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diff --git a/commons-math3-3.6.1/docs/apidocs/org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html b/commons-math3-3.6.1/docs/apidocs/org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html new file mode 100644 index 0000000..17eabd3 --- /dev/null +++ b/commons-math3-3.6.1/docs/apidocs/org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html @@ -0,0 +1,429 @@ +<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> +<!-- NewPage --> +<html lang="en"> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> +<title>AdamsNordsieckTransformer (Apache Commons Math 3.6.1 API)</title> +<link rel="stylesheet" type="text/css" href="../../../../../../stylesheet.css" title="Style"> +</head> +<body> +<script type="text/javascript"><!-- + if (location.href.indexOf('is-external=true') == -1) { + parent.document.title="AdamsNordsieckTransformer (Apache Commons Math 3.6.1 API)"; + } +//--> +</script> +<noscript> +<div>JavaScript is disabled on your browser.</div> +</noscript> +<!-- ========= START OF TOP NAVBAR ======= --> +<div class="topNav"><a name="navbar_top"> +<!-- --> +</a><a href="#skip-navbar_top" title="Skip navigation links"></a><a name="navbar_top_firstrow"> +<!-- --> +</a> +<ul class="navList" title="Navigation"> +<li><a href="../../../../../../overview-summary.html">Overview</a></li> +<li><a href="package-summary.html">Package</a></li> +<li class="navBarCell1Rev">Class</li> +<li><a href="class-use/AdamsNordsieckTransformer.html">Use</a></li> +<li><a href="package-tree.html">Tree</a></li> +<li><a href="../../../../../../deprecated-list.html">Deprecated</a></li> +<li><a href="../../../../../../index-all.html">Index</a></li> +<li><a href="../../../../../../help-doc.html">Help</a></li> +</ul> +<div class="aboutLanguage"><em><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script></em></div> +</div> +<div class="subNav"> +<ul class="navList"> +<li><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="class in org.apache.commons.math3.ode.nonstiff"><span class="strong">Prev Class</span></a></li> +<li><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdaptiveStepsizeFieldIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><span class="strong">Next Class</span></a></li> +</ul> +<ul class="navList"> +<li><a href="../../../../../../index.html?org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html" target="_top">Frames</a></li> +<li><a href="AdamsNordsieckTransformer.html" target="_top">No Frames</a></li> +</ul> +<ul class="navList" id="allclasses_navbar_top"> +<li><a href="../../../../../../allclasses-noframe.html">All Classes</a></li> +</ul> +<div> +<script type="text/javascript"><!-- + allClassesLink = document.getElementById("allclasses_navbar_top"); + if(window==top) { + allClassesLink.style.display = "block"; + } + else { + allClassesLink.style.display = "none"; + } + //--> +</script> +</div> +<div> +<ul class="subNavList"> +<li>Summary: </li> +<li>Nested | </li> +<li>Field | </li> +<li>Constr | </li> +<li><a href="#method_summary">Method</a></li> +</ul> +<ul class="subNavList"> +<li>Detail: </li> +<li>Field | </li> +<li>Constr | </li> +<li><a href="#method_detail">Method</a></li> +</ul> +</div> +<a name="skip-navbar_top"> +<!-- --> +</a></div> +<!-- ========= END OF TOP NAVBAR ========= --> +<!-- ======== START OF CLASS DATA ======== --> +<div class="header"> +<div class="subTitle">org.apache.commons.math3.ode.nonstiff</div> +<h2 title="Class AdamsNordsieckTransformer" class="title">Class AdamsNordsieckTransformer</h2> +</div> +<div class="contentContainer"> +<ul class="inheritance"> +<li><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">java.lang.Object</a></li> +<li> +<ul class="inheritance"> +<li>org.apache.commons.math3.ode.nonstiff.AdamsNordsieckTransformer</li> +</ul> +</li> +</ul> +<div class="description"> +<ul class="blockList"> +<li class="blockList"> +<hr> +<br> +<pre>public class <span class="strong">AdamsNordsieckTransformer</span> +extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre> +<div class="block">Transformer to Nordsieck vectors for Adams integrators. + <p>This class is used by <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsBashforthIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Adams-Bashforth</code></a> and + <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Adams-Moulton</code></a> integrators to convert between + classical representation with several previous first derivatives and Nordsieck + representation with higher order scaled derivatives.