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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 Hessian automatic differentiation ： ウィキペディア英語版 Hessian automatic differentiation In applied mathematics, Hessian automatic differentiation are techniques based on automatic differentiation (AD) that calculate the second derivative of a $n$-dimensional function, known as the Hessian Matrix. When examining a function in a neighborhood of a point, one can discard many complicated global aspects of the function and accurately approximate it with simpler functions. The quadratic approximation is the best-fitting quadratic in the neighborhood of a point, and is frequently used in engineering and science. To calculate the quadratic approximation, one must first calculate its gradient and Hessian matrix. Let $f:\; \backslash mathbb^n\; \backslash rightarrow\; \backslash mathbb$, for each $x\; \backslash in\; \backslash mathbb^n$ the Hessian matrix $H(x)\; \backslash in\; \backslash mathbb^$ is the second order derivative and is a symmetric matrix. See the article on Hessian matrices for more on the definition. == Reverse Hessian-vector products == For a given $u\; \backslash in\; \backslash mathbb^n$, this method efficiently calculates the Hessian-vector product $H(x)u$. Thus can be used to calculate the entire Hessian by calculating $H(x)e\_i$, for $i\; =\; 1,\; \backslash ldots,\; n$.〔 The method works by first using forward AD to perform $f(x)\; \backslash rightarrow\; u^T\backslash nabla\; f(x)$, subsequently the method then calculates the gradient of $u^T\; \backslash nabla\; f(x)$ using Reverse AD to yield $\backslash nabla\; \backslash left(\; u\; \backslash cdot\; \backslash nabla\; f(x)\backslash right)\; =\; u^T\; H(x)\; =\; (H(x)u)^T$. Both of these two steps come at a time cost proportional to evaluating the function, thus the entire Hessian can be evaluated at a cost proportional to n evaluations of the function. 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「Hessian automatic differentiation」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.