提出一种用于解决递推估计问题的优化算法,该算法基于递推粒子群优化.递推估计问题获得的数据并非一次性获得,而是陆续获得.在递推的粒子群算法中,粒子基于过去的数据信息和新获取的数据递推地更新自己位置.实验结果表明,基于递推算法的径向基函数网络模拟系统只需要较少的径向基函数,同时在解决动态问题时能够比传统粒子群算法获得更准确的结果.

A Recursive Particle Swarm Optimization (R-PSO)is proposed to solve dynamic optimization problems where the data is ob-tained not once but one by one.In R-PSO,the position of each particle swarm is updated recursively based on the continuous data and the historical knowledge.The experiment results indicate that RPSO-based radial basis function networks needs fewer radial basis func-tions and meanwhile gives more accurate results than traditional PSO in solving dynamic problems.

结合实际问题对“穷举”和“递推”算法的基本思想进行分析，研究算法的实现方法，借助C语言将算法转化为可执行的程序代码。“穷举”算法的关键是找到数据的测试范围和测试标准，并对测试标准的转化技巧、测试范围的控制方法进行了阐述。“递推”算法的归结到对递推初值、递推法则、递推次数三个条件的研究，并从顺推、逆推两个方向分别进行了阐述。

This paper analyzes the basic idea of"exhaustive"and"recursive"algorithm and implementation of the algorithm. The algorithm converted into executable code by C language. "Exhaustive" algorithm focuses find the test data ranges and testing standards. Conversion techniques and testing standards, test range control methods are described. "Recursive" algorithm attributed to the initial value of the recursion, recursive rule, the number of recursive three conditions studied, and in both directions along the push, the reverse push expounded respectively.

基于一类数据具有不断放大的特性，应用外P-集合的结构与动态特性，将P-集合串的基集合动态化，提出外-递推数据概念，获得外-递推数据的结构、生成原理与生成准则；给出外-递推数据依赖与还原相关定理及应用。

By using the structure and dynamic characteristic of outer P-set based on a class of data which has a charac-teristic of recurrence amplification, the concept of outward-recursion data is proposed.Also, the paper gives the struc-ture, the generation principle and the criterion as well as the interrelated theorems of dependence and reduction for the outward recursion data.Finally, the application of the outward-recursion information is presented.

二维Otsu法是一种经典的图像分割算法，在其基础上发展的降维算法由于存在大量冗余计算限制了其实际应用，针对这一问题，提出二维Otsu法的降维递推综合改进算法。先把二维阈值形式降低到一维，计算区域改为对角线两侧的带形区域；然后采用递推思想给出递推公式，应用递推公式得到图像分割阈值，去除了原有的冗余计算。结合降维与递推，时间复杂度由降维算法的O（L3）降低为O（ L2）。实验结果表明，该种算法在保证分割效果的同时大大提高了降维算法的计算效率。

Two-dimensional Otsu algorithm is widely used in classic image segmentation.The dimensionality reduction algorithm derived from the Otsu algorithm limits its application due to a large number of redundant computations.In order to solve this problem, there comes into being an improved dimensionality reduction recursive synthesis algorithm based on the 2D Otsu method.This algorithm reduces the 2D threshold form to 1D and improves calculation regions on both sides of the diagonal band-shaped area, after that, the recursion formula can be deduced according to the recursive thought, which gets rid of the original redundant computation.Through the ingenious combination of dimensionality reduction and recursive thoughts, the algorithm time complexity is reduced from O ( L3 ) of dimensionality reduction algorithm to O ( L2 ) . Experimental results show that this improved algorithm can save a great deal of running time as well as enhance the computational efficiency of the dimensional

针对双率系统，采用基于辅助模型的改进随机牛顿递推算法辨识输出误差模型。若当前参数估计对应的估计系统不稳定，则出现中间不可测时刻输出估计发散，辨识过程停止。增加非线性模型与常规辅助模型一起为下步递推提供信息估计，确保递推进行。为避免出现输入不充分或者广泛时Hessian阵奇异或者接近奇异的情况，在Hessian阵的递推中增加对称正定矩阵。最后给出了所提出辨识算法的一致收敛性证明。

An auxiliary-model-based improved stochastic Newton recursive algorithm is utilized to identify the output-error model for the dual-rate system. The output estimations at the time between two slow-sampled periods will diverge when the unstable estimated system arises and the identification process will cease. Therefore, a nonlinear model is proposed as the sub-auxiliary model. Together with the general auxiliary model, the proposed model provides estimated information for the next recursion. When the input is not general enough, the Hessian matrix will be singular or nearly singular. A positive definite symmetric matrix is added to the recursion for Hessian matrix to ensure it positive definite. Finally, the uniform convergence of the proposed algorithm is proved.