Leo Zhang

A simple man with my own ideal

Cholesky分解

1、为什么要进行矩阵分解

      个人认为,首先,当数据量很大时,将一个矩阵分解为若干个矩阵的乘积可以大大降低存储空间;其次,可以减少真正进行问题处理时的计算量,毕竟算法扫描的元素越少完成任务的速度越快,这个时候矩阵的分解是对数据的一个预处理;再次,矩阵分解可以高效和有效的解决某些问题;最后,矩阵分解可以提高算法数值稳定性,关于这一点可以有进一步的说明,

借用一个上学时老师给的例子:

有方程组:

                 \begin{equation}

\begin{cases}

5x_1+7x_2=0.7\\

7x_1+10x_2=1

\end{cases}

\end{equation}

令             A= 
\begin{pmatrix} 
5 & 7 \\ 
7 & 10 
\end{pmatrix},                 x=
\begin{pmatrix} 
x_1 \\ 
x_2 
\end{pmatrix},                 b=
\begin{pmatrix} 
0.7\\ 
1
\end{pmatrix}

解方程组可得:

                 x= 
\begin{pmatrix} 
0.0 \\ 
0.1
\end{pmatrix}

现在对b进行微小扰动:

                 b= 
\begin{pmatrix} 
0.69 \\ 
1.01
\end{pmatrix},扰动项为:
\begin{pmatrix} 
-0.01 \\ 
0.01 
\end{pmatrix}

此时相应的解为:

                 x= 
\begin{pmatrix} 
-0.17 \\ 
0.22 
\end{pmatrix}

        这个例子说明,当方程组常数项发生微小变动的时候会导致求出的结果差别相当大,而导致这种差别的并不是求解方法,而是方程组系数矩阵本身的问题,这会给我们解决问题带来很大危害,例如,我们在用计算机求解这类问题时难以避免在计算当中出现舍入误差,如果矩阵本身性质不好会直接导致所答非所问。

        对常数向量b和矩阵A进行一个简单的扰动分析:

        1)、扰动b,原方程组为:

                 Ax=b (式子1),(b\neq0,A非奇异)

扰动后为:

                 A(x+\delta x)=b+\delta b(式子2)

把式子1带入式子2得:A\delta x=\delta b,用2-范式来衡量这种变化得:||A\delta x|| =||\delta b||,由于||Ax||\leq ||A||||x||,于是得到:

                 \frac{||\delta x||}{||x||}=\frac{||A^{-1}\delta b||}{||x||}\leq \frac{||A^{-1}||||\delta b||}{||x||}

而利用式子1同理可得||x||\geq\frac{||b||}{||A||},整理后得:

                             \frac{||\delta x||}{||x||}\leq \frac{||A||\quad ||A^{-1}||\quad ||\delta b||}{||b||},可见b的扰动对解的影响由||A||\quad ||A^{-1}||决定。

        2)、扰动A,扰动后为:

                (A+\delta A)(x+\delta x)=b(式子3),(b\neq0,A非奇异)

稍微做一下变换:

                A(x+\delta x)+\delta Ax+\delta A\delta x=b

把式子1带入后得到:

                \delta x=A^{-1}(-\delta Ax-\delta A\delta x)

对两边同时取2-范式有:

                 ||\delta x||=||A^{-1}(-\delta Ax-\delta A\delta x)|| \leq ||A^{-1}||\quad ||\delta Ax|| \quad ||\delta A|| \quad ||\delta x||

于是有:

                 ||\delta x||=||A^{-1}(-\delta Ax-\delta A\delta x)|| \leq ||A^{-1}||\quad ||\delta A||\quad||x|| \quad ||\delta A|| \quad ||\delta x||,整理一下就是:

                 
\frac{||\delta x||}{||x||}\leq \frac{||A^{-1}|| \quad ||A||\frac{||\delta A||}{||A||}}{1-||A^{-1}|| \quad ||A||\frac{||\delta A||}{||A||}},A的扰动对解的影响依然是由||A||\quad ||A^{-1}||决定。

