最小二乘和总体最小二乘问题的条件数研究

发布时间：2018-08-30 07:38

【摘要】：众所周知,最小二乘(LS)和总体最小二乘(TLS)是科学计算中的两种重要方法.条件数刻画了一个问题的解对于输入数据小扰动的敏感程度,关于条件数的研究是矩阵扰动分析和数值分析的一个重要课题.近年来,一大批学者在LS和TLS问题的条件数方面做了大量的工作.本文继续研究LS和TLS问题的条件数,主要工作包括以下五个部分:第一部分讨论了Tikhonov正则化解的条件数.首先,我们给出了当系数矩阵、正则化矩阵和右端项向量同时扰动时,Tikhonov正则化解的相对范数型、混合型和分量型条件数,推广了[Chu et al.,Numer.Linear Algebra Appl.2011,18(1):87-103]中的结果.其次,我们研究了当系数矩阵A具有线性结构时Tikhonov正则化解的结构化条件数.第二部分研究了多右端项LS问题的条件数理论.我们分别在系数矩阵是列满秩矩阵和秩亏矩阵的假设条件下,研究了多右端项LS问题的范数型、混合型和分量型条件数.所得结果推广了单右端项LS问题条件数的结果.第三部分研究了单右端项TLS问题的条件数.首先,给出了单右端项TLS问题的混合型和分量型条件数的精确表达式和上界;然后,当单右端项TLS问题的系数矩阵具有线性结构(如下三角、Toeplitz或Hankel结构)和Vandermonde、Cauchy非线性结构时,我们给出了其结构化的条件数.数值例子表明结构化的条件数确实比无结构的条件数小,甚至可以小很多.第四部分讨论了截断TLS(T-TLS)解的线性函数的条件数.我们给出了T-TLS解的线性函数LTxk的范数型、混合型和分量型条件数的精确表达式及上界,其中xk是截断水平为k的T-TLS解.本部分所得的结果推广或改进了已知文献中的结果.另外,我们还给出了LTxk的绝对范数型条件数的两个统计估计.数值例子表明条件数的上界和这两个统计估计确实是相应真实值的很好的估计.第五部分研究了多右端项TLS问题的条件数.据我们所知,目前尚无文献讨论过这个问题.首先,当多右端项TLS问题有唯一解时,我们给出了其范数型、混合型和分量型条件数的精确表达式及上界,这些结果推广了单右端项TLS问题的条件数理论.另外,我们给出了如何利用幂方法计算绝对范数型条件数.接着,我们给出了当多右端项TLS问题有无数多解时其极小Frobenius范数解的范数型、混合型和分量型条件数的上界,数值结果表明这些上界是紧的.
[Abstract]:It is well known that the least square (LS) and the total least squares (TLS) are two important methods in scientific calculation. The condition number characterizes the sensitivity of the solution of a problem to the small perturbation of the input data. The study of the condition number is an important subject of matrix perturbation analysis and numerical analysis. In recent years, a large number of scholars have done a lot of work on the condition number of LS and TLS problems. In this paper, we continue to study the conditional numbers of LS and TLS problems. The main work includes the following five parts: in the first part, we discuss the conditional numbers of Tikhonov regularization solutions. First of all, we give the relative norm type, mixed type and component type condition number of Tikhonov regularization solution when the coefficient matrix, regularization matrix and right term vector are disturbed at the same time, and generalize the results in [Chu et al.,Numer.Linear Algebra Appl.2011,18 (1): 87-103]. Secondly, we study the structural condition number of Tikhonov regularization solution when the coefficient matrix A has a linear structure. In the second part, we study the condition number theory of LS problem with multiple right end terms. Under the assumption that the coefficient matrix is a column full rank matrix and a rank deficient matrix, we study the norm type, mixed type and component type condition number of the LS problem with multiple right end terms, respectively. The results generalize the condition number of LS problem with single right end term. In the third part, the condition number of TLS problem with single right end term is studied. First of all, the exact expressions and upper bounds of the mixed and component type condition numbers of the single right term TLS problem are given, and then, when the coefficient matrix of the single right term TLS problem has a linear structure (the following triangulation Toeplitz or Hankel structure) and Vandermonde,Cauchy nonlinear structure, We give its structured condition number. Numerical examples show that the structured conditional number is indeed smaller or even much smaller than the unstructured condition number. In the fourth part, we discuss the condition number of linear function truncating TLS (T-TLS) solution. We give the exact expressions and upper bounds of the norm type, mixed type and component type condition number of the linear function LTxk of the T-TLS solution, where xk is the T-TLS solution with truncated level k. The results obtained in this part extend or improve the results in the known literature. In addition, we give two statistical estimates of LTxk's condition numbers of absolute norm type. Numerical examples show that the upper bound of the conditional number and these two statistical estimates are indeed good estimates of the corresponding true values. In the fifth part, we study the condition number of multi-right TLS problem. As far as we know, there is no literature on this issue. First, when there is a unique solution to the multi-right term TLS problem, we give the exact expression and upper bound of the norm type, mixed type and component type condition number. These results generalize the conditional number theory of the single right end TLS problem. In addition, we give how to use the power method to calculate the condition number of absolute norm type. Then, we give the upper bounds of the norm type, mixed type and component type condition number of the minimal Frobenius norm solution for the multi-right term TLS problem with innumerable multiple solutions. The numerical results show that these upper bounds are compact.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.5

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