The weighted GS-PIA algorithm for cubic B-spline curve interpolations and convergence analysis

The weighted Gauss–Seidel-progressive iterative approximation (WGS-PIA) algorithm for cubic B-spline curve interpolations is considered in this paper. The convergence of the WGS-PIA algorithm is analyzed, and an upper bound which is strictly smaller than one for the contraction factor of this WGS-PIA algorithm is derived. It is shown that for cubic B-spline curve interpolations, the GS-PIA algorithm converges faster than the Jacobi-PIA (J-PIA) algorithm, and that there always exists a positive weight \(\omega \) $\omega$ such that the WGS-PIA converges faster than GS-PIA. Particularly, we derive a formula for the effective weight \(\omega ^{\star }\) ${\omega }^{\star }$ and the “theoretically optimal” weight \(\omega _{m}\) ${\omega }_m$ , which significantly improves the performance of the WGS-PIA algorithm with minimal additional cost. The numerical experiments are shown that for a given termination tolerance, the number of iterations and the CPU time required by the WGS-PIA algorithm are less than those required by the GS-PIA algorithm.