        3)、对于同时扰动A和b的情况偶就不推了,最后的结果依然是,扰动对解的影响依然由||A||\quad ||A^{-1}||决定。

        定义矩阵的条件数cond(A)=||A||||A^{-1}||来描述矩阵的病态程度,一般认为条件数小于100为良态,条件数在100到1000之间为中等程度的病态,条件数超过1000存在严重病态。以上面的矩阵A为例,采用2-范数表示的条件数为:cond(A)=222.9955,看来矩阵处于中等病态程度。

        矩阵其实就是一个给定的线性变换,特征向量描述了这个线性变换的主要方向,而特征值描述了一个特征向量的长度在该线性变换下缩放的比例,有关特征值和特征向量的相关概念可查看http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors,对开篇的例子进一步观察发现,A是个对称正定矩阵,A的特征值分别为\lambda_1:14.93303437 和\lambda_2:0.06696563,两个特征值在数量级上相差很大,这意味着b发生扰动时,向量x在这两个特征向量方向上的移动距离是相差很大的——对于\lambda_1对应的特征向量只需要微小的移动就可到达b的新值,而对于\lambda_2,由于它比起\lambda_1太小了,因此需要x做大幅度移动才能到达b的新值,于是悲剧就发生了……………..。

        关于矩阵可以有以下各种分解方式,①矩阵的三角分解(Cholesky分解、LU分解等),②矩阵的正交三角分解(QR分解等),③矩阵的满秩分解,④矩阵的奇异值分解(SVD)(关于SVD可以查看高人LeftNotEasyhttp://www.cnblogs.com/LeftNotEasy/archive/2011/01/19/svd-and-applications.html#2038925一文以及他提供的参考资料)。

        再看矩阵A,它是个对称正定矩阵,对这种矩阵都可以进行Cholesky分解,也就是将矩阵A分解为:A=L\quad L^T,其中L为一个下三角矩阵,具体操作随后讨论,回头看方程组Ax=b,它就变成了:

                 Ax=L\quad L^T\quad x=b

将它看成两个方程组:

                  Ly=bL^Tx=y,其中:L=
\begin{pmatrix}
2.236068 & 0.000000\\
3.1304952 & 0.4472136

\end{pmatrix},特征值为\lambda_1:2.2360680和\lambda_2:0.4472136

此时采用2-范数表示的条件数为cond(L)=5,显然上面这两个方程组也都是良态的且只需要存储矩阵L的下三角部分即可,矩阵分解的优点可见一斑。

2、实正定Hermit矩阵的完全Cholesky分解

      设矩阵A有如下形式:

                                        A=
\begin{pmatrix}
a_{1,1} & b_1^*\\
b_1&B^1\\
\end{pmatrix}

      借用http://en.wikipedia.org/wiki/Cholesky_decomposition中的推导:

A^{(1)}:=A,在第i次迭代时有:

                                       A^{(i)}=
\begin{pmatrix}
I_{i-1}&0&0\\
0&a_{i,i}&b_i^*\\
0&b_i&B^{(i)}
\end{pmatrix},其中I_{i-1}为i-1维单位矩阵

定义矩阵L_i

                                       L_{i}=
\begin{pmatrix}
I_{i-1}&0&0\\
0&\sqrt{a_{i,i}}&0\\
0&\frac{b_i}{\sqrt{a_{i,i}}}&I_{n-i}
\end{pmatrix}

于是要满足A^{(i)}=L_iA^{(i+1)}L^*_i,就有:

                                       A^{(i+1)}=
\begin{pmatrix}
I_{i-1}&0&0\\
0&1&0\\
0&0&B^{(i)}-\frac{b_i\quad b_i^*}{a_{i,i}}
\end{pmatrix}

这样一直迭代下去,直到A^{(i+1)}=I,也就是有:A=L_1\quad  L_2\quad L_3\quad ...L_n\quad L_n^*\quad  ...L_3^*\quad  L_2^*\quad  L_1^*\quad =L\quad L^*,最后得到分解后的下三角矩阵L的元素:

                                       L_{j,j}=\sqrt{A_{j,j}-\sum\limit_{k=1}^{j-1}L_{j,k}^2}

                                      L_{i,j}=\frac{A_{i,j}-{\sum\limit_{k=1}^{j-1}{L_{i,k}\quad L_{j,k}}}}{L_{j,j}}      i>j

