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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 Piecewise linear continuation ： ウィキペディア英語版 Piecewise linear continuation Simplicial continuation, or piecewise linear continuation (Allgower and Georg),〔Eugene L. Allgower, K. Georg, "Introduction to Numerical Continuation Methods", ''SIAM Classics in Applied Mathematics'' 45, 2003.〕〔E. L. Allgower, K. Georg, "Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations", ''SIAM Review'', Volume 22, 28-85, 1980.〕 is a one parameter continuation method which is well suited to small to medium embedding spaces. The algorithm has been generalized to compute higher-dimensional manifolds by (Allgower and Gnutzman)〔Eugene L. Allgower, Stefan Gnutzmann, "An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces", ''SIAM Journal on Numerical Analysis'', Volume 24, Number 2, 452-469, 1987.〕 and (Allgower and Schmidt).〔Eugene L. Allgower, Phillip H. Schmidt, "An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold", ''SIAM Journal on Numerical Analysis'', Volume 22, Number 2, 322-346, April 1985.〕 The algorithm for drawing contours is a simplicial continuation algorithm, and since it is easy to visualize, it serves as a good introduction to the algorithm. == Contour plotting == The contour plotting problem is to find the zeros (contours) of $f(x,y)=0\backslash ,$ ($f(\backslash cdot)\backslash ,$ a smooth scalar valued function) in the square $0\backslash leq\; x\; \backslash leq\; 1,\; 0\backslash leq\; y\; \backslash leq\; 1\backslash ,$, The square is divided into small triangles, usually by introducing points at the corners of a regular square mesh $ih\_x\backslash leq\; x\backslash leq\; (i+1)h\_x\backslash ,$, $jh\_y\backslash leq\; y\; \backslash leq\; (j+1)h\_y\backslash ,$, making a table of the values of $f(x\_i,y\_j)\backslash ,$ at each corner $(i,j)\backslash ,$, and then dividing each square into two triangles. The value of $f(x\_i,y\_j)\backslash ,$ at the corners of the triangle defines a unique Piecewise Linear interpolant $lf(x,y)\backslash ,$ to $f(\backslash cdot)\backslash ,$ over each triangle. One way of writing this interpolant on the triangle with corners $(x\_0,y\_0),~(x\_1,y\_1),~(x\_2,y\_2)\backslash ,$ is as the set of equations : $(x,y)\; =\; (x\_0,y\_0)+(x\_1-x\_0,y\_1-y\_0)s+(x\_2-x\_0,y\_2-y\_0)t\backslash ,$ : $0\backslash leq\; s\backslash ,$ : $0\backslash leq\; t\backslash ,$ : $s+t\; \backslash leq\; 1\backslash ,$ : $lf(x,y)\; =\; f(x\_0,y\_0)+(f(x\_1,y\_1)-f(x\_0,y\_0))s+(f(x\_2,y\_2)-f(x\_0,y\_0))t\backslash ,$ The first four equations can be solved for $(s,t)\backslash ,$ (this maps the original triangle to a right unit triangle), then the remaining equation gives the interpolated value of $f(\backslash cdot)\backslash ,$. Over the whole mesh of triangles, this piecewise linear interpolant is continuous. The contour of the interpolant on an individual triangle is a line segment (it is an interval on the intersection of two planes). The equation for the line can be found, however the points where the line crosses the edges of the triangle are the endpoints of the line segment. The contour of the piecewise linear interpolant is a set of curves made up of these line segments. Any point on the edge connecting $(x\_0,y\_0)\backslash ,$ and $(x\_1,y\_1)\backslash ,$ can be written as : $(x,y)\; =\; (x\_0,y\_0)\; +\; t\; (x\_1-x\_0,y\_1-y\_0),\backslash ,$ with $t\backslash ,$ in $(0,1)\backslash ,$, and the linear interpolant over the edge is : $f\; \backslash sim\; f\_0\; +\; t\; (f\_1-f\_0)\backslash ,$ So setting $f\; =\; 0\backslash ,$ : $t\; =\; -f\_0/(f\_1-f\_0)\backslash ,$ and $(x,y)\; =\; (x\_0,y\_0)-f\_0$ *(x_1-x_0,y_1-y_0)/(f_1-f_0)\, Since this only depends on values on the edge, every triangle which shares this edge will produce the same point, so the contour will be continuous. Each triangle can be tested independently, and if all are checked the entire set of contour curves can be found. 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「Piecewise linear continuation」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.