Abstract: By employing matrix block theory and eigendecomposition techniques, a detailed investigation of the local subdivision matrix structure for a class of non-stationary subdivision schemes is conducted. A closed-form expression for the chain product of subdivision matrices is derived, uncovering the algebraic structure of this product. Building on this foundation, a subdivision algorithm is proposed that accurately interpolates limit curves, thereby improving the accuracy and quality of the subdivision schemes. The effectiveness of the algorithm is proved by provided modeling examples.