本文選題：二次特征值問(wèn)題 + 條件數(shù)�。� 參考：《華東理工大學(xué)》2017年碩士論文

【摘要】：矩陣特征值的條件數(shù)反映了特征值對(duì)于矩陣元素變化的敏感性,它對(duì)于衡量特征值問(wèn)題數(shù)值算法的穩(wěn)定性有重要作用。本文以正則二次特征值問(wèn)題半單特征值的解析擾動(dòng)為基礎(chǔ),研究了正則二次特征值問(wèn)題半單特征值的條件數(shù)。我們從半單特征值的方向?qū)?shù)出發(fā),給出了正則二次特征值問(wèn)題半單特征值條件數(shù)的多種定義。利用奇異值分解和酉不變范數(shù)的性質(zhì),導(dǎo)出了條件數(shù)的計(jì)算表達(dá)式。和已有結(jié)果相比較,本文定義的條件數(shù)不僅可以衡量重特征值擾動(dòng)的最壞情形,而且能反映重特征值擾動(dòng)后產(chǎn)生的不同特征值的相應(yīng)靈敏度。另一方面,本文還研究了二次特征值問(wèn)題半單特征值的病態(tài)擾動(dòng),給出了半單特征值重?cái)?shù)發(fā)生改變時(shí)系數(shù)矩陣的擾動(dòng)上界。
[Abstract]:The condition number of matrix eigenvalue reflects the sensitivity of eigenvalue to the change of matrix element, and it plays an important role in evaluating the stability of numerical algorithm for eigenvalue problem. Based on the analytic perturbation of semi-simple eigenvalues of regular quadratic eigenvalue problems, the condition number of semi-simple eigenvalues for regular quadratic eigenvalue problems is studied. Based on the directional derivatives of semi-simple eigenvalues, we give several definitions of semi-simple eigenvalue conditions for regular quadratic eigenvalue problems. By using the properties of singular value decomposition and unitary invariant norm, the expression of conditional number is derived. Compared with the existing results, the condition number defined in this paper can not only measure the worst-case of the repeated eigenvalue perturbation, but also reflect the sensitivity of different eigenvalues produced by the repeated eigenvalue perturbation. On the other hand, we also study the ill-conditioned perturbation of semi-simple eigenvalue for quadratic eigenvalue problems, and give the upper bound of the perturbation of coefficient matrix when the multiplicity of semi-simple eigenvalue changes.
【學(xué)位授予單位】：華東理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

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本文編號(hào)：1872208