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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 logarithmic norm ： ウィキペディア英語版 logarithmic norm In mathematics, the logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic norm was independently introduced by Germund Dahlquist〔Germund Dahlquist, "Stability and error bounds in the numerical integration of ordinary differential equations", Almqvist & Wiksell, Uppsala 1958〕 and Sergei Lozinskiĭ in 1958, for square matrices. It has since been extended to nonlinear operators and unbounded operators as well.〔Gustaf Söderlind, "The logarithmic norm. History and modern theory", ''BIT Numerical Mathematics'', 46(3):631-652, 2006〕 The logarithmic norm has a wide range of applications, in particular in matrix theory, differential equations and numerical analysis. In the finite dimensional setting it is also referred to as the matrix measure. ==Original definition== Let $A$ be a square matrix and $\backslash |\; \backslash cdot\; \backslash |$ be an induced matrix norm. The associated logarithmic norm $\backslash mu$ of $A$ is defined :$\backslash mu(A)\; =\; \backslash lim\; \backslash limits\_\; \backslash frac$ Here $I$ is the identity matrix of the same dimension as $A$, and $h$ is a real, positive number. The limit as $h\backslash rightarrow\; 0^-$ equals $-\backslash mu(-A)$, and is in general different from the logarithmic norm $\backslash mu(A)$, as $-\backslash mu(-A)\; \backslash leq\; \backslash mu(A)$ for all matrices. The matrix norm $\backslash |A\backslash |$ is always positive if $A\backslash neq\; 0$, but the logarithmic norm $\backslash mu(A)$ may also take negative values, e.g. when $A$ is negative definite. Therefore, the logarithmic norm does not satisfy the axioms of a norm. The name ''logarithmic norm,'' which does not appear in the original reference, seems to originate from estimating the logarithm of the norm of solutions to the differential equation :$\backslash dot\; x\; =\; Ax.$ The maximal growth rate of $\backslash log\; \backslash |x\backslash |$ is $\backslash mu(A)$. This is expressed by the differential inequality :$\backslash frac\; \backslash log\; \backslash |x\backslash |\; \backslash leq\; \backslash mu(A),$ where $\backslash mathrm\; d/\backslash mathrm\; dt^+$ is the upper right Dini derivative. Using logarithmic differentiation the differential inequality can also be written :$\backslash frac\; \backslash leq\; \backslash mu(A)\backslash cdot\; \backslash |x\backslash |,$ showing its direct relation to Grönwall's lemma. In fact, it can be shown that the norm of the state transition matrix $\backslash Phi(t,\; t\_0)$ associated to the differential equation $\backslash dot\; x\; =\; A(t)x$ is bounded by :$\backslash exp\backslash left(-\backslash int\_^\; \backslash mu(-A(s))\; ds\; \backslash right)\; \backslash le\; \backslash |\backslash Phi(t,t\_0)\backslash |\; \backslash le\; \backslash exp\backslash left(\backslash int\_^\; \backslash mu(A(s))\; ds\; \backslash right)$ for all $t\; \backslash ge\; t\_0$. 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「logarithmic norm」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.