Matrix::LUPDecomposition

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.

Attributes

pivots[R]

Returns the pivoting indices

Public Class Methods

new(a)

 
               # File matrix/lup_decomposition.rb, line 154
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
end

Public Instance Methods

det()

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

 
               # File matrix/lup_decomposition.rb, line 79
def det
  if (@row_count != @column_count)
    Matrix.Raise Matrix::ErrDimensionMismatch
  end
  d = @pivot_sign
  @column_count.times do |j|
    d *= @lu[j][j]
  end
  d
end

Also aliased as: determinant

determinant()

Alias for: det

l()

 
               # File matrix/lup_decomposition.rb, line 22
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

p()

Returns the permutation matrix P

 
               # File matrix/lup_decomposition.rb, line 48
def p
  rows = Array.new(@row_count){Array.new(@row_count, 0)}
  @pivots.each_with_index{|p, i| rows[i][p] = 1}
  Matrix.send :new, rows, @row_count
end

singular?()

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

 
               # File matrix/lup_decomposition.rb, line 67
def singular? ()
  @column_count.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

 
               # File matrix/lup_decomposition.rb, line 95
def solve b
  if (singular?)
    Matrix.Raise Matrix::ErrNotRegular, "Matrix is singular."
  end
  if b.is_a? Matrix
    if (b.row_count != @row_count)
      Matrix.Raise Matrix::ErrDimensionMismatch
    end

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

    # Solve L*Y = P*b
    @column_count.times do |k|
      (k+1).upto(@column_count-1) do |i|
        nx.times do |j|
          m[i][j] -= m[k][j]*@lu[i][k]
        end
      end
    end
    # Solve U*m = Y
    (@column_count-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_count)
      Matrix.Raise Matrix::ErrDimensionMismatch
    end

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

    # Solve L*Y = P*b
    @column_count.times do |k|
      (k+1).upto(@column_count-1) do |i|
        m[i] -= m[k]*@lu[i][k]
      end
    end
    # Solve U*m = Y
    (@column_count-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

 
               # File matrix/lup_decomposition.rb, line 56
def to_ary
  [l, u, p]
end

Also aliased as: to_a

u()

Returns the upper triangular factor U

 
               # File matrix/lup_decomposition.rb, line 36
def u
  Matrix.build([@column_count, @row_count].min, @column_count) do |i, j|
    if (i <= j)
      @lu[i][j]
    else
      0
    end
  end
end