 4 unknown coefficients. This results in a system with 4*(N-1) unknowns. By expressing each cubic polynomial in a specific form and applying various constraints, we can determine these polynomials by solving N systems of equations with N unknown coefficients. The implementation typically involves constructing a tridiagonal matrix system and using efficient algorithms like Thomas algorithm for solving the linear equations.