【摘要】：眾所周知,最小二乘(LS)和總體最小二乘(TLS)是科學(xué)計(jì)算中的兩種重要方法.條件數(shù)刻畫了一個(gè)問題的解對(duì)于輸入數(shù)據(jù)小擾動(dòng)的敏感程度,關(guān)于條件數(shù)的研究是矩陣擾動(dòng)分析和數(shù)值分析的一個(gè)重要課題.近年來,一大批學(xué)者在LS和TLS問題的條件數(shù)方面做了大量的工作.本文繼續(xù)研究LS和TLS問題的條件數(shù),主要工作包括以下五個(gè)部分:第一部分討論了Tikhonov正則化解的條件數(shù).首先,我們給出了當(dāng)系數(shù)矩陣、正則化矩陣和右端項(xiàng)向量同時(shí)擾動(dòng)時(shí),Tikhonov正則化解的相對(duì)范數(shù)型、混合型和分量型條件數(shù),推廣了[Chu et al.,Numer.Linear Algebra Appl.2011,18(1):87-103]中的結(jié)果.其次,我們研究了當(dāng)系數(shù)矩陣A具有線性結(jié)構(gòu)時(shí)Tikhonov正則化解的結(jié)構(gòu)化條件數(shù).第二部分研究了多右端項(xiàng)LS問題的條件數(shù)理論.我們分別在系數(shù)矩陣是列滿秩矩陣和秩虧矩陣的假設(shè)條件下,研究了多右端項(xiàng)LS問題的范數(shù)型、混合型和分量型條件數(shù).所得結(jié)果推廣了單右端項(xiàng)LS問題條件數(shù)的結(jié)果.第三部分研究了單右端項(xiàng)TLS問題的條件數(shù).首先,給出了單右端項(xiàng)TLS問題的混合型和分量型條件數(shù)的精確表達(dá)式和上界;然后,當(dāng)單右端項(xiàng)TLS問題的系數(shù)矩陣具有線性結(jié)構(gòu)(如下三角、Toeplitz或Hankel結(jié)構(gòu))和Vandermonde、Cauchy非線性結(jié)構(gòu)時(shí),我們給出了其結(jié)構(gòu)化的條件數(shù).數(shù)值例子表明結(jié)構(gòu)化的條件數(shù)確實(shí)比無結(jié)構(gòu)的條件數(shù)小,甚至可以小很多.第四部分討論了截?cái)郥LS(T-TLS)解的線性函數(shù)的條件數(shù).我們給出了T-TLS解的線性函數(shù)LTxk的范數(shù)型、混合型和分量型條件數(shù)的精確表達(dá)式及上界,其中xk是截?cái)嗨綖閗的T-TLS解.本部分所得的結(jié)果推廣或改進(jìn)了已知文獻(xiàn)中的結(jié)果.另外,我們還給出了LTxk的絕對(duì)范數(shù)型條件數(shù)的兩個(gè)統(tǒng)計(jì)估計(jì).數(shù)值例子表明條件數(shù)的上界和這兩個(gè)統(tǒng)計(jì)估計(jì)確實(shí)是相應(yīng)真實(shí)值的很好的估計(jì).第五部分研究了多右端項(xiàng)TLS問題的條件數(shù).據(jù)我們所知,目前尚無文獻(xiàn)討論過這個(gè)問題.首先,當(dāng)多右端項(xiàng)TLS問題有唯一解時(shí),我們給出了其范數(shù)型、混合型和分量型條件數(shù)的精確表達(dá)式及上界,這些結(jié)果推廣了單右端項(xiàng)TLS問題的條件數(shù)理論.另外,我們給出了如何利用冪方法計(jì)算絕對(duì)范數(shù)型條件數(shù).接著,我們給出了當(dāng)多右端項(xiàng)TLS問題有無數(shù)多解時(shí)其極小Frobenius范數(shù)解的范數(shù)型、混合型和分量型條件數(shù)的上界,數(shù)值結(jié)果表明這些上界是緊的.
[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.
【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O241.5

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