当然L也可以是上三角矩阵,此时

                                       L_{i}^*=
\begin{pmatrix}
I_{i-1}&0&0\\
0&\sqrt{a_{i,i}}&0\\
0&\frac{b_i}{\sqrt{a_{i,i}}}&I_{n-i}
\end{pmatrix}

                                       A=L_1^*\quad L_2^*\quad L_3^*\quad ...L_n^*\quad L_n\quad  ...L_3\quad  L_2\quad  L_1\quad =L^*\quad L

所以L的元素是:

                                       L_{j,j}=\sqrt{A_{j,j}-\sum\limit_{k=1}^{j-1}L_{j,k}^2}

                                       L_{i,j}=\frac{A_{i,j}-{\sum\limit_{k=1}^{j-1}{L_{k,i}\quad L_{k,j}}}}{L_{i,i}}   j>i

 

一种比较直白的分解算法可以是这样的:

设有实对称矩阵:

     A=(a_{i,j}) \quad \quad \quad \quad 1 \leq i,j \leq n

以及向量w和保存分解结果的矩阵L,算法伪码描述如下:

   1: for i:=1 to n do                                        //逐行计算L的值,w向量初始状态为实hermit矩阵A的上三角部分的一行
   2:     w(i .. n) := a(i,i .. n);
   3:     for k:=1 to i-1 do                                  
   4:         temp := L(k,i);
   5:         if(temp != 0) then
   6:             w(i .. n) := w(i .. n) - temp*L(k,i .. n);  //计算L(i,j)公式的分子部分
   7:         endif;
   8:     endfor;    
   9:     w(i) := sqrt(w(i));                                 //L矩阵对角线元素计算
  10:     w(i+1 .. n) := w(i+1 .. n) / w(i);                  //L矩阵非对角线元素计算
  11:  
  12:     L(i,i .. n) := w(i .. n);                           //更新L矩阵
  13:     w(i .. n) :=0;                                      //清空w向量
  14: endfor;

 

一个简单的实现如下:

   1: //串行完全Chlolesky分解,时间复杂度(O(n^3))
   2: double **complete_cholesky_decompose(double **A)
   3: {
   4:     if(NULL==A)
   5:         return NULL;
   6:  
   7:     double **L=malloc_matrix();
   8:     clear_matrix(L);
   9:  
  10:     int i,j,k,m;
  11:  
  12:     double *w=malloc_vector();
  13:  
  14:     clear_vector(w);//清除向量
  15:  
  16:     for(i=0;i<LEN;i++){
  17:         for(m=i;m<LEN;m++){
  18:             w[m]=A[i][m];
  19:         }
  20:  
  21:         for(k=0;k<i;k++){
  22:             double temp=L[k][i];
  23:             if(temp!=0){
  24:                 for(m=i;m<LEN;m++){
  25:                     w[m] -= temp*L[k][m];
  26:                 }
  27:             }
  28:         }
  29:  
  30:         w[i]=sqrt(w[i]);
  31:         for(m=i+1;m<LEN;m++){
  32:             w[m] /=w[i];
  33:         }
  34:  
  35:         for(m=i;m<LEN;m++){
  36:             L[i][m]=w[m];
  37:         }
  38:  
  39:         clear_vector(w);
  40:     }
  41:  
  42:     return L;
  43: }

 

代码可以在这里下载到。

在实践中,完全Cholesky分解有时并不是必须的,为了提高效率或者有利于编写并行算法等原因,不完全Cholesky(ICF)有时更加实用,关于ICF有很多算法,还有待学习。

3、实用工具和网址

(1)、R

         我认为R是进行关于矩阵相关操作的较好工具,它是免费的和开源的(我记的是),功能不亚于matlab,当然它能做的事情远远不止矩阵计算,还可以做分类、聚类、回归、统计分析等等等等,R可以从http://www.r-project.org/获取到,同时有linux版本和windows版本。

关于矩阵的一些简单命令我稍微列一下:

1)、创建向量

数据无规律:

   1: >  x=c(0,1,3,4,6,8,9)  
   2: > x 
   3:     [1] 0 1  3 4  6  8 9

数据有简单规律(从1~3以0.1的间隔输出):

   1: > x=seq(1,3,by=0.1) 
   2: > x 
   3: [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 

数据有复杂规律(从0~5,每个数字重复2次):

