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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert matrix ： ウィキペディア英語版 Hilbert matrix In linear algebra, a Hilbert matrix, introduced by , is a square matrix with entries being the unit fractions :$H\_\; =\; \backslash frac.$ For example, this is the 5 × 5 Hilbert matrix: :$H\; =\; \backslash begin$ 1 & \frac & \frac & \frac & \frac \\() \frac & \frac & \frac & \frac & \frac \\() \frac & \frac & \frac & \frac & \frac \\() \frac & \frac & \frac & \frac & \frac \\() \frac & \frac & \frac & \frac & \frac \end. The Hilbert matrix can be regarded as derived from the integral :$H\_\; =\; \backslash int\_^\; x^\; \backslash ,\; dx,$ that is, as a Gramian matrix for powers of ''x''. It arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8 · 105. ==Historical note== introduced the Hilbert matrix to study the following question in approximation theory: "Assume that , is a real interval. Is it then possible to find a non-zero polynomial ''P'' with integral coefficients, such that the integral :$\backslash int\_^b\; P(x)^2\; dx$ is smaller than any given bound ''ε'' > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the determinant of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length of the interval is smaller than 4. 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「Hilbert matrix」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.