Class: Matrix::LUPDecomposition

Inherits:
Object
  • Object
show all
Includes:
ConversionHelper
Defined in:
lib/backports/1.9.2/stdlib/matrix/lup_decomposition.rb

Overview

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.

Instance Attribute Summary collapse

Instance Method Summary collapse

Constructor Details

#initialize(a) ⇒ LUPDecomposition

Returns a new instance of LUPDecomposition.

Raises:

  • (TypeError)

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# File 'lib/backports/1.9.2/stdlib/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 Attribute Details

#pivotsObject (readonly)

Returns the pivoting indices


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# File 'lib/backports/1.9.2/stdlib/matrix/lup_decomposition.rb', line 62

def pivots
  @pivots
end

Instance Method Details

#detObject Also known as: determinant

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


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# File 'lib/backports/1.9.2/stdlib/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

#lObject


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# File 'lib/backports/1.9.2/stdlib/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

#pObject

Returns the permutation matrix P


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# File 'lib/backports/1.9.2/stdlib/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?Boolean

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

Returns:

  • (Boolean)

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# File 'lib/backports/1.9.2/stdlib/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) ⇒ Object

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


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# File 'lib/backports/1.9.2/stdlib/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_aryObject Also known as: to_a

Returns L, U, P in an array


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# File 'lib/backports/1.9.2/stdlib/matrix/lup_decomposition.rb', line 55

def to_ary
  [l, u, p]
end

#uObject

Returns the upper triangular factor U


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# File 'lib/backports/1.9.2/stdlib/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