   1: > x=rep(0:5,rep(2,6)) 
   2: > x 
   3: [1] 0 0 1 1 2 2 3 3 4 4 5 5

2)、创建矩阵

函数原型:

   1: > matrix
   2: function (data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL) 
   3: {
   4:     data <- as.vector(data)
   5:     if (missing(nrow)) 
   6:         nrow <- ceiling(length(data)/ncol)
   7:     else if (missing(ncol)) 
   8:         ncol <- ceiling(length(data)/nrow)
   9:     .Internal(matrix(data, nrow, ncol, byrow, dimnames))
  10: }
  11: <environment: namespace:base>

data存储矩阵数据,nrow和ncol分别为行数和列数,byrows默认为FALSE,表示矩阵数据生成时以列为主序,例如:

   1: > data=c( 
   2: + 0.0100,0.1710,0.0967,0.1661,0.1254,0.0728,0.0928,0.1272,0.1041,0.0644, 
   3: + 0.1710,3.3743,2.3896,3.5556,2.6312,1.6968,1.8502,2.6637,2.5208,1.2792, 
   4: + 0.0967,2.3896,2.5098,3.0234,2.1745,1.6661,1.5959,2.5601,2.6507,1.0512, 
   5: + 0.1661,3.5556,3.0234,4.3530,3.4144,2.5287,2.3244,3.6691,3.5036,1.6727, 
   6: + 0.1254,2.6312,2.1745,3.4144,2.9375,2.4152,1.9473,3.0911,2.9016,1.6776, 
   7: + 0.0728,1.6968,1.6661,2.5287,2.4152,2.5097,1.7161,2.8719,2.6663,2.0713, 
   8: + 0.0928,1.8502,1.5959,2.3244,1.9473,1.7161,2.0160,3.2206,2.3529,1.8967, 
   9: + 0.1272,2.6637,2.5601,3.6691,3.0911,2.8719,3.2206,5.8295,4.0840,3.4258, 
  10: + 0.1041,2.5208,2.6507,3.5036,2.9016,2.6663,2.3529,4.0840,3.8915,2.7323, 
  11: + 0.0644,1.2792,1.0512,1.6727,1.6776,2.0713,1.8967,3.4258,2.7323,4.3582 
  12: + ) 
  13: > a=matrix(data,nrow=10,ncol=10,byrow=TRUE) 
  14: > a 
  15:         [,1]   [,2]   [,3]   [,4]   [,5]   [,6]   [,7]   [,8]   [,9]  [,10] 
  16: [1,] 0.0100 0.1710 0.0967 0.1661 0.1254 0.0728 0.0928 0.1272 0.1041 0.0644 
  17: [2,] 0.1710 3.3743 2.3896 3.5556 2.6312 1.6968 1.8502 2.6637 2.5208 1.2792 
  18: [3,] 0.0967 2.3896 2.5098 3.0234 2.1745 1.6661 1.5959 2.5601 2.6507 1.0512 
  19: [4,] 0.1661 3.5556 3.0234 4.3530 3.4144 2.5287 2.3244 3.6691 3.5036 1.6727 
  20: [5,] 0.1254 2.6312 2.1745 3.4144 2.9375 2.4152 1.9473 3.0911 2.9016 1.6776 
  21: [6,] 0.0728 1.6968 1.6661 2.5287 2.4152 2.5097 1.7161 2.8719 2.6663 2.0713 
  22: [7,] 0.0928 1.8502 1.5959 2.3244 1.9473 1.7161 2.0160 3.2206 2.3529 1.8967 
  23: [8,] 0.1272 2.6637 2.5601 3.6691 3.0911 2.8719 3.2206 5.8295 4.0840 3.4258 
  24: [9,] 0.1041 2.5208 2.6507 3.5036 2.9016 2.6663 2.3529 4.0840 3.8915 2.7323 
  25: [10,] 0.0644 1.2792 1.0512 1.6727 1.6776 2.0713 1.8967 3.4258 2.7323 4.3582

3)、矩阵转置

   1: > A=matrix(0:11,nrow=3,ncol=4,byrow=TRUE) 
   2: > A 
   3:      [,1] [,2] [,3] [,4] 
   4: [1,]    0    1    2    3 
   5: [2,]    4    5    6    7 
   6: [3,]    8    9   10   11 
   7: > t(A) 
   8:      [,1] [,2] [,3] 
   9: [1,]    0    4    8 
  10: [2,]    1    5    9 
  11: [3,]    2    6   10 
  12: [4,]    3    7   11 

