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66
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67 <h5>User Guide</h5>
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68 <ul>
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69
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70 <li class="none">
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71 <a href="../userguide/index.html">Contents</a>
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72 </li>
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73
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74 <li class="none">
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75 <a href="../userguide/overview.html">Overview</a>
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84 </li>
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85
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86 <li class="none">
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87 <strong>Linear Algebra</strong>
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88 </li>
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91 <a href="../userguide/analysis.html">Numerical Analysis</a>
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103 <a href="../userguide/complex.html">Complex Numbers</a>
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104 </li>
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124 </li>
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127 <a href="../userguide/ode.html">Ordinary Differential Equations</a>
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133 </ul>
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136 </a>
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137
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138
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145 </div>
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146 </div>
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147 <div id="bodyColumn">
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148 <div id="contentBox">
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149 <div class="section"><h2><a name="a3_Linear_Algebra"></a>3 Linear Algebra</h2>
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150 <div class="section"><h3><a name="a3.1_Overview"></a>3.1 Overview</h3>
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151 <p>
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152 Linear algebra support in commons-math provides operations on real matrices
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153 (both dense and sparse matrices are supported) and vectors. It features basic
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154 operations (addition, subtraction ...) and decomposition algorithms that can
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155 be used to solve linear systems either in exact sense and in least squares sense.
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156 </p>
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157 </div>
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158 <div class="section"><h3><a name="a3.2_Real_matrices"></a>3.2 Real matrices</h3>
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159 <p>
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160 The <a href="../apidocs/org/apache/commons/math/linear/RealMatrix.html">
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161 RealMatrix</a> interface represents a matrix with real numbers as
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162 entries. The following basic matrix operations are supported:
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163 <ul><li>Matrix addition, subtraction, multiplication</li>
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164 <li>Scalar addition and multiplication</li>
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165 <li>transpose</li>
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166 <li>Norm and Trace</li>
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167 <li>Operation on a vector</li>
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168 </ul>
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169 </p>
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170 <p>
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171 Example:
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172 <div class="source"><pre>
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173 // Create a real matrix with two rows and three columns
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174 double[][] matrixData = { {1d,2d,3d}, {2d,5d,3d}};
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175 RealMatrix m = new Array2DRowRealMatrix(matrixData);
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176
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177 // One more with three rows, two columns
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178 double[][] matrixData2 = { {1d,2d}, {2d,5d}, {1d, 7d}};
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179 RealMatrix n = new Array2DRowRealMatrix(matrixData2);
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180
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181 // Note: The constructor copies the input double[][] array.
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182
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183 // Now multiply m by n
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184 RealMatrix p = m.multiply(n);
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185 System.out.println(p.getRowDimension()); // 2
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186 System.out.println(p.getColumnDimension()); // 2
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187
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188 // Invert p, using LU decomposition
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189 RealMatrix pInverse = new LUDecompositionImpl(p).getSolver().getInverse();
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190 </pre>
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191 </div>
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192 </p>
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193 <p>
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194 The three main implementations of the interface are <a href="../apidocs/org/apache/commons/math/linear/Array2DRowRealMatrix.html">
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195 Array2DRowRealMatrix</a> and <a href="../apidocs/org/apache/commons/math/linear/BlockRealMatrix.html">
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196 BlockRealMatrix</a> for dense matrices (the second one being more suited to
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197 dimensions above 50 or 100) and <a href="../apidocs/org/apache/commons/math/linear/SparseRealMatrix.html">
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198 SparseRealMatrix</a> for sparse matrices.
