Adaptive weighted progressive iterative approximation based on coordinate decomposition

During the iterative process of the progressive iterative approximation, it is necessary to calculate the difference between the current interpolation curve and the corresponding data points, known as the adjustment vector. To achieve more precise adjustments of control points, this paper decomposes the adjustment vector into its coordinate components and introduces a weight for each component. By dynamically adjusting these weights, we can accelerate the convergence of iterations and enhance approximation accuracy. During the iteration, the weight coefficients are flexibly adjusted based on the error of the current iteration step, demonstrating the flexibility and precision of the geometric iterative method in addressing geometric approximation problems. Numerical experiment results indicate that this vector decomposition technique is a critical mathematical operation for improving algorithm efficiency and precisely adjusting the shapes of curves or surfaces to approximate the data set.