4)、矩阵加减

   1: > A=matrix(0:11,nrow=3,ncol=4,byrow=TRUE) 
   2: > B=matrix(1:12,nrow=3,ncol=4,byrow=TRUE) 
   3: > B-A 
   4:      [,1] [,2] [,3] [,4] 
   5: [1,]    1    1    1    1 
   6: [2,]    1    1    1    1 
   7: [3,]    1    1    1    1 
   8: > B+A 
   9:      [,1] [,2] [,3] [,4] 
  10: [1,]    1    3    5    7 
  11: [2,]    9   11   13   15 
  12: [3,]   17   19   21   23 

5)、矩阵相乘

   1: > A=matrix(0:11,nrow=3,ncol=4,byrow=TRUE) 
   2: > B=matrix(1:12,nrow=4,ncol=3,byrow=TRUE) 
   3: > A%*%B 
   4:      [,1] [,2] [,3] 
   5: [1,]   48   54   60 
   6: [2,]  136  158  180 
   7: [3,]  224  262  300 

6)、取方阵对角元素

   1: > A=matrix(0:8,nrow=3,ncol=3,byrow=TRUE) 
   2: > A 
   3:      [,1] [,2] [,3] 
   4: [1,]    0    1    2 
   5: [2,]    3    4    5 
   6: [3,]    6    7    8 
   7: > diag(A) 
   8: [1] 0 4 8 

7)、矩阵求逆

   1: > A=matrix(c(1,2,3,0,4,5,0,0,6),nrow=3,ncol=3,byrow=TRUE) 
   2: > solve(A) 
   3:      [,1]  [,2]        [,3] 
   4: [1,]    1 -0.50 -0.08333333 
   5: [2,]    0  0.25 -0.20833333 
   6: [3,]    0  0.00  0.16666667 

8)、特征值与特征向量

   1: > A=matrix(c(1,2,3,0,4,5,0,0,6),nrow=3,ncol=3,byrow=TRUE) 
   2: > eigen(A) 
   3: $values 
   4: [1] 6 4 1
   5:  
   6: $vectors 
   7:           [,1]      [,2] [,3] 
   8: [1,] 0.5108407 0.5547002    1 
   9: [2,] 0.7981886 0.8320503    0 
  10: [3,] 0.3192754 0.0000000    0 
  11:  

9)、正定hermit矩阵的完全Cholesky分解

   1: > data=c( 
   2: + 0.0100,0.1710,0.0967,0.1661,0.1254,0.0728,0.0928,0.1272,0.1041,0.0644, 
   3: + 0.1710,3.3743,2.3896,3.5556,2.6312,1.6968,1.8502,2.6637,2.5208,1.2792, 
   4: + 0.0967,2.3896,2.5098,3.0234,2.1745,1.6661,1.5959,2.5601,2.6507,1.0512, 
   5: + 0.1661,3.5556,3.0234,4.3530,3.4144,2.5287,2.3244,3.6691,3.5036,1.6727, 
   6: + 0.1254,2.6312,2.1745,3.4144,2.9375,2.4152,1.9473,3.0911,2.9016,1.6776, 
   7: + 0.0728,1.6968,1.6661,2.5287,2.4152,2.5097,1.7161,2.8719,2.6663,2.0713, 
   8: + 0.0928,1.8502,1.5959,2.3244,1.9473,1.7161,2.0160,3.2206,2.3529,1.8967, 
   9: + 0.1272,2.6637,2.5601,3.6691,3.0911,2.8719,3.2206,5.8295,4.0840,3.4258, 
  10: + 0.1041,2.5208,2.6507,3.5036,2.9016,2.6663,2.3529,4.0840,3.8915,2.7323, 
  11: + 0.0644,1.2792,1.0512,1.6727,1.6776,2.0713,1.8967,3.4258,2.7323,4.3582 
  12: + ) 
  13: > a=matrix(data,nrow=10,ncol=10,byrow=TRUE) 
  14: > chol(a) 
  15:       [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]      [,8]      [,9]     [,10] 
  16: [1,]  0.1 1.7100000 0.9670000 1.6610000 1.2540000 0.7280000 0.9280000 1.2720000 1.0410000 0.6440000 
  17: [2,]  0.0 0.6709694 1.0969650 1.0660545 0.7256068 0.6735329 0.3924471 0.7281703 1.1039102 0.2652282 
  18: [3,]  0.0 0.0000000 0.6094086 0.4066049 0.2722586 0.3663913 0.4398089 0.8718268 0.7106926 0.2256384 
  19: [4,]  0.0 0.0000000 0.0000000 0.5406286 0.8273109 0.8369753 0.3436621 0.7871389 0.5710010 0.4226980 
  20: [5,]  0.0 0.0000000 0.0000000 0.0000000 0.2826847 0.7831198 0.3352446 0.2797313 0.4573777 0.9425268 
  21: [6,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.2793255 0.2322540 0.9241146 0.2450380 0.8923532 
  22: [7,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.7231424 1.0953043 0.3243910 0.5908204 
  23: [8,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4119291 0.4303230 0.3608438 
  24: [9,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4454546 0.8321836 
  25: [10,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8871720 

