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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 Linear approximation ： ウィキペディア英語版 Linear approximation In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. ==Definition== Given a twice continuously differentiable function $f$ of one real variable, Taylor's theorem for the case $n\; =\; 1$ states that :$f(x)\; =\; f(a)\; +\; f\text{'}(a)(x\; -\; a)\; +\; R\_2\backslash$ where $R\_2$ is the remainder term. The linear approximation is obtained by dropping the remainder: :$f(x)\; \backslash approx\; f(a)\; +\; f\text{'}(a)(x\; -\; a)$. This is a good approximation for $x$ when it is close enough to $a$; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of $f$ at $(a,f(a))$. For this reason, this process is also called the tangent line approximation. If $f$ is concave down in the interval between $x$ and $a$, the approximation will be an overestimate (since the derivative is decreasing in that interval). If $f$ is concave up, the approximation will be an underestimate.〔(【引用サイトリンク】url=http://math.mit.edu/classes/18.013A/HTML/chapter12/section01.html )〕 Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function $f(x,\; y)$ with real values, one can approximate $f(x,\; y)$ for $(x,\; y)$ close to $(a,\; b)$ by the formula :$f\backslash left(x,y\backslash right)\backslash approx\; f\backslash left(a,b\backslash right)+\backslash frac\backslash left(a,b\backslash right)\backslash left(x-a\backslash right)+\backslash frac\backslash left(a,b\backslash right)\backslash left(y-b\backslash right).$ The right-hand side is the equation of the plane tangent to the graph of $z=f(x,\; y)$ at $(a,\; b).$ In the more general case of Banach spaces, one has :$f(x)\; \backslash approx\; f(a)\; +\; Df(a)(x\; -\; a)$ where $Df(a)$ is the Fréchet derivative of $f$ at $a$. 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「Linear approximation」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.