Explicit Gaussian Quadrature Rules for $$C^1$$ C 1 Cubic Splines with Non-uniform Knot Sequences

This paper provides the explicit and optimal quadrature rules for the cubic $$C^1$$ spline space, which is the extension of the results in Ait-Haddou et al. (J Comput Appl Math 290:543–552, 2015) for less restricted non-uniform knot values. The rules are optimal in the sense that there exist no other quadrature rules with fewer quadrature points to exactly integrate the functions in the given spline space. The explicit means that the quadrature nodes and weights are derived via an explicit recursive formula. Numerical experiments and the error estimations of the quadrature rules are also presented in the end.