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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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  30. Class to obtain eigenvalues and eigenvectors of a real matrix</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>

  41. <li><a href="../packages/JAMA%0D%0AFor%20an%20m-by-n%20matrix%20A%20with%20m%20&gt;=%20n,%20the%20singular%20value%20decomposition%20is%0D%0Aan%20m-by-n%20orthogonal%20matrix%20U,%20an%20n-by-n%20diagonal%20matrix%20S,%20and%0D%0Aan%20n-by-n%20orthogonal%20matrix%20V%20so%20that%20A%20=%20U*S*V'.html"><i class="icon-folder-open"></i> JAMA

  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>

  45. <li><a href="../packages/JAMA%0D%0APythagorean%20Theorem:%0D%0Aa%20=%203%0D%0Ab%20=%204%0D%0Ar%20=%20sqrt(square(a)%20+%20square(b))%0D%0Ar%20=%205%0D%0Ar%20=%20sqrt(a%5E2%20+%20b%5E2)%20without%20under.overflow.html"><i class="icon-folder-open"></i> JAMA

  46. Pythagorean Theorem:

  47. a = 3

  48. b = 4

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

  50. r = 5

  51. r = sqrt(a^2 + b^2) without under/overflow</a></li>

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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 :: Construct the singular value decomposition"><span class="description">Construct the singular value decomposition</span><pre>__construct()</pre></a></li>

  96. <li class="method public "><a href="#method_cond" title="cond :: Two norm condition number"><span class="description">Two norm condition number</span><pre>cond()</pre></a></li>

  97. <li class="method public "><a href="#method_getS" title="getS :: Return the diagonal matrix of singular values"><span class="description">Return the diagonal matrix of singular values</span><pre>getS()</pre></a></li>

  98. <li class="method public "><a href="#method_getSingularValues" title="getSingularValues :: Return the one-dimensional array of singular values"><span class="description">Return the one-dimensional array of singular values</span><pre>getSingularValues()</pre></a></li>

  99. <li class="method public "><a href="#method_getU" title="getU :: Return the left singular vectors"><span class="description">Return the left singular vectors</span><pre>getU()</pre></a></li>

  100. <li class="method public "><a href="#method_getV" title="getV :: Return the right singular vectors"><span class="description">Return the right singular vectors</span><pre>getV()</pre></a></li>

  101. <li class="method public "><a href="#method_norm2" title="norm2 :: Two norm"><span class="description">Two norm</span><pre>norm2()</pre></a></li>

  102. <li class="method public "><a href="#method_rank" title="rank :: Effective numerical matrix rank"><span class="description">Effective numerical matrix rank</span><pre>rank()</pre></a></li>

  103. </ul>

  104. </li>

  105. <li class="nav-header">

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

  107.                     <ul></ul>

  108. </li>

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

  110.                     <ul>

  111. <li class="property private "><a href="#property_U" title="$U :: Internal storage of U."><span class="description"></span><pre>$U</pre></a></li>

  112. <li class="property private "><a href="#property_V" title="$V :: Internal storage of V."><span class="description"></span><pre>$V</pre></a></li>

  113. <li class="property private "><a href="#property_m" title="$m :: Row dimension."><span class="description"></span><pre>$m</pre></a></li>

  114. <li class="property private "><a href="#property_n" title="$n :: Column dimension."><span class="description"></span><pre>$n</pre></a></li>

  115. <li class="property private "><a href="#property_s" title="$s :: Internal storage of singular values."><span class="description"></span><pre>$s</pre></a></li>

  116. </ul>

  117. </li>

  118. </ul>

  119. </div>

  120. <div class="span8">

  121. <a id="\SingularValueDecomposition"></a><ul class="breadcrumb">

  122. <li>

  123. <a href="../index.html"><i class="icon-custom icon-class"></i></a><span class="divider">\</span>

  124. </li>

  125. <li><a href="../namespaces/global.html">global</a></li>

  126. <li class="active">

  127. <span class="divider">\</span><a href="../classes/SingularValueDecomposition.html">SingularValueDecomposition</a>

  128. </li>

  129. </ul>

  130. <div class="element class">

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

  132. <div class="details">

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

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

  135. <tr>

  136. <th>package</th>

  137. <td><a href="../packages/JAMA%0D%0AFor%20an%20m-by-n%20matrix%20A%20with%20m%20&gt;=%20n,%20the%20singular%20value%20decomposition%20is%0D%0Aan%20m-by-n%20orthogonal%20matrix%20U,%20an%20n-by-n%20diagonal%20matrix%20S,%20and%0D%0Aan%20n-by-n%20orthogonal%20matrix%20V%20so%20that%20A%20=%20U*S*V'.%0D%0AThe%20singular%20values,%20sigma%5B%24k%5D%20=%20S%5B%24k%5D%5B%24k%5D,%20are%20ordered%20so%20that%0D%0Asigma%5B0%5D%20&gt;=%20sigma%5B1%5D%20&gt;=%20...%20&gt;=%20sigma%5Bn-1%5D.%0D%0AThe%20singular%20value%20decompostion%20always%20exists,%20so%20the%20constructor%20will%0D%0Anever%20fail.%20%20The%20matrix%20condition%20number%20and%20the%20effective%20numerical%0D%0Arank%20can%20be%20computed%20from%20this%20decomposition..html">JAMA

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

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

  140. an n-by-n orthogonal matrix V so that A = U*S*V'.

