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\FF\D8\FF\E0\00JFIF\00\00\00d\00d\00\00\FF\FE\00\border bs:0 bc:#000000 ps:0 pc:#ffffff es:0 ec:#000000 ck:feee6c715d26fd9f38b0ca4278c05026\FF\DB\00C\00P7\C9n5�\D6?\BD�\9Ds\EBp\9F[`8m\B7)o\B5\E8\E6I\99\FE3]]A2\BA\8Cw\D6E\93\\DEv\C8\009\F2\F1NI?uc\\F5\EA\96k\xN<~buv\EA\C8\D7 \8B\84\CEcxI\BBg\AE\9E=\D6+n\EC\80\C8A\8C\AE\EB\CF\D5\DA\E9"2\A4\B9j5\EB\F3W\B63\96\B30Yu\DA\FC8\ED\DF\E7Ms\FB\F1\8E\B3\FA\EA\E8\E6(\883zs\F2_\8DFk\8Bh \00\8C\DCw\D3R\B5+6X\BA\B2\C4j\AB0\B4\FCMw\C2I\8E\9B\E3\A9~9u\FA\D3l\80\C8%p\EE\FDn2� \00$\FEj\C4e\A9\DB\~\95\A7\A5\80EK\BB\8DDsP\00@@AD'k\CF\E8\DB\D2(\80\9AK\D3\85\D6lb\F2\BA\8C*\80\00)\95 59\A3R:\F3\CE"\B6\80\88\00\00i1u4�\E9\F2�\A6\A2\FACM\93WMb*\E0*\00\00\00\00\00(\A8\80\00\00\80\00\00\00\00\00\00\00\00\00\FF\D9 C/// # frozen_string_literal: false 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 end end end # Returns the permutation matrix +P+ 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 # Returns +L+, +U+, +P+ in an array def to_ary [l, u, p] end alias_method :to_a, :to_ary # Returns the pivoting indices attr_reader :pivots # Returns +true+ if +U+, and hence +A+, is singular. def singular? @column_count.times do |j| if (@lu[j][j] == 0) return true end end false end # Returns the determinant of +A+, calculated efficiently # from the factorization. def det if (@row_count != @column_count) raise Matrix::ErrDimensionMismatch end d = @pivot_sign @column_count.times do |j| d *= @lu[j][j] end d end alias_method :determinant, :det # Returns +m+ so that A*m = b, # or equivalently so that L*U*m = P*b # +b+ can be a Matrix or a Vector def solve b if (singular?) raise Matrix::ErrNotRegular, "Matrix is singular." end if b.is_a? Matrix if (b.row_count != @row_count) 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) 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|