减少排梯队形式答案

【问题标题】：Reduced row echelon form减少排梯队形式
【发布时间】：2011-03-08 19:01:19
【问题描述】：

R 中是否有生成矩阵的reduced row echelon form 的函数？ This 参考说没有。你同意吗？

【问题讨论】：

【解决方案1】：

看起来没有内置的，但我在this 页面上找到了这个 rref 函数。

 rref <- function(A, tol=sqrt(.Machine$double.eps),verbose=FALSE,
                 fractions=FALSE){
  ## A: coefficient matrix
  ## tol: tolerance for checking for 0 pivot
  ## verbose: if TRUE, print intermediate steps
  ## fractions: try to express nonintegers as rational numbers
  ## Written by John Fox
  if (fractions) {
    mass <- require(MASS)
    if (!mass) stop("fractions=TRUE needs MASS package")
  }
  if ((!is.matrix(A)) || (!is.numeric(A)))
    stop("argument must be a numeric matrix")
  n <- nrow(A)
  m <- ncol(A)
  for (i in 1:min(c(m, n))){
    col <- A[,i]
    col[1:n < i] <- 0
    # find maximum pivot in current column at or below current row
    which <- which.max(abs(col))
    pivot <- A[which, i]
    if (abs(pivot) <= tol) next     # check for 0 pivot
    if (which > i) A[c(i, which),] <- A[c(which, i),]  # exchange rows
    A[i,] <- A[i,]/pivot            # pivot
    row <- A[i,]
    A <- A - outer(A[,i], row)      # sweep
    A[i,] <- row                    # restore current row
    if (verbose)
      if (fractions) print(fractions(A))
      else print(round(A,round(abs(log(tol,10)))))
  }
  for (i in 1:n)
    if (max(abs(A[i,1:m])) <= tol)
      A[c(i,n),] <- A[c(n,i),] # 0 rows to bottom
  if (fractions) fractions (A)
  else round(A, round(abs(log(tol,10))))
}

【讨论】：

【解决方案2】：

pracma 包还包含一个实现。见 pracma::rref。

【讨论】：

【解决方案3】：

我没有足够的代表发表评论，但士兵.moth 在接受的答案 [编辑 2018: 不再接受的答案] 中给出的 ~~above~~ 函数是错误的 - 它没有处理 RREF 解的主对角线上为零的矩阵。试试例如

并注意输出不在 RREF 中。

我认为我可以正常工作，但您可能想自己检查输出：

rref <- function(A, tol=sqrt(.Machine$double.eps),verbose=FALSE,
                 fractions=FALSE){
  ## A: coefficient matrix
  ## tol: tolerance for checking for 0 pivot
  ## verbose: if TRUE, print intermediate steps
  ## fractions: try to express nonintegers as rational numbers
  ## Written by John Fox
  # Modified by Geoffrey Brent 2014-12-17 to fix a bug
  if (fractions) {
    mass <- require(MASS)
    if (!mass) stop("fractions=TRUE needs MASS package")
  }
  if ((!is.matrix(A)) || (!is.numeric(A)))
    stop("argument must be a numeric matrix")
  n <- nrow(A)
  m <- ncol(A)
  x.position<-1
  y.position<-1
  # change loop:
  while((x.position<=m) & (y.position<=n)){
    col <- A[,x.position]
    col[1:n < y.position] <- 0
    # find maximum pivot in current column at or below current row
    which <- which.max(abs(col))
    pivot <- col[which]
    if (abs(pivot) <= tol) x.position<-x.position+1     # check for 0 pivot
    else{
      if (which > y.position) { A[c(y.position,which),]<-A[c(which,y.position),] } # exchange rows
      A[y.position,]<-A[y.position,]/pivot # pivot
      row <-A[y.position,]
      A <- A - outer(A[,x.position],row) # sweep
      A[y.position,]<-row # restore current row
      if (verbose)
        if (fractions) print(fractions(A))
        else print(round(A,round(abs(log(tol,10)))))
      x.position<-x.position+1
      y.position<-y.position+1
    }
  }
  for (i in 1:n)
    if (max(abs(A[i,1:m])) <= tol)
      A[c(i,n),] <- A[c(n,i),] # 0 rows to bottom
  if (fractions) fractions (A)
  else round(A, round(abs(log(tol,10))))
}