10)、矩阵奇异值分解

   1: > data=c(
   2: + 0.0100,0.1710,0.0967,0.1661,0.1254,0.0728,0.0928,0.1272,0.1041,0.0644,
   3: + 0.1710,3.3743,2.3896,3.5556,2.6312,1.6968,1.8502,2.6637,2.5208,1.2792,
   4: + 0.0967,2.3896,2.5098,3.0234,2.1745,1.6661,1.5959,2.5601,2.6507,1.0512,
   5: + 0.1661,3.5556,3.0234,4.3530,3.4144,2.5287,2.3244,3.6691,3.5036,1.6727,
   6: + 0.1254,2.6312,2.1745,3.4144,2.9375,2.4152,1.9473,3.0911,2.9016,1.6776,
   7: + 0.0728,1.6968,1.6661,2.5287,2.4152,2.5097,1.7161,2.8719,2.6663,2.0713,
   8: + 0.0928,1.8502,1.5959,2.3244,1.9473,1.7161,2.0160,3.2206,2.3529,1.8967,
   9: + 0.1272,2.6637,2.5601,3.6691,3.0911,2.8719,3.2206,5.8295,4.0840,3.4258,
  10: + 0.1041,2.5208,2.6507,3.5036,2.9016,2.6663,2.3529,4.0840,3.8915,2.7323,
  11: + 0.0644,1.2792,1.0512,1.6727,1.6776,2.0713,1.8967,3.4258,2.7323,4.3582
  12: + )
  13: > a=matrix(data,nrow=10,ncol=10,byrow=TRUE)
  14: > svd(a)
  15: $d
  16:  [1] 2.419118e+01 4.155429e+00 1.305297e+00 1.042439e+00 7.271958e-01 1.724779e-01 1.188910e-01 7.550273e-02 6.811257e-04 4.025360e-04
  17:  
  18: $u
  19:              [,1]        [,2]        [,3]          [,4]        [,5]         [,6]        [,7]        [,8]         [,9]        [,10]
  20:  [1,] -0.01423809 -0.01770852  0.01046421  0.0440060554 -0.03708011  0.003989202  0.02212364 -0.02965763  0.609656748 -0.789301225
  21:  [2,] -0.30655677 -0.38844316  0.21540302  0.5860837223 -0.20831346 -0.071321462  0.09280202 -0.53705509  0.058902791  0.127479773
  22:  [3,] -0.27500938 -0.28653235 -0.04046986 -0.0006133085  0.62767935 -0.424262650 -0.38187184  0.16164557  0.260582126  0.163683441
  23:  [4,] -0.39289768 -0.37181967  0.10218229  0.0430729100 -0.04830660  0.258172491 -0.19145885  0.37788201 -0.530454225 -0.406528598
  24:  [5,] -0.32434239 -0.18416219  0.18854340 -0.2694300561 -0.38073924  0.273128771  0.08003428  0.37039768  0.491516620  0.384701079
  25:  [6,] -0.28065826  0.08302171  0.24277466 -0.6198550162 -0.27053693 -0.402153630 -0.20235690 -0.39991126 -0.144858619 -0.119997665
  26:  [7,] -0.26573534  0.09522360 -0.29346909  0.1374724777 -0.21528535 -0.600459122  0.52653396  0.34671820 -0.089612967 -0.053976475
  27:  [8,] -0.44372190  0.30810327 -0.69462776  0.0748652314 -0.12439310  0.238340862 -0.32629727 -0.18852619  0.059652991  0.047118889
  28:  [9,] -0.38271998  0.06649679  0.03894330 -0.2531095648  0.52173933  0.307376912  0.59639765 -0.24228533 -0.036358521 -0.033403328
  29: [10,] -0.27906174  0.69225888  0.52609547  0.3276170825  0.08203487 -0.008974789 -0.14718647  0.17374370  0.008992724  0.007135959
  30:  
  31: $v
  32:              [,1]        [,2]        [,3]          [,4]        [,5]         [,6]        [,7]        [,8]         [,9]        [,10]
  33:  [1,] -0.01423809 -0.01770852  0.01046421  0.0440060554 -0.03708011  0.003989202  0.02212364 -0.02965763  0.609656748 -0.789301225
  34:  [2,] -0.30655677 -0.38844316  0.21540302  0.5860837223 -0.20831346 -0.071321462  0.09280202 -0.53705509  0.058902791  0.127479773
  35:  [3,] -0.27500938 -0.28653235 -0.04046986 -0.0006133085  0.62767935 -0.424262650 -0.38187184  0.16164557  0.260582126  0.163683441
  36:  [4,] -0.39289768 -0.37181967  0.10218229  0.0430729100 -0.04830660  0.258172491 -0.19145885  0.37788201 -0.530454225 -0.406528598
  37:  [5,] -0.32434239 -0.18416219  0.18854340 -0.2694300561 -0.38073924  0.273128771  0.08003428  0.37039768  0.491516620  0.384701079
  38:  [6,] -0.28065826  0.08302171  0.24277466 -0.6198550162 -0.27053693 -0.402153630 -0.20235690 -0.39991126 -0.144858619 -0.119997665
  39:  [7,] -0.26573534  0.09522360 -0.29346909  0.1374724777 -0.21528535 -0.600459122  0.52653396  0.34671820 -0.089612967 -0.053976475
  40:  [8,] -0.44372190  0.30810327 -0.69462776  0.0748652314 -0.12439310  0.238340862 -0.32629727 -0.18852619  0.059652991  0.047118889
  41:  [9,] -0.38271998  0.06649679  0.03894330 -0.2531095648  0.52173933  0.307376912  0.59639765 -0.24228533 -0.036358521 -0.033403328
  42: [10,] -0.27906174  0.69225888  0.52609547  0.3276170825  0.08203487 -0.008974789 -0.14718647  0.17374370  0.008992724  0.007135959

