#
# Linear Equation (without pivoting)
#
def gj_pivot(x)
  row=x.size 
  col=row+1
  pivot=Array.new(row){|i| i}
  a=Array.new(row){|i| x[i].dup}
  for k in 0..(row-1)
    # find absolute-maximal coefficient 
    # among a[k+1][k]..a[row-1][k]
    max = k
    for i in k+1..row-1
      if a[pivot[i]][k].abs > a[pivot[max]][k].abs
        max = i
      end
    end
    # swap pivot[k] and pivot[max]
    tmp=pivot[max]
    pivot[max]=pivot[k]
    pivot[k]=tmp
    # normalize a[k][i] s.t. a[k][k]=1
    akk = a[pivot[k]][k]
    for i in 0..(col-1)
      a[pivot[k]][i]=a[pivot[k]][i]/akk
    end
    # adjust i-th row s.t. a[i][k]=0,  skipping k-th row
    
    for i in 0..(row-1)
      if i != k
        aik = a[pivot[i]][k]
        for j in k..(col-1)
          a[pivot[i]][j] = a[pivot[i]][j] - aik * a[pivot[k]][j]
        end
      end
    end
  end
  b=Array.new(row)
  for i in 0..(row-1)
    b[i]=a[pivot[i]][row]
  end
  b
end