翻訳と辞書 Words near each other ・ Wilkinson matrix ・ Wilkinson Memorial Trail ・ Wilkinson Microwave Anisotropy Probe ・ Wilkinson Peaks ・ Wilkinson power divider ・ Wilkinson River ・ Wilkinson Street tram stop ・ Wilkinson Sword ・ Wilkinson TMC ・ Wilkinson Township, Cass County, Minnesota ・ Wilkinson v ASB Bank Ltd ・ Wilkinson v Downton ・ Wilkinson v. Austin ・ Wilkinson v. United States ・ Wilkinson's catalyst ・ Wilkinson's polynomial ・ Wilkinson, Illinois ・ Wilkinson, Indiana ・ Wilkinson, Mississippi ・ Wilkinson, West Virginia ・ Wilkinson, Wisconsin ・ Wilkinson-Boineau House ・ Wilkinson-Dozier House ・ Wilkinson-Martin House ・ Wilkinsonellus ・ Wilkinsons ・ Wilknicki Młyn ・ Wilkniss Mountains ・ Wilknity ・ Wilko Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Wilkinson's polynomial ： ウィキペディア英語版 Wilkinson's polynomial In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial. The polynomial is :$w(x)\; =\; \backslash prod\_^\; (x\; -\; i)\; =\; (x-1)(x-2)\; \backslash ldots\; (x-20).$ Sometimes, the term ''Wilkinson's polynomial'' is also used to refer to some other polynomials appearing in Wilkinson's discussion. ==Background== Wilkinson's polynomial arose in the study of algorithms for finding the roots of a polynomial :$p(x)\; =\; \backslash sum\_^n\; c\_i\; x^i.$ It is a natural question in numerical analysis to ask whether the problem of finding the roots of ''p'' from the coefficients ''c''''i'' is well-conditioned. That is, we hope that a small change in the coefficients will lead to a small change in the roots. Unfortunately, this is not the case here. The problem is ill-conditioned when the polynomial has a multiple root. For instance, the polynomial ''x''2 has a double root at ''x'' = 0. However, the polynomial ''x''2−ε (a perturbation of size ε) has roots at ±√ε, which is much bigger than ε when ε is small. It is therefore natural to expect that ill-conditioning also occurs when the polynomial has zeros which are very close. However, the problem may also be extremely ill-conditioned for polynomials with well-separated zeros. Wilkinson used the polynomial ''w''(''x'') to illustrate this point (Wilkinson 1963). In 1984, he described the personal impact of this discovery: :''Speaking for myself I regard it as the most traumatic experience in my career as a numerical analyst.''〔 〕 Wilkinson's polynomial is often used to illustrate the undesirability of naively computing eigenvalues of a matrix by first calculating the coefficients of the matrix's characteristic polynomial and then finding its roots, since using the coefficients as an intermediate step may introduce an extreme ill-conditioning even if the original problem was well conditioned. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wilkinson's polynomial」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.