You are an expert in writing code in many different languages. Your goal is to perform code completion for the following code, keeping in mind the rest of the code and the file meta data.
    Metadata details:
    filename: lup_decomposition.rb
    path of file: ./repos/powerstation/powerstation/lib/compiled-jruby/lib/ruby/stdlib/matrix
    
    the code of the file until where you have to start completion: class Matrix
  # Adapted from JAMA: http://math.nist.gov/javanumerics/jama/

  #
  # For an m-by-n matrix A with m >= n, the LU decomposition is an m-by-n
  # unit lower triangular matrix L, an n-by-n upper triangular matrix U,
  # and a m-by-m permutation matrix P so that L*U = P*A.
  # If m < n, then L is m-by-m and U is m-by-n.
  #
  # The LUP decomposition with pivoting always exists, even if the matrix is
  # singular, so the constructor will never fail.  The primary use of the
  # LU decomposition is in the solution of square systems of simultaneous
  # linear equations.  This will fail if singular? returns true.
  #

  class LUPDecomposition
    # Returns the lower triangular factor +L+

    include Matrix::ConversionHelper

    def l
      Matrix.build(@row_count, [@column_count, @row_count].min) do |i, j|
        if (i > j)
          @lu[i][j]
        elsif (i == j)
          1
        else
          0
        end
      end
    end

    # Returns the upper triangular factor +U+

    def u
      Matrix.build([@column_count, @row_count].min, @column_count) do |i, j|
        if (i <= j)
          @lu[i][j]
        else
          0
        e
    
    def initialize a
      raise TypeError, "Expected Matrix but got #{a.class}" unless a.is_a?(Matrix)
      # Use a "left-looking", dot-product, Crout/Doolittle algorithm.
      @lu = a.to_a
      @row_count = a.row_count
      @column_count = a.column_count
      @pivots = Array.new(@row_count)
      @row_count.times do |i|
         @pivots[i] = i
      end
      @pivot_sign = 1
      lu_col_j = Array.new(@row_count)

      # Outer loop.

      @column_count.times do |j|

        # Make a copy of the j-th column to localize references.

        @row_count.times do |i|
          lu_col_j[i] = @lu[i][j]
        end

        # Apply previous transformations.

        @row_count.times do |i|
          lu_row_i = @lu[i]

          # Most of the time is spent in the following dot product.

          kmax = [i, j].min
          s = 0
          kmax.times do |k|
            s += lu_row_i[k]*lu_col_j[k]
          end

          lu_row_i[j] = lu_col_j[i] -= s
        end

        # Find pivot and exchange if necessary.

        p = j
        (j+1).upto(@row_count-1) do |i|
          if (lu_col_j[i].abs > lu_col_j[p].abs)
            p = i
          end
        end
        if (p != j)
          @column_count.times do |k|
            t = @lu[p][k]; @lu[p][k] = @lu[j][k]; @lu[j][k] = t
          end
          k = @pivots[p]; @pivots[p] = @pivots[j]; @pivots[j] = k
          @pivot_sign = -@pivot_sign
        end

        # Compute multipliers.

        if (j < @row_count && @lu[j][j] != 0)
          (j+1).upto(@row_count-1) do |i|
            @lu[i][j] = @lu[i][j].quo(@lu[j][j])
          end
        end
      end