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  26. Cholesky decomposition class

  27. For a symmetric, positive definite matrix A, the Cholesky decomposition

  28. is an lower triangular matrix L so that A = L*L'</a></li>

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  34. For an m-by-n matrix A with m &gt;= n, the LU decomposition is an m-by-n

  35. unit lower triangular matrix L, an n-by-n upper triangular matrix U,

  36. and a permutation vector piv of length m so that A(piv,:) = L*U</a></li>

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  38. For an m-by-n matrix A with m &gt;= n, the QR decomposition is an m-by-n

  39. orthogonal matrix Q and an n-by-n upper triangular matrix R so that

  40. A = Q*R</a></li>

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  42. For an m-by-n matrix A with m &gt;= n, the singular value decomposition is

  43. an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and

  44. an n-by-n orthogonal matrix V so that A = U*S*V'</a></li>

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  46. Pythagorean Theorem:

  47. a = 3

  48. b = 4

  49. r = sqrt(square(a) + square(b))

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  91. <ul class="side-nav nav nav-list">

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  93. <i class="icon-custom icon-method"></i> Methods

  94.                     <ul>

  95. <li class="method public "><a href="#method___construct" title="__construct :: CholeskyDecomposition"><span class="description">CholeskyDecomposition</span><pre>__construct()</pre></a></li>

  96. <li class="method public "><a href="#method_getL" title="getL :: getL"><span class="description">getL</span><pre>getL()</pre></a></li>

  97. <li class="method public "><a href="#method_isSPD" title="isSPD :: Is the matrix symmetric and positive definite?"><span class="description">Is the matrix symmetric and positive definite?</span><pre>isSPD()</pre></a></li>

  98. <li class="method public "><a href="#method_solve" title="solve :: Solve A*X = B"><span class="description">Solve A*X = B</span><pre>solve()</pre></a></li>

  99. </ul>

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  102. <i class="icon-custom icon-property"></i> Properties

  103.                     <ul></ul>

  104. </li>

  105. <li class="nav-header private">» Private

  106.                     <ul>

  107. <li class="property private "><a href="#property_L" title="$L :: Decomposition storage"><span class="description"></span><pre>$L</pre></a></li>

  108. <li class="property private "><a href="#property_isspd" title="$isspd :: Symmetric positive definite flag"><span class="description"></span><pre>$isspd</pre></a></li>

  109. <li class="property private "><a href="#property_m" title="$m :: Matrix row and column dimension"><span class="description"></span><pre>$m</pre></a></li>

  110. </ul>

  111. </li>

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  121. <span class="divider">\</span><a href="../classes/CholeskyDecomposition.html">CholeskyDecomposition</a>

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  123. </ul>

  124. <div class="element class">

  125. <p class="short_description"></p>

  126. <div class="details">

  127. <div class="long_description"></div>

  128. <table class="table table-bordered">

  129. <tr>

  130. <th>package</th>

  131. <td><a href="../packages/JAMA%0D%0ACholesky%20decomposition%20class%0D%0AFor%20a%20symmetric,%20positive%20definite%20matrix%20A,%20the%20Cholesky%20decomposition%0D%0Ais%20an%20lower%20triangular%20matrix%20L%20so%20that%20A%20=%20L*L'.%0D%0AIf%20the%20matrix%20is%20not%20symmetric%20or%20positive%20definite,%20the%20constructor%0D%0Areturns%20a%20partial%20decomposition%20and%20sets%20an%20internal%20flag%20that%20may%0D%0Abe%20queried%20by%20the%20isSPD()%20method..html">JAMA

  132. Cholesky decomposition class

  133. For a symmetric, positive definite matrix A, the Cholesky decomposition

  134. is an lower triangular matrix L so that A = L*L'.

