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We provide explicit expressions for quadrature rules on the space of $C^1$cubic splines with non-uniform, symmetrically stretched knot sequences. Thequadrature nodes and weights are derived via an explicit recursion that avoidsan intervention of any numerical solver and the rule is optimal, that is, itrequires minimal number of nodes. Numerical experiments validating thetheoretical results and the error estimates of the quadrature rules are alsopresented.

Explicit Gaussian quadrature rules for cubic splines with non-uniform knot sequences