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引用本文:	袁利国，余荣忠，旷菊红.递归(分片)仿射分形插值数值模拟与盒维数研究(J/M/D/N,J:杂志，M：书，D：论文，N：报纸).期刊名称,2021，38（3）：122-128
	CHEN X. Adap tive slidingmode contr ol for discrete2ti me multi2inputmulti2 out put systems[ J ]. Aut omatica, 2006, 42(6): 4272-435

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递归(分片)仿射分形插值数值模拟与盒维数研究
袁利国，余荣忠，旷菊红^1,2,3
1.华南农业大学数学与信息学院,广州 510640;2.九江学院理学院,江西九江 332005;3.五邑大学数学与计算科学学院,广东江门 529020

摘要:

分析了Barnsley M F的经典文献中递归仿射分形插值的数值模拟问题，利用Matlab与随机迭代算法，给出了递归仿射分形插值算例的随机迭代算法，得到相应分形插值曲线。递归仿射分形插值把较长的原像区间Ji′=［xl,xm］压缩映射到更短的像区间Ji=［xi－1,xi］时，有3种情况:(1) JiJi′; (2) JiJi′且Ji∩Ji′=Φ;(3) JiJi′，但是Ji∩Ji′≠Φ。根据递归仿射分形插值与分片分形插值，对以上3种情形进行了随机迭代数值模拟，给出了算法流程与详细的程序代码，这些数值分析是对分形插值理论的补充。最后，利用粒子群最优化算法给出仿射分形插值函数的盒维数最优解。

关键词: 递归仿射分形插值分片仿射分形插值随机迭代数值模拟盒维数

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Research on Box Dimension and Numerical Simulations of Recurrent (Piecewise) Affine Fractal Interpolation

YUAN Li-guo, YU Rong-zhong, KUANG Ju-hong^1,2,3

1.School of Mathematics and Informatics, South China Agricultural University, Guangzhou 510640,China;2.School of Science, Jiujiang University,Jiangxi Jiujiang 332005,China;3.School of Mathematics and Computational Science, Wuyi University,Guangdong Jiangmen 529020,China

Abstract:

This paper analyzes the numerical simulation of recurrent affine fractal interpolation in M.F.Barnsley’s classical literature［1］, Recurrent Iterated Function Systems.By using MATLAB and random iterative algorithm, the random iterative algorithm of recurrent affine fractal interpolation example is given, and the corresponding fractal interpolation curve is obtained.When the recurrent affine fractal interpolation maps the longer original image range Ji′=［xl,xm］ compression to the shorter image range Ji=［xi-1,xi］,there are three situations:(1)JiJi′， (2)JiJi′ and Ji∩Ji′=Φ,(3)JiJi′andJi∩Ji′≠Φ。According to the recurrent affine fractal interpolation and piecewise fractal interpolation, the random iterative numerical simulation of the above three cases is carried out, and the algorithm process and detailed program codes are given.The numerical analysis is a supplement to the fractal interpolation theory.Finally, the optimal solution of box dimension of affine fractal interpolation function is given by particle swarm optimization algorithm.

Key words: recurrent affine fractal interpolation piecewise affine fractal interpolation random iteration numerical simulation box dimension