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Progressive-iterative Approximation Method
In mathematics, the progressive-iterative approximation method is an iterative method of Curve fitting, data fitting with geometric meanings. Given a set of data points to be fitted, the method obtains a series of fitting curves (or surfaces) by iteratively updating the control points, and the Limit (mathematics), limit curve (surface) can Interpolation, interpolate or Approximation theory, approximate the given data points. It avoids solving a linear system of equations directly and allows flexibility in adding constraints during the iterative process. Therefore, it has been widely used in geometric design and related fields. The study of the iterative method with geometric meaning can be traced back to the work of scholars such as Dongxu Qi and Carl R. de Boor, Carl de Boor in the 1970s. In 1975, Qi et al. developed and proved the "profit and loss" algorithm for uniform cubic B-spline curves, and in 1979, de Boor independently proposed this algorithm. In 2004, Hongwei Lin and co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with Algorithm#Termination, termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or Quasi-Newton method, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno algorithm, BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called ''Convergent series, convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |