翻訳と辞書 Words near each other ・ System Management Unit ・ System manager ・ System Manager (HP LX) ・ System migration ・ System model ・ System Monitor ・ System monitor ・ System monitoring ・ System N2 ・ System Object Model (file format) ・ System of 5 ・ System of a Down ・ System of a Down (album) ・ System of a Down discography ・ System of a Down Reunion Tour ・ System of bilinear equations ・ System of concepts to support continuity of care ・ System of Cooperation Among the American Air Forces ・ System of Environmental and Economic Accounting for Water ・ System of Government under the Holy Prophet ・ System of imprimitivity ・ System of Integrated Environmental and Economic Accounting ・ System of linear equations ・ System of measurement ・ System of National Accounts ・ System of parameters ・ System of polynomial equations ・ System of record ・ System of Rice Intensification ・ System of Survival Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク System of bilinear equations ： ウィキペディア英語版 System of bilinear equations look like the followingy^TA_ix=g_i for i=1,2,\ldots,r for some integer r where A_i are matrices and g_i are some real numbers. These arise in many subjects like engineering, biology, statistics etc.==Solving in integers==We consider here the solution theory for bilinear equations in integers. Let the given system of bilinear equation be:\begin ax_1x_2+bx_1y_2+cx_2y_1+dy_1y_2&=&\alpha\\ ex_1x_2+fx_1y_2+gx_2y_1+hy_1y_2&=&\beta\endThis system can be written as: \begina&b&c&d\\e&f&g&h\end\beginx_1x_2\\x_1y_2\\y_1x_2\\y_1y_2\end=\begin\alpha\\\beta\endOnce we solve this linear system of equations then by using rank factorization below, we can get a solution for the given bilinear system.: mat(\beginx_1x_2\\x_1y_2\\y_1x_2\\y_1y_2\end)=\beginx_1x_2&x_1y_2\\y_1x_2&y_1y_2\end=\beginx_1\\y_1\end\beginx_2&y_2\endNow we solve first equation by using smith normal form, given any m\times n matrix A, we can get two matrices U and V in \mbox_m(\mathbb) and \mbox_n(\mathbb), respectively such that UAV=D, where D is as follows:: D=\begind_1&0&0&\ldots&0\\0&d_2&0&\ldots&0\\\vdots&&&d_s&0&\\0&0&0&\ldots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\end_where d_i>0 and d_i|d_ for i=1,2,\ldots,s-1. It is immediate to note that given a system A\textbf=\textbf, we can rewrite it as D\textbf=\textbf, where V\textbf=\textbf and \textbf=U\textbf. Solving D\textbf=\textbf is easier as the matrix D is somewhat diagonal. Since we are multiplying with some nonsingular matrices we have the two system of equations to be equivalent in the sense that the solutions of one system have one to one correspondence with the solutions of another system. We solve D\textbf=\textbf, and take \textbf=V\textbf.Let the solution of D\textbf=\textbf is: \textbf=\begina_1\\b_1\\s\\t\endwhere s,t\in\mathbb are free integers and these are all solutions of D\textbf=\textbf. So, any solution of A\textbf=\textbf is V\textbf. Let V be given by: V=\begina_&a_&a_&a_\\a_&a_&a_&a_\\a_&a_&a_&a_\\a_&a_&a_&a_\end=\beginA_1&B_1\\C_1&D_1\endThen \textbf is: M=mat(\textbf)=\begina_a_1+a_b_1+a_s+a_t&a_a_1+a_b_1+a_s+a_t\\a_a_1+a_b_1+a_s+a_t&a_a_1+a_b_1+a_s+a_t\endWe want matrix M to have rank 1 so that the factorization given in second equation can be done. Solving quadratic equations in 2 variables in integers will give us the solutions for a bilinear systems. This method can be extended to any dimension, but at higher dimension solutions become more complicated. This algorithm can be applied in Sage or Matlab to get to the equations at end. System of bilinear equations look like the following $y^TA\_ix=g\_i$ for $i=1,2,\backslash ldots,r$ for some integer $r$ where $A\_i$ are matrices and $g\_i$ are some real numbers. These arise in many subjects like engineering, biology, statistics etc. ==Solving in integers== We consider here the solution theory for bilinear equations in integers. Let the given system of bilinear equation be :$\backslash begin$ ax_1x_2+bx_1y_2+cx_2y_1+dy_1y_2&=&\alpha\\ ex_1x_2+fx_1y_2+gx_2y_1+hy_1y_2&=&\beta \end This system can be written as :$$ \begina&b&c&d\\e&f&g&h\end\beginx_1x_2\\x_1y_2\\y_1x_2\\y_1y_2\end=\begin\alpha\\\beta\end Once we solve this linear system of equations then by using rank factorization below, we can get a solution for the given bilinear system. :$$ mat(\beginx_1x_2\\x_1y_2\\y_1x_2\\y_1y_2\end)=\beginx_1x_2&x_1y_2\\y_1x_2&y_1y_2\end=\beginx_1\\y_1\end\beginx_2&y_2\end Now we solve first equation by using smith normal form, given any $m\backslash times\; n$ matrix $A$, we can get two matrices $U$ and $V$ in $\backslash mbox\_m(\backslash mathbb)$ and $\backslash mbox\_n(\backslash mathbb)$, respectively such that $UAV=D$, where $D$ is as follows: :$$ D=\begind_1&0&0&\ldots&0\\0&d_2&0&\ldots&0\\\vdots&&&d_s&0&\\0&0&0&\ldots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\end_ where $d\_i>0$ and $d\_i|d\_$ for $i=1,2,\backslash ldots,s-1$. It is immediate to note that given a system $A\backslash textbf=\backslash textbf$, we can rewrite it as $D\backslash textbf=\backslash textbf$, where $V\backslash textbf=\backslash textbf$ and $\backslash textbf=U\backslash textbf$. Solving $D\backslash textbf=\backslash textbf$ is easier as the matrix $D$ is somewhat diagonal. Since we are multiplying with some nonsingular matrices we have the two system of equations to be equivalent in the sense that the solutions of one system have one to one correspondence with the solutions of another system. We solve $D\backslash textbf=\backslash textbf$, and take $\backslash textbf=V\backslash textbf$. Let the solution of $D\backslash textbf=\backslash textbf$ is :$$ \textbf=\begina_1\\b_1\\s\\t\end where $s,t\backslash in\backslash mathbb$ are free integers and these are all solutions of $D\backslash textbf=\backslash textbf$. So, any solution of $A\backslash textbf=\backslash textbf$ is $V\backslash textbf$. Let $V$ be given by :$$ V=\begina_&a_&a_&a_\\a_&a_&a_&a_\\a_&a_&a_&a_\\a_&a_&a_&a_\end=\beginA_1&B_1\\C_1&D_1\end Then $\backslash textbf$ is :$$ M=mat(\textbf)=\begina_a_1+a_b_1+a_s+a_t&a_a_1+a_b_1+a_s+a_t\\a_a_1+a_b_1+a_s+a_t&a_a_1+a_b_1+a_s+a_t\end We want matrix $M$ to have rank 1 so that the factorization given in second equation can be done. Solving quadratic equations in 2 variables in integers will give us the solutions for a bilinear systems. This method can be extended to any dimension, but at higher dimension solutions become more complicated. This algorithm can be applied in Sage or Matlab to get to the equations at end. 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「System of bilinear equations look like the followingy^TA_ix=g_i for i=1,2,\ldots,r for some integer r where A_i are matrices and g_i are some real numbers. These arise in many subjects like engineering, biology, statistics etc.==Solving in integers==We consider here the solution theory for bilinear equations in integers. Let the given system of bilinear equation be:\begin ax_1x_2+bx_1y_2+cx_2y_1+dy_1y_2&=&\alpha\\ ex_1x_2+fx_1y_2+gx_2y_1+hy_1y_2&=&\beta\endThis system can be written as: \begina&b&c&d\\e&f&g&h\end\beginx_1x_2\\x_1y_2\\y_1x_2\\y_1y_2\end=\begin\alpha\\\beta\endOnce we solve this linear system of equations then by using rank factorization below, we can get a solution for the given bilinear system.: mat(\beginx_1x_2\\x_1y_2\\y_1x_2\\y_1y_2\end)=\beginx_1x_2&x_1y_2\\y_1x_2&y_1y_2\end=\beginx_1\\y_1\end\beginx_2&y_2\endNow we solve first equation by using smith normal form, given any m\times n matrix A, we can get two matrices U and V in \mbox_m(\mathbb) and \mbox_n(\mathbb), respectively such that UAV=D, where D is as follows:: D=\begind_1&0&0&\ldots&0\\0&d_2&0&\ldots&0\\\vdots&&&d_s&0&\\0&0&0&\ldots&0\\\vdots&\vdots&\vdots&\vdots&\vdots\end_where d_i>0 and d_i|d_ for i=1,2,\ldots,s-1. It is immediate to note that given a system A\textbf=\textbf, we can rewrite it as D\textbf=\textbf, where V\textbf=\textbf and \textbf=U\textbf. Solving D\textbf=\textbf is easier as the matrix D is somewhat diagonal. Since we are multiplying with some nonsingular matrices we have the two system of equations to be equivalent in the sense that the solutions of one system have one to one correspondence with the solutions of another system. We solve D\textbf=\textbf, and take \textbf=V\textbf.Let the solution of D\textbf=\textbf is: \textbf=\begina_1\\b_1\\s\\t\endwhere s,t\in\mathbb are free integers and these are all solutions of D\textbf=\textbf. So, any solution of A\textbf=\textbf is V\textbf. Let V be given by: V=\begina_&a_&a_&a_\\a_&a_&a_&a_\\a_&a_&a_&a_\\a_&a_&a_&a_\end=\beginA_1&B_1\\C_1&D_1\endThen \textbf is: M=mat(\textbf)=\begina_a_1+a_b_1+a_s+a_t&a_a_1+a_b_1+a_s+a_t\\a_a_1+a_b_1+a_s+a_t&a_a_1+a_b_1+a_s+a_t\endWe want matrix M to have rank 1 so that the factorization given in second equation can be done. Solving quadratic equations in 2 variables in integers will give us the solutions for a bilinear systems. This method can be extended to any dimension, but at higher dimension solutions become more complicated. This algorithm can be applied in Sage or Matlab to get to the equations at end.」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.