發(fā)布時(shí)間：2018-02-12 10:51

  本文關(guān)鍵詞： 非線(xiàn)性矩陣方程(方程組) Hermitian 正定解 不動(dòng)點(diǎn)迭代 免逆迭代 最小二乘解　出處：《青島科技大學(xué)》2017年碩士論文　論文類(lèi)型：學(xué)位論文

【摘要】：非線(xiàn)性矩陣方程的求解問(wèn)題是近年來(lái)數(shù)值代數(shù)領(lǐng)域討論的重要課題之一,它在最優(yōu)控制理論、梯形網(wǎng)絡(luò)、動(dòng)態(tài)規(guī)劃、隨機(jī)過(guò)濾等領(lǐng)域均有廣泛的應(yīng)用.迭代法是求解非線(xiàn)性矩陣方程常用的方法,但在采用迭代法求解非線(xiàn)性矩陣方程時(shí),經(jīng)常會(huì)出現(xiàn)解的收斂速度緩慢、計(jì)算量大的問(wèn)題.近年來(lái),我們較多采用不動(dòng)點(diǎn)迭代法和免逆迭代法求解非線(xiàn)性矩陣方程,其中免逆迭代法大大地簡(jiǎn)化了計(jì)算的復(fù)雜度.基于Kronecker積的性質(zhì),首先得到了非線(xiàn)性矩陣方程X + A*(Im(?)X-C)~(-t)A = Q(t0)存在Hermitian正定解的充分必要條件;其次,運(yùn)用有界序列的收斂原理,分別提出了求解方程的不動(dòng)點(diǎn)迭代法和免逆迭代法;最后,通過(guò)數(shù)值例子驗(yàn)證了這兩種迭代方法的有效性.我們也考慮非線(xiàn)性矩陣方程X~s+A*X~(-t_1) A+B*X~(-t_2)B = I(s,t_1,t_20).首先得到方程存在Hermitian正定解的一些新的條件和唯一 Hermitian正定解存在的充分條件,并通過(guò)對(duì)s,t_1,t_2取值范圍的討論,給出了方程解的存在區(qū)間;其次,構(gòu)造了求解方程的不動(dòng)點(diǎn)迭代法;最后,通過(guò)數(shù)值例子驗(yàn)證了迭代方法是行之有效的.進(jìn)而,我們研究了非線(xiàn)性矩陣方程組首先得到方程組存在正定解的條件;其次,提出了求解方程組的不動(dòng)點(diǎn)迭代法;最后,通過(guò)數(shù)值例子驗(yàn)證了迭代方法的有效性.最后,我們研究了非線(xiàn)性矩陣方程組分別運(yùn)用最速下降法和Newton法求解方程組的最小二乘解,并且通過(guò)具體的數(shù)值例子驗(yàn)證了 Newton法的有效性.
[Abstract]:The problem of solving the nonlinear matrix equation is one of the important issues discussed in numerical algebra field in recent years, it is in the optimal control theory, ladder networks, dynamic programming, stochastic filtering etc. were applied widely. The iterative method is a method of solving the nonlinear matrix equation, but in using the iterative method of solving the nonlinear matrix equation, often appear the convergence speed is slow, the problem of large amount of calculation. In recent years, we are using fixed point iteration method and free inverse iteration method for solving nonlinear matrix equation, the free inverse iteration method greatly simplifies the computational complexity. Based on the properties of product Kronecker, has been the first X + A* (Im (nonlinear matrix equation X-C?)) ~ (-t) A = Q (T0) Hermitian positive definite solution of the necessary and sufficient condition; secondly, based on the convergence principle of bounded sequence, are proposed respectively fixed point iteration method for solving equations and free inverse iteration On behalf of the law; finally, through numerical examples verify the effectiveness of the two kinds of iterative methods. We also consider the nonlinear matrix equation X~s+A*X~ (-t_1) A+B*X~ (-t_2) B = I (s, t_1, t_20). We obtain sufficient conditions for Hermitian positive definite solutions of some new conditions and only Hermitian positive definite solutions of the problems based on the equation, s, t_1, t_2 discussed the range interval equations are presented; secondly, we construct fixed point iteration method for solving equations; finally, a numerical example shows the iterative method is effective. Then, we study the nonlinear matrix equations firstly obtained equations the existence of positive definite solutions the conditions; secondly, the fixed point iteration method for solving equations; finally, through numerical examples verify the effectiveness of the iterative method. Finally, we study the nonlinear matrix equations using the steepest descent method and Newton The method is used to solve the least square solution of the equation group, and the validity of the Newton method is verified by a specific numerical example.

【學(xué)位授予單位】：青島科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.6

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本文編號(hào)：1505460