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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>

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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>

  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 :: Constructor: Check for symmetry, then construct the eigenvalue decomposition"><span class="description">Constructor: Check for symmetry, then construct the eigenvalue decomposition</span><pre>__construct()</pre></a></li>

  96. <li class="method public "><a href="#method_getD" title="getD :: Return the block diagonal eigenvalue matrix"><span class="description">Return the block diagonal eigenvalue matrix</span><pre>getD()</pre></a></li>

  97. <li class="method public "><a href="#method_getImagEigenvalues" title="getImagEigenvalues :: Return the imaginary parts of the eigenvalues"><span class="description">Return the imaginary parts of the eigenvalues</span><pre>getImagEigenvalues()</pre></a></li>

  98. <li class="method public "><a href="#method_getRealEigenvalues" title="getRealEigenvalues :: Return the real parts of the eigenvalues"><span class="description">Return the real parts of the eigenvalues</span><pre>getRealEigenvalues()</pre></a></li>

  99. <li class="method public "><a href="#method_getV" title="getV :: Return the eigenvector matrix"><span class="description">Return the eigenvector matrix</span><pre>getV()</pre></a></li>

  100. </ul>

  101. </li>

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

  103.                     <ul>

  104. <li class="method private "><a href="#method_cdiv" title="cdiv :: Performs complex division."><span class="description">Performs complex division.</span><pre>cdiv()</pre></a></li>

  105. <li class="method private "><a href="#method_hqr2" title="hqr2 :: Nonsymmetric reduction from Hessenberg to real Schur form."><span class="description">Nonsymmetric reduction from Hessenberg to real Schur form.</span><pre>hqr2()</pre></a></li>

  106. <li class="method private "><a href="#method_orthes" title="orthes :: Nonsymmetric reduction to Hessenberg form."><span class="description">Nonsymmetric reduction to Hessenberg form.</span><pre>orthes()</pre></a></li>

  107. <li class="method private "><a href="#method_tql2" title="tql2 :: Symmetric tridiagonal QL algorithm."><span class="description">Symmetric tridiagonal QL algorithm.</span><pre>tql2()</pre></a></li>

  108. <li class="method private "><a href="#method_tred2" title="tred2 :: Symmetric Householder reduction to tridiagonal form."><span class="description">Symmetric Householder reduction to tridiagonal form.</span><pre>tred2()</pre></a></li>

  109. </ul>

  110. </li>

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

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

  113.                     <ul></ul>

  114. </li>

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

  116.                     <ul>

  117. <li class="property private "><a href="#property_H" title="$H :: Array for internal storage of nonsymmetric Hessenberg form."><span class="description"></span><pre>$H</pre></a></li>

  118. <li class="property private "><a href="#property_V" title="$V :: Array for internal storage of eigenvectors."><span class="description"></span><pre>$V</pre></a></li>

  119. <li class="property private "><a href="#property_cdivi" title="$cdivi :: "><span class="description"></span><pre>$cdivi</pre></a></li>

  120. <li class="property private "><a href="#property_cdivr" title="$cdivr :: Used for complex scalar division."><span class="description"></span><pre>$cdivr</pre></a></li>

  121. <li class="property private "><a href="#property_d" title="$d :: Arrays for internal storage of eigenvalues."><span class="description"></span><pre>$d</pre></a></li>

  122. <li class="property private "><a href="#property_e" title="$e :: "><span class="description"></span><pre>$e</pre></a></li>

  123. <li class="property private "><a href="#property_issymmetric" title="$issymmetric :: Internal symmetry flag."><span class="description"></span><pre>$issymmetric</pre></a></li>

  124. <li class="property private "><a href="#property_n" title="$n :: Row and column dimension (square matrix)."><span class="description"></span><pre>$n</pre></a></li>

  125. <li class="property private "><a href="#property_ort" title="$ort :: Working storage for nonsymmetric algorithm."><span class="description"></span><pre>$ort</pre></a></li>

  126. </ul>

  127. </li>

  128. </ul>

  129. </div>

  130. <div class="span8">

  131. <a id="\EigenvalueDecomposition"></a><ul class="breadcrumb">

  132. <li>

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

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

  136. <li class="active">

  137. <span class="divider">\</span><a href="../classes/EigenvalueDecomposition.html">EigenvalueDecomposition</a>