</p> + + <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: + <pre> + s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative + s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative + s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative + ... + s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative + </pre></p> + + <p>With the previous definition, the classical representation of multistep methods + uses first derivatives only, i.e. it handles y<sub>n</sub>, s<sub>1</sub>(n) and + q<sub>n</sub> where q<sub>n</sub> is defined as: + <pre> + q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup> + </pre> + (we omit the k index in the notation for clarity).</p> + + <p>Another possible representation uses the Nordsieck vector with + higher degrees scaled derivatives all taken at the same step, i.e it handles y<sub>n</sub>, + s<sub>1</sub>(n) and r<sub>n</sub>) where r<sub>n</sub> is defined as: + <pre> + r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> + </pre> + (here again we omit the k index in the notation for clarity) + </p> + + <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be + computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact + for degree k polynomials. + <pre> + s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n) + </pre> + The previous formula can be used with several values for i to compute the transform between + classical representation and Nordsieck vector at step end. The transform between r<sub>n</sub> + and q<sub>n</sub> resulting from the Taylor series formulas above is: + <pre> + q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> + </pre> + where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built + with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being + the column number starting from 1: + <pre> + [ -2 3 -4 5 ... ] + [ -4 12 -32 80 ... ] + P = [ -6 27 -108 405 ... ] + [ -8 48 -256 1280 ... ] + [ ... ] + </pre></p> + + <p>Changing -i into +i in the formula above can be used to compute a similar transform between + classical representation and Nordsieck vector at step start. The resulting matrix is simply + the absolute value of matrix P.</p> + + <p>For <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsBashforthIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Adams-Bashforth</code></a> method, the Nordsieck vector + at step n+1 is computed from the Nordsieck vector at step n as follows: + <ul> + <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> + <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> + <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li> + </ul> + where A is a rows shifting matrix (the lower left part is an identity matrix): + <pre> + [ 0 0 ... 0 0 | 0 ] + [ ---------------+---] + [ 1 0 ... 0 0 | 0 ] + A = [ 0 1 ... 0 0 | 0 ] + [ ... | 0 ] + [ 0 0 ... 1 0 | 0 ] + [ 0 0 ... 0 1 | 0 ] + </pre></p> + + <p>For <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsMoultonIntegrator.html" title="class in org.apache.commons.math3.ode.nonstiff"><code>Adams-Moulton</code></a> method, the predicted Nordsieck vector + at step n+1 is computed from the Nordsieck vector at step n as follows: + <ul> + <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> + <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> + <li>R<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li> + </ul> + From this predicted vector, the corrected vector is computed as follows: + <ul> + <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li> + <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> + <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li> + </ul> + where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the + predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> + represent the corrected states.</p> + + <p>We observe that both methods use similar update formulas. In both cases a P<sup>-1</sup>u + vector and a P<sup>-1</sup> A P matrix are used that do not depend on the state, + they only depend on k. This class handles these transformations.</p></div> +<dl><dt><span class="strong">Since:</span></dt> + <dd>2.0</dd></dl> +</li> +</ul> +</div> +<div class="summary"> +<ul class="blockList"> +<li class="blockList"> +<!-- ========== METHOD SUMMARY =========== --> +<ul class="blockList"> +<li class="blockList"><a name="method_summary"> +<!