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.

Methods
D
L
N
P
S
T
U
Attributes
[R] pivots

Returns the pivoting indices
Class Public methods
new(a)

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 153
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_size = a.row_size
  @col_size = a.column_size
  @pivots = Array.new(@row_size)
  @row_size.times do |i|
     @pivots[i] = i
  end
  @pivot_sign = 1
  lu_col_j = Array.new(@row_size)

  # Outer loop.

  @col_size.times do |j|

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

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

    # Apply previous transformations.

    @row_size.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_size-1) do |i|
      if (lu_col_j[i].abs > lu_col_j[p].abs)
        p = i
      end
    end
    if (p != j)
      @col_size.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_size && @lu[j][j] != 0)
      (j+1).upto(@row_size-1) do |i|
        @lu[i][j] = @lu[i][j].quo(@lu[j][j])
      end
    end
  end
end
Instance Public methods
det()

Returns the determinant of A, calculated efficiently from the factorization.

Also aliased as: determinant

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 78
def det
  if (@row_size != @col_size)
    Matrix.Raise Matrix::ErrDimensionMismatch unless square?
  end
  d = @pivot_sign
  @col_size.times do |j|
    d *= @lu[j][j]
  end
  d
end
determinant()
Alias for: det
l()

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 21
def l
  Matrix.build(@row_size, @col_size) do |i, j|
    if (i > j)
      @lu[i][j]
    elsif (i == j)
      1
    else
      0
    end
  end
end
p()

Returns the permutation matrix P

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 47
def p
  rows = Array.new(@row_size){Array.new(@col_size, 0)}
  @pivots.each_with_index{|p, i| rows[i][p] = 1}
  Matrix.send :new, rows, @col_size
end
singular?()

Returns true if U, and hence A, is singular.

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 66
def singular? ()
  @col_size.times do |j|
    if (@lu[j][j] == 0)
      return true
    end
  end
  false
end
solve(b)

Returns m so that A*m = b, or equivalently so that L*U*m = P*b b can be a Matrix or a Vector

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 94
def solve b
  if (singular?)
    Matrix.Raise Matrix::ErrNotRegular, "Matrix is singular."
  end
  if b.is_a? Matrix
    if (b.row_size != @row_size)
      Matrix.Raise Matrix::ErrDimensionMismatch
    end

    # Copy right hand side with pivoting
    nx = b.column_size
    m = @pivots.map{|row| b.row(row).to_a}

    # Solve L*Y = P*b
    @col_size.times do |k|
      (k+1).upto(@col_size-1) do |i|
        nx.times do |j|
          m[i][j] -= m[k][j]*@lu[i][k]
        end
      end
    end
    # Solve U*m = Y
    (@col_size-1).downto(0) do |k|
      nx.times do |j|
        m[k][j] = m[k][j].quo(@lu[k][k])
      end
      k.times do |i|
        nx.times do |j|
          m[i][j] -= m[k][j]*@lu[i][k]
        end
      end
    end
    Matrix.send :new, m, nx
  else # same algorithm, specialized for simpler case of a vector
    b = convert_to_array(b)
    if (b.size != @row_size)
      Matrix.Raise Matrix::ErrDimensionMismatch
    end

    # Copy right hand side with pivoting
    m = b.values_at(*@pivots)

    # Solve L*Y = P*b
    @col_size.times do |k|
      (k+1).upto(@col_size-1) do |i|
        m[i] -= m[k]*@lu[i][k]
      end
    end
    # Solve U*m = Y
    (@col_size-1).downto(0) do |k|
      m[k] = m[k].quo(@lu[k][k])
      k.times do |i|
        m[i] -= m[k]*@lu[i][k]
      end
    end
    Vector.elements(m, false)
  end
end
to_a()
Alias for: to_ary
to_ary()

Returns L, U, P in an array

Also aliased as: to_a

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 55
def to_ary
  [l, u, p]
end
u()

Returns the upper triangular factor U

Source: show | on GitHub 

# File ../ruby/lib/matrix/lup_decomposition.rb, line 35
def u
  Matrix.build(@col_size, @col_size) do |i, j|
    if (i <= j)
      @lu[i][j]
    else
      0
    end
  end
end