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199 </p>
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200 </div>
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201 <div class="section"><h3><a name="a3.3_Real_vectors"></a>3.3 Real vectors</h3>
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202 <p>
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203 The <a href="../apidocs/org/apache/commons/math/linear/RealVector.html">
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204 RealVector</a> interface represents a vector with real numbers as
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205 entries. The following basic matrix operations are supported:
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206 <ul><li>Vector addition, subtraction</li>
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207 <li>Element by element multiplication, division</li>
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208 <li>Scalar addition, subtraction, multiplication, division and power</li>
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209 <li>Mapping of mathematical functions (cos, sin ...)</li>
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210 <li>Dot product, outer product</li>
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211 <li>Distance and norm according to norms L1, L2 and Linf</li>
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212 </ul>
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213 </p>
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214 <p>
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215 The <a href="../apidocs/org/apache/commons/math/linear/RealVectorFormat.html">
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216 RealVectorFormat</a> class handles input/output of vectors in a customizable
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217 textual format.
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218 </p>
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219 </div>
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220 <div class="section"><h3><a name="a3.4_Solving_linear_systems"></a>3.4 Solving linear systems</h3>
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221 <p>
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222 The <code>solve()</code> methods of the <a href="../apidocs/org/apache/commons/math/linear/DecompositionSolver.html">DecompositionSolver</a>
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223 interface support solving linear systems of equations of the form AX=B, either
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224 in linear sense or in least square sense. A <code>RealMatrix</code> instance is
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225 used to represent the coefficient matrix of the system. Solving the system is a
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226 two phases process: first the coefficient matrix is decomposed in some way and
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227 then a solver built from the decomposition solves the system. This allows to
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228 compute the decomposition and build the solver only once if several systems have
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229 to be solved with the same coefficient matrix.
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230 </p>
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231 <p>
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232 For example, to solve the linear system
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233 <pre>
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234 2x + 3y - 2z = 1
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235 -x + 7y + 6x = -2
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236 4x - 3y - 5z = 1
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237 </pre>
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238 Start by decomposing the coefficient matrix A (in this case using LU decomposition)
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239 and build a solver
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240 <div class="source"><pre>
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241 RealMatrix coefficients =
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242 new Array2DRowRealMatrix(new double[][] { { 2, 3, -2 }, { -1, 7, 6 }, { 4, -3, -5 } },
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243 false);
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244 DecompositionSolver solver = new LUDecompositionImpl(coefficients).getSolver();
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245 </pre>
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246 </div>
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247
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248 Next create a <code>RealVector</code> array to represent the constant
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249 vector B and use <code>solve(RealVector)</code> to solve the system
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250 <div class="source"><pre>
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251 RealVector constants = new RealVectorImpl(new double[] { 1, -2, 1 }, false);
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252 RealVector solution = solver.solve(constants);
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253 </pre>
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254 </div>
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255
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256 The <code>solution</code> vector will contain values for x
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257 (<code>solution.getEntry(0)</code>), y (<code>solution.getEntry(1)</code>),
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258 and z (<code>solution.getEntry(2)</code>) that solve the system.
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259 </p>
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260 <p>
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261 Each type of decomposition has its specific semantics and constraints on
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262 the coefficient matrix as shown in the following table. For algorithms that
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263 solve AX=B in least squares sense the value returned for X is such that the
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264 residual AX-B has minimal norm. If an exact solution exist (i.e. if for some
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265 X the residual AX-B is exactly 0), then this exact solution is also the solution
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266 in least square sense. This implies that algorithms suited for least squares
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267 problems can also be used to solve exact problems, but the reverse is not true.