  141. The singular values, sigma[$k] = S[$k][$k], are ordered so that

  142. sigma[0] &gt;= sigma[1] &gt;= ... &gt;= sigma[n-1].

  143. The singular value decompostion always exists, so the constructor will

  144. never fail.  The matrix condition number and the effective numerical

  145. rank can be computed from this decomposition.</a></td>

  146. </tr>

  147. <tr>

  148. <th>author</th>

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

  150. </tr>

  151. <tr>

  152. <th>license</th>

  153. <td><a href="">PHP v3.0</a></td>

  154. </tr>

  155. <tr>

  156. <th>version</th>

  157. <td>1.1</td>

  158. </tr>

  159. </table>

  160. <h3>

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

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

  163. <h2>Construct the singular value decomposition</h2>

  164. <pre>__construct($Arg) : \Structure</pre>

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

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

  167. <div class="long_description"><p>Derived from LINPACK code.</p></div>

  168. <h3>Parameters</h3>

  169. <div class="subelement argument"><h4>$Arg</h4></div>

  170. <h3>Returns</h3>

  171. <div class="subelement response">

  172. <code>\Structure</code>to access U, S and V.</div>

  173. </div></div>

  174. </div>

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

  176. <h2>Two norm condition number</h2>

  177. <pre>cond() : \max(S)/min(S)</pre>

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

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

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

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

  182. <th>access</th>

  183. <td>public</td>

  184. </tr></table>

  185. <h3>Returns</h3>

  186. <div class="subelement response"><code>\max(S)/min(S)</code></div>

  187. </div></div>

  188. </div>

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

  190. <h2>Return the diagonal matrix of singular values</h2>

  191. <pre>getS() : \S</pre>

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

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

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

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

  196. <th>access</th>

  197. <td>public</td>

  198. </tr></table>

  199. <h3>Returns</h3>

  200. <div class="subelement response"><code>\S</code></div>

  201. </div></div>

  202. </div>

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

  204. <h2>Return the one-dimensional array of singular values</h2>

  205. <pre>getSingularValues() : \diagonal</pre>

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

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

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

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

  210. <th>access</th>

  211. <td>public</td>

  212. </tr></table>

  213. <h3>Returns</h3>

  214. <div class="subelement response">

  215. <code>\diagonal</code>of S.</div>

  216. </div></div>

  217. </div>

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

  219. <h2>Return the left singular vectors</h2>

  220. <pre>getU() : \U</pre>

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

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

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

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

  225. <th>access</th>

  226. <td>public</td>

  227. </tr></table>

  228. <h3>Returns</h3>

  229. <div class="subelement response"><code>\U</code></div>

  230. </div></div>

  231. </div>

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

  233. <h2>Return the right singular vectors</h2>

  234. <pre>getV() : \V</pre>

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

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

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

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

  239. <th>access</th>

  240. <td>public</td>

  241. </tr></table>

  242. <h3>Returns</h3>

  243. <div class="subelement response"><code>\V</code></div>

  244. </div></div>

  245. </div>

  246. <a id="method_norm2"></a><div class="element clickable method public method_norm2" data-toggle="collapse" data-target=".method_norm2 .collapse">

  247. <h2>Two norm</h2>

  248. <pre>norm2() : \max(S)</pre>

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

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

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

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

  253. <th>access</th>

  254. <td>public</td>

  255. </tr></table>

  256. <h3>Returns</h3>

  257. <div class="subelement response"><code>\max(S)</code></div>

  258. </div></div>

  259. </div>

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

  261. <h2>Effective numerical matrix rank</h2>

  262. <pre>rank() : \Number</pre>

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

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

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

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

  267. <th>access</th>

  268. <td>public</td>

  269. </tr></table>

  270. <h3>Returns</h3>

  271. <div class="subelement response">

  272. <code>\Number</code>of nonnegligible singular values.</div>

  273. </div></div>

  274. </div>

  275. <h3>

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

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

  278. <h2></h2>

  279. <pre>$U : array</pre>

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

  281. <div class="row collapse"><div class="detail-description"><div class="long_description"></div></div></div>

  282. </div>

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

  284. <h2></h2>

  285. <pre>$V : array</pre>

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

  287. <div class="row collapse"><div class="detail-description"><div class="long_description"></div></div></div>

  288. </div>

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

  290. <h2></h2>

  291. <pre>$m : int</pre>

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

  293. <div class="row collapse"><div class="detail-description"><div class="long_description"></div></div></div>

  294. </div>

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

  296. <h2></h2>

  297. <pre>$n : int</pre>

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

  299. <div class="row collapse"><div class="detail-description"><div class="long_description"></div></div></div>

  300. </div>

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

  302. <h2></h2>

  303. <pre>$s : array</pre>

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

  305. <div class="row collapse"><div class="detail-description"><div class="long_description"></div></div></div>

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