  138. </li>

  139. </ul>

  140. <div class="element class">

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

  142. <div class="details">

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

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

  145. <tr>

  146. <th>package</th>

  147. <td><a href="../packages/JAMA%0D%0AClass%20to%20obtain%20eigenvalues%20and%20eigenvectors%20of%20a%20real%20matrix.%0D%0AIf%20A%20is%20symmetric,%20then%20A%20=%20V*D*V'%20where%20the%20eigenvalue%20matrix%20D%0D%0Ais%20diagonal%20and%20the%20eigenvector%20matrix%20V%20is%20orthogonal%20(i.e.%0D%0AA%20=%20V.times(D.times(V.transpose()))%20and%20V.times(V.transpose())%0D%0Aequals%20the%20identity%20matrix).%0D%0AIf%20A%20is%20not%20symmetric,%20then%20the%20eigenvalue%20matrix%20D%20is%20block%20diagonal%0D%0Awith%20the%20real%20eigenvalues%20in%201-by-1%20blocks%20and%20any%20complex%20eigenvalues,%0D%0Alambda%20+%20i*mu,%20in%202-by-2%20blocks,%20%5Blambda,%20mu;%20-mu,%20lambda%5D.%20%20The%0D%0Acolumns%20of%20V%20represent%20the%20eigenvectors%20in%20the%20sense%20that%20A*V%20=%20V*D,%0D%0Ai.e.%20A.times(V)%20equals%20V.times(D).%20%20The%20matrix%20V%20may%20be%20badly%0D%0Aconditioned,%20or%20even%20singular,%20so%20the%20validity%20of%20the%20equation%0D%0AA%20=%20V*D*inverse(V)%20depends%20upon%20V.cond()..html">JAMA

  148. Class to obtain eigenvalues and eigenvectors of a real matrix.

  149. If A is symmetric, then A = V*D*V' where the eigenvalue matrix D

  150. is diagonal and the eigenvector matrix V is orthogonal (i.e.

  151. A = V.times(D.times(V.transpose())) and V.times(V.transpose())

  152. equals the identity matrix).

  153. If A is not symmetric, then the eigenvalue matrix D is block diagonal

  154. with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,

  155. lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The

  156. columns of V represent the eigenvectors in the sense that A*V = V*D,

  157. i.e. A.times(V) equals V.times(D).  The matrix V may be badly

  158. conditioned, or even singular, so the validity of the equation

  159. A = V*D*inverse(V) depends upon V.cond().</a></td>

  160. </tr>

  161. <tr>

  162. <th>author</th>

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

  164. </tr>

  165. <tr>

  166. <th>license</th>

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

  168. </tr>

  169. <tr>

  170. <th>version</th>

  171. <td>1.1</td>

  172. </tr>

  173. </table>

  174. <h3>

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

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

  177. <h2>Constructor: Check for symmetry, then construct the eigenvalue decomposition</h2>

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

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

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

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

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

  183. <th>access</th>

  184. <td>public</td>

  185. </tr></table>

  186. <h3>Parameters</h3>

  187. <div class="subelement argument">

  188. <h4>$Arg</h4>

  189. <code>\A</code><p>Square matrix</p></div>

  190. <h3>Returns</h3>

  191. <div class="subelement response">

  192. <code>\Structure</code>to access D and V.</div>

  193. </div></div>

  194. </div>

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

  196. <h2>Return the block diagonal eigenvalue matrix</h2>

  197. <pre>getD() : \D</pre>

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

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

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

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

  202. <th>access</th>

  203. <td>public</td>

  204. </tr></table>

  205. <h3>Returns</h3>

  206. <div class="subelement response"><code>\D</code></div>

  207. </div></div>

  208. </div>

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

  210. <h2>Return the imaginary parts of the eigenvalues</h2>

  211. <pre>getImagEigenvalues() : \imag(diag(D))</pre>

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

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

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

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

  216. <th>access</th>

  217. <td>public</td>

  218. </tr></table>

  219. <h3>Returns</h3>

  220. <div class="subelement response"><code>\imag(diag(D))</code></div>

  221. </div></div>

  222. </div>

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

  224. <h2>Return the real parts of the eigenvalues</h2>

  225. <pre>getRealEigenvalues() : \real(diag(D))</pre>

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

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

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

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

  230. <th>access</th>

  231. <td>public</td>

  232. </tr></table>

  233. <h3>Returns</h3>

  234. <div class="subelement response"><code>\real(diag(D))</code></div>

  235. </div></div>

  236. </div>

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

  238. <h2>Return the eigenvector matrix</h2>

  239. <pre>getV() : \V</pre>

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

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

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

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

  244. <th>access</th>

  245. <td>public</td>

  246. </tr></table>

  247. <h3>Returns</h3>

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

  249. </div></div>

  250. </div>

  251. <a id="method_cdiv"></a><div class="element clickable method private method_cdiv" data-toggle="collapse" data-target=".method_cdiv .collapse">