11)、矩阵QR分解

   1: > A=matrix(1:12,3,4)
   2: > qr(A)
   3: $qr
   4:            [,1]      [,2]          [,3]          [,4]
   5: [1,] -3.7416574 -8.552360 -1.336306e+01 -1.817376e+01
   6: [2,]  0.5345225  1.963961  3.927922e+00  5.891883e+00
   7: [3,]  0.8017837  0.988693  3.443426e-16 -3.439089e-16
   8:  
   9: $rank
  10: [1] 2
  11:  
  12: $qraux
  13: [1] 1.267261e+00 1.149954e+00 3.443426e-16 3.439089e-16
  14:  
  15: $pivot
  16: [1] 1 2 3 4
  17:  
  18: attr(,"class")
  19: [1] "qr"

(2)、找论文的好去处

1)、http://www.pdfgratis.com/一个非常强大的网站,能下载到各种论文;

2)、http://jeffhuang.com/best_paper_awards.html 

(3)、编写latex公式的工具

         最初是在小桥流水的博客上发现的,地址如下:http://www.cnblogs.com/youwang/archive/2010/04/04/1704099.html,对他的插件做了一点修改,增加的功能是:可以还原公式,例如:用该工具写公式, a=b+1
,选中该公式,然后单击插件,可以看到公式的latex代码,源码可以在这里下载到。

posted on 2011-03-22 15:28  Leo Zhang  阅读(19080)  评论(21编辑  收藏

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