  135. If the matrix is not symmetric or positive definite, the constructor

  136. returns a partial decomposition and sets an internal flag that may

  137. be queried by the isSPD() method.</a></td>

  138. </tr>

  139. <tr>

  140. <th>author</th>

  141. <td><a href="">Paul Meagher</a></td>

  142. </tr>

  143. <tr>

  144. <th>author</th>

  145. <td><a href="">Michael Bommarito</a></td>

  146. </tr>

  147. <tr>

  148. <th>version</th>

  149. <td>1.2</td>

  150. </tr>

  151. </table>

  152. <h3>

  153. <i class="icon-custom icon-method"></i> Methods</h3>

  154. <a id="method___construct"></a><div class="element clickable method public method___construct" data-toggle="collapse" data-target=".method___construct .collapse">

  155. <h2>CholeskyDecomposition</h2>

  156. <pre>__construct(mixed $A) </pre>

  157. <div class="labels"></div>

  158. <div class="row collapse"><div class="detail-description">

  159. <div class="long_description"><p>Class constructor - decomposes symmetric positive definite matrix</p></div>

  160. <h3>Parameters</h3>

  161. <div class="subelement argument">

  162. <h4>$A</h4>

  163. <code>mixed</code><p>Matrix square symmetric positive definite matrix</p></div>

  164. </div></div>

  165. </div>

  166. <a id="method_getL"></a><div class="element clickable method public method_getL" data-toggle="collapse" data-target=".method_getL .collapse">

  167. <h2>getL</h2>

  168. <pre>getL() : \Matrix</pre>

  169. <div class="labels"></div>

  170. <div class="row collapse"><div class="detail-description">

  171. <div class="long_description"><p>Return triangular factor.</p></div>

  172. <h3>Returns</h3>

  173. <div class="subelement response">

  174. <code>\Matrix</code>Lower triangular matrix</div>

  175. </div></div>

  176. </div>

  177. <a id="method_isSPD"></a><div class="element clickable method public method_isSPD" data-toggle="collapse" data-target=".method_isSPD .collapse">

  178. <h2>Is the matrix symmetric and positive definite?</h2>

  179. <pre>isSPD() : boolean</pre>

  180. <div class="labels"></div>

  181. <div class="row collapse"><div class="detail-description">

  182. <div class="long_description"></div>

  183. <h3>Returns</h3>

  184. <div class="subelement response"><code>boolean</code></div>

  185. </div></div>

  186. </div>

  187. <a id="method_solve"></a><div class="element clickable method public method_solve" data-toggle="collapse" data-target=".method_solve .collapse">

  188. <h2>Solve A*X = B</h2>

  189. <pre>solve($B) : \Matrix</pre>

  190. <div class="labels"></div>

  191. <div class="row collapse"><div class="detail-description">

  192. <div class="long_description"></div>

  193. <h3>Parameters</h3>

  194. <div class="subelement argument">

  195. <h4>$B</h4><p>Row-equal matrix</p>

  196. </div>

  197. <h3>Returns</h3>

  198. <div class="subelement response">

  199. <code>\Matrix</code>L * L' * X = B</div>

  200. </div></div>

  201. </div>

  202. <h3>

  203. <i class="icon-custom icon-property"></i> Properties</h3>

  204. <a id="property_L"> </a><div class="element clickable property private property_L" data-toggle="collapse" data-target=".property_L .collapse">

  205. <h2></h2>

  206. <pre>$L : array</pre>

  207. <div class="labels"></div>

  208. <div class="row collapse"><div class="detail-description">

  209. <div class="long_description"></div>

  210. <table class="table table-bordered"><tr>

  211. <th>access</th>

  212. <td>private</td>

  213. </tr></table>

  214. </div></div>

  215. </div>

  216. <a id="property_isspd"> </a><div class="element clickable property private property_isspd" data-toggle="collapse" data-target=".property_isspd .collapse">

  217. <h2></h2>

  218. <pre>$isspd : boolean</pre>

  219. <div class="labels"></div>

  220. <div class="row collapse"><div class="detail-description">

  221. <div class="long_description"></div>

  222. <table class="table table-bordered"><tr>

  223. <th>access</th>

  224. <td>private</td>

  225. </tr></table>

  226. </div></div>

  227. </div>

  228. <a id="property_m"> </a><div class="element clickable property private property_m" data-toggle="collapse" data-target=".property_m .collapse">

  229. <h2></h2>

  230. <pre>$m : int</pre>

  231. <div class="labels"></div>

  232. <div class="row collapse"><div class="detail-description">

  233. <div class="long_description"></div>

  234. <table class="table table-bordered"><tr>

  235. <th>access</th>

  236. <td>private</td>

  237. </tr></table>

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