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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 ``` ``````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) 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, @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(@column_count, 0)} @pivots.each_with_index{|p, i| rows[i][p] = 1} Matrix.send :new, rows, @column_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) Matrix.Raise Matrix::ErrDimensionMismatch unless square? 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?) 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 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 end end ``````