# class 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) click to toggle source

    # File lib/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() click to toggle source

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

    # File lib/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() click to toggle source

    # File lib/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() click to toggle source

Returns the permutation matrix `P`

    # File lib/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?() click to toggle source

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

    # File lib/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) click to toggle source

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 lib/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() click to toggle source

Returns `L`, `U`, `P` in an array

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

Also aliased as: to_a
u() click to toggle source

Returns the upper triangular factor `U`

    # File lib/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