  252. <h2>Performs complex division.</h2>

  253. <pre>cdiv($xr, $xi, $yr, $yi) </pre>

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

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

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

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

  258. <th>access</th>

  259. <td>private</td>

  260. </tr></table>

  261. <h3>Parameters</h3>

  262. <div class="subelement argument"><h4>$xr</h4></div>

  263. <div class="subelement argument"><h4>$xi</h4></div>

  264. <div class="subelement argument"><h4>$yr</h4></div>

  265. <div class="subelement argument"><h4>$yi</h4></div>

  266. </div></div>

  267. </div>

  268. <a id="method_hqr2"></a><div class="element clickable method private method_hqr2" data-toggle="collapse" data-target=".method_hqr2 .collapse">

  269. <h2>Nonsymmetric reduction from Hessenberg to real Schur form.</h2>

  270. <pre>hqr2() </pre>

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

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

  273. <div class="long_description"><p>Code is derived from the Algol procedure hqr2,

  274. by Martin and Wilkinson, Handbook for Auto. Comp.,

  275. Vol.ii-Linear Algebra, and the corresponding

  276. Fortran subroutine in EISPACK.</p></div>

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

  278. <th>access</th>

  279. <td>private</td>

  280. </tr></table>

  281. </div></div>

  282. </div>

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

  284. <h2>Nonsymmetric reduction to Hessenberg form.</h2>

  285. <pre>orthes() </pre>

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

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

  288. <div class="long_description"><p>This is derived from the Algol procedures orthes and ortran,

  289. by Martin and Wilkinson, Handbook for Auto. Comp.,

  290. Vol.ii-Linear Algebra, and the corresponding

  291. Fortran subroutines in EISPACK.</p></div>

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

  293. <th>access</th>

  294. <td>private</td>

  295. </tr></table>

  296. </div></div>

  297. </div>

  298. <a id="method_tql2"></a><div class="element clickable method private method_tql2" data-toggle="collapse" data-target=".method_tql2 .collapse">

  299. <h2>Symmetric tridiagonal QL algorithm.</h2>

  300. <pre>tql2() </pre>

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

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

  303. <div class="long_description"><p>This is derived from the Algol procedures tql2, by

  304. Bowdler, Martin, Reinsch, and Wilkinson, Handbook for

  305. Auto. Comp., Vol.ii-Linear Algebra, and the corresponding

  306. Fortran subroutine in EISPACK.</p></div>

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

  308. <th>access</th>

  309. <td>private</td>

  310. </tr></table>

  311. </div></div>

  312. </div>

  313. <a id="method_tred2"></a><div class="element clickable method private method_tred2" data-toggle="collapse" data-target=".method_tred2 .collapse">

  314. <h2>Symmetric Householder reduction to tridiagonal form.</h2>

  315. <pre>tred2() </pre>

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

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

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

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

  320. <th>access</th>

  321. <td>private</td>

  322. </tr></table>

  323. </div></div>

  324. </div>

  325. <h3>

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

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

  328. <h2></h2>

  329. <pre>$H : array</pre>

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

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

  332. </div>

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

  334. <h2></h2>

  335. <pre>$V : array</pre>

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

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

  338. </div>

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

  340. <h2></h2>

  341. <pre>$cdivi </pre>

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

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

  344. </div>

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

  346. <h2></h2>

  347. <pre>$cdivr : float</pre>

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

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

  350. </div>

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

  352. <h2></h2>

  353. <pre>$d : array</pre>

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

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

  356. </div>

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

  358. <h2></h2>

  359. <pre>$e </pre>

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

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

  362. </div>

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

  364. <h2></h2>

  365. <pre>$issymmetric : int</pre>

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

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

  368. </div>

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

  370. <h2></h2>

  371. <pre>$n : int</pre>

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

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

  374. </div>

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

  376. <h2></h2>

  377. <pre>$ort : array</pre>

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

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

  380. </div>

  381. </div>

  382. </div>

  383. </div>

  384. </div>

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  388.             generated on 2014-03-02T15:27:36Z.<br></footer></div>

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