【讨论】：

这没有提供问题的答案。要批评或要求作者澄清，请在他们的帖子下方发表评论 - 您可以随时评论自己的帖子，一旦您有足够的reputation，您就可以comment on any post。
对不起，我是新来的，可能遗漏了一些东西，但是：上面由士兵.moth 提供的“接受的答案”是错误的（当我自己尝试使用它时发现了困难的方法！）所以我认为标记它很重要。我没有足够的代表直接评论士兵.moth 的答案，所以我创建了一个新答案 - 我应该在这里做什么？
获得足够的声誉以先发表评论？
耸耸肩我真的不认为有人会因为我指出一个不明显的错误而烦恼，该错误已经四年没有更正，并且考虑到错误的“解决方案” 上面发布的内容被接受为答案，我很困惑为什么认为相同代码的固定版本不那么重要。
@GeoffreyBrent，欢迎使用 Stackoverflow。规则取代常识。

【解决方案4】：

最近还开发了一个用于教学线性代数的软件包 (matlib)，它既可以计算矩阵的梯形形式，也可以显示沿途使用的步骤。

来自reference docs的示例：

library('matlib')
A <- matrix(c(2, 1, -1,-3, -1, 2,-2, 1, 2), 3, 3, byrow=TRUE)
b <- c(8, -11, -3)
echelon(A, b, verbose=TRUE, fractions=TRUE)

Initial matrix:
     [,1] [,2] [,3] [,4]
[1,]   2    1   -1    8 
[2,]  -3   -1    2  -11 
[3,]  -2    1    2   -3 

row: 1 

 exchange rows 1 and 2 
     [,1] [,2] [,3] [,4]
[1,]  -3   -1    2  -11 
[2,]   2    1   -1    8 
[3,]  -2    1    2   -3 

 multiply row 1 by -1/3 
     [,1] [,2] [,3] [,4]
[1,]    1  1/3 -2/3 11/3
[2,]    2    1   -1    8
[3,]   -2    1    2   -3

 multiply row 1 by 2 and subtract from row 2 
     [,1] [,2] [,3] [,4]
[1,]    1  1/3 -2/3 11/3
[2,]    0  1/3  1/3  2/3
[3,]   -2    1    2   -3

 multiply row 1 by 2 and add to row 3 
     [,1] [,2] [,3] [,4]
[1,]    1  1/3 -2/3 11/3
[2,]    0  1/3  1/3  2/3
[3,]    0  5/3  2/3 13/3

row: 2 

 exchange rows 2 and 3 
     [,1] [,2] [,3] [,4]
[1,]    1  1/3 -2/3 11/3
[2,]    0  5/3  2/3 13/3
[3,]    0  1/3  1/3  2/3

 multiply row 2 by 3/5 
     [,1] [,2] [,3] [,4]
[1,]    1  1/3 -2/3 11/3
[2,]    0    1  2/5 13/5
[3,]    0  1/3  1/3  2/3

 multiply row 2 by 1/3 and subtract from row 1 
     [,1] [,2] [,3] [,4]
[1,]    1    0 -4/5 14/5
[2,]    0    1  2/5 13/5
[3,]    0  1/3  1/3  2/3

 multiply row 2 by 1/3 and subtract from row 3 
     [,1] [,2] [,3] [,4]
[1,]    1    0 -4/5 14/5
[2,]    0    1  2/5 13/5
[3,]    0    0  1/5 -1/5

row: 3 

 multiply row 3 by 5 
     [,1] [,2] [,3] [,4]
[1,]    1    0 -4/5 14/5
[2,]    0    1  2/5 13/5
[3,]    0    0    1   -1

 multiply row 3 by 4/5 and add to row 1 
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    2
[2,]    0    1  2/5 13/5
[3,]    0    0    1   -1

 multiply row 3 by 2/5 and subtract from row 2 
     [,1] [,2] [,3] [,4]
[1,]  1    0    0    2  
[2,]  0    1    0    3  
[3,]  0    0    1   -1

【讨论】：