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268 </p>
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269 <p><table class="bodyTable"><tr class="a"><td><font size="+2">Decomposition algorithms</font></td>
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270 </tr>
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271 <tr class="b"><font size="+1"><td>Name</td>
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272 <td>coefficients matrix</td>
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273 <td>problem type</td>
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274 </font></tr>
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275 <tr class="a"><td><a href="../apidocs/org/apache/commons/math/linear/LUDecomposition.html">LU</a></td>
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276 <td>square</td>
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277 <td>exact solution only</td>
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278 </tr>
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279 <tr class="b"><td><a href="../apidocs/org/apache/commons/math/linear/CholeskyDecomposition.html">Cholesky</a></td>
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280 <td>symmetric positive definite</td>
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281 <td>exact solution only</td>
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282 </tr>
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283 <tr class="a"><td><a href="../apidocs/org/apache/commons/math/linear/QRDecomposition.html">QR</a></td>
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284 <td>any</td>
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285 <td>least squares solution</td>
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286 </tr>
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287 <tr class="b"><td><a href="../apidocs/org/apache/commons/math/linear/EigenDecomposition.html">eigen decomposition</a></td>
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288 <td>square</td>
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289 <td>exact solution only</td>
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290 </tr>
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291 <tr class="a"><td><a href="../apidocs/org/apache/commons/math/linear/SingularValueDecomposition.html">SVD</a></td>
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292 <td>any</td>
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293 <td>least squares solution</td>
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294 </tr>
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295 </table>
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296 </p>
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297 <p>
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298 It is possible to use a simple array of double instead of a <code>RealVector</code>.
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299 In this case, the solution will be provided also as an array of double.
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300 </p>
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301 <p>
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302 It is possible to solve multiple systems with the same coefficient matrix
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303 in one method call. To do this, create a matrix whose column vectors correspond
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304 to the constant vectors for the systems to be solved and use <code>solve(RealMatrix),</code>
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305 which returns a matrix with column vectors representing the solutions.
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306 </p>
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307 </div>
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308 <div class="section"><h3><a name="a3.5_Eigenvalueseigenvectors_and_singular_valuessingular_vectors"></a>3.5 Eigenvalues/eigenvectors and singular values/singular vectors</h3>
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309 <p>
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310 Decomposition algorithms may be used for themselves and not only for linear system solving.
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311 This is of prime interest with eigen decomposition and singular value decomposition.
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312 </p>
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313 <p>
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314 The <code>getEigenvalue()</code>, <code>getEigenvalues()</code>, <code>getEigenVector()</code>,
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315 <code>getV()</code>, <code>getD()</code> and <code>getVT()</code> methods of the
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316 <code>EigenDecomposition</code> interface support solving eigenproblems of the form
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317 AX = lambda X where lambda is a real scalar.
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318 </p>
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319 <p>The <code>getSingularValues()</code>, <code>getU()</code>, <code>getS()</code> and
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320 <code>getV()</code> methods of the <code>SingularValueDecomposition</code> interface
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321 allow to solve singular values problems of the form AXi = lambda Yi where lambda is a
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322 real scalar, and where the Xi and Yi vectors form orthogonal bases of their respective
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323 vector spaces (which may have different dimensions).
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324 </p>
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325 </div>
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326 <div class="section"><h3><a name="a3.6_Non-real_fields_complex_fractions_..."></a>3.6 Non-real fields (complex, fractions ...)</h3>
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327 <p>
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328 In addition to the real field, matrices and vectors using non-real <a href="../apidocs/org/apache/commons/math/FieldElement.html">field elements</a> can be used.
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329 The fields already supported by the library are:
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330 <ul><li><a href="../apidocs/org/apache/commons/math/complex/Complex.html">Complex</a></li>
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331 <li><a href="../apidocs/org/apache/commons/math/fraction/Fraction.html">Fraction</a></li>
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332 <li><a href="../apidocs/org/apache/commons/math/fraction/BigFraction.html">BigFraction</a></li>
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333 <li><a href="../apidocs/org/apache/commons/math/util/BigReal.html">BigReal</a></li>
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334 </ul>
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335 </p>
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336 </div>
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337 </div>
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338
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339 </div>
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340 </div>
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341 <div class="clear">
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342 <hr/>
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343 </div>
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344 <div id="footer">
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345 <div class="xright">©
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346 2003-2010
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347
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354
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355
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356 </div>
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357 <div class="clear">
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358 <hr/>
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361 </body>
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362 </html>
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