-- --> +</a> +<h3>Method Summary</h3> +<table class="overviewSummary" border="0" cellpadding="3" cellspacing="0" summary="Method Summary table, listing methods, and an explanation"> +<caption><span>Methods</span><span class="tabEnd"> </span></caption> +<tr> +<th class="colFirst" scope="col">Modifier and Type</th> +<th class="colLast" scope="col">Method and Description</th> +</tr> +<tr class="altColor"> +<td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsNordsieckTransformer</a></code></td> +<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#getInstance(int)">getInstance</a></strong>(int nSteps)</code> +<div class="block">Get the Nordsieck transformer for a given number of steps.</div> +</td> +</tr> +<tr class="rowColor"> +<td class="colFirst"><code>int</code></td> +<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#getNSteps()">getNSteps</a></strong>()</code> +<div class="block"><strong>Deprecated.</strong> +<div class="block"><i>as of 3.6, this method is not used anymore</i></div> +</div> +</td> +</tr> +<tr class="altColor"> +<td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a></code></td> +<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#initializeHighOrderDerivatives(double,%20double[],%20double[][],%20double[][])">initializeHighOrderDerivatives</a></strong>(double h, + double[] t, + double[][] y, + double[][] yDot)</code> +<div class="block">Initialize the high order scaled derivatives at step start.</div> +</td> +</tr> +<tr class="rowColor"> +<td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a></code></td> +<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#updateHighOrderDerivativesPhase1(org.apache.commons.math3.linear.Array2DRowRealMatrix)">updateHighOrderDerivativesPhase1</a></strong>(<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a> highOrder)</code> +<div class="block">Update the high order scaled derivatives for Adams integrators (phase 1).</div> +</td> +</tr> +<tr class="altColor"> +<td class="colFirst"><code>void</code></td> +<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#updateHighOrderDerivativesPhase2(double[],%20double[],%20org.apache.commons.math3.linear.Array2DRowRealMatrix)">updateHighOrderDerivativesPhase2</a></strong>(double[] start, + double[] end, + <a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a> highOrder)</code> +<div class="block">Update the high order scaled derivatives Adams integrators (phase 2).</div> +</td> +</tr> +</table> +<ul class="blockList"> +<li class="blockList"><a name="methods_inherited_from_class_java.lang.Object"> +<!-- --> +</a> +<h3>Methods inherited from class java.lang.<a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></h3> +<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long,%20int)" title="class or interface in java.lang">wait</a></code></li> +</ul> +</li> +</ul> +</li> +</ul> +</div> +<div class="details"> +<ul class="blockList"> +<li class="blockList"> +<!-- ============ METHOD DETAIL ========== --> +<ul class="blockList"> +<li class="blockList"><a name="method_detail"> +<!-- --> +</a> +<h3>Method Detail</h3> +<a name="getInstance(int)"> +<!-- --> +</a> +<ul class="blockList"> +<li class="blockList"> +<h4>getInstance</h4> +<pre>public static <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsNordsieckTransformer</a> getInstance(int nSteps)</pre> +<div class="block">Get the Nordsieck transformer for a given number of steps.</div> +<dl><dt><span class="strong">Parameters:</span></dt><dd><code>nSteps</code> - number of steps of the multistep method + (excluding the one being computed)</dd> +<dt><span class="strong">Returns:</span></dt><dd>Nordsieck transformer for the specified number of steps</dd></dl> +</li> +</ul> +<a name="getNSteps()"> +<!-- --> +</a> +<ul class="blockList"> +<li class="blockList"> +<h4>getNSteps</h4> +<pre><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Deprecated.html?is-external=true" title="class or interface in java.lang">@Deprecated</a> +public int getNSteps()</pre> +<div class="block"><span class="strong">Deprecated.</span> <i>as of 3.6, this method is not used anymore</i></div> +<div class="block">Get the number of steps of the method + (excluding the one being computed).</div> +<dl><dt><span class="strong">Returns:</span></dt><dd>number of steps of the method + (excluding the one being computed)</dd></dl> +</li> +</ul> +<a name="initializeHighOrderDerivatives(double, double[], double[][], double[][])"> +<!-- --> +</a> +<ul class="blockList"> +<li class="blockList"> +<h4>initializeHighOrderDerivatives</h4> +<pre>public <a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a> initializeHighOrderDerivatives(double h, + double[] t, + double[][] y, + double[][] yDot)</pre> +<div class="block">Initialize the high order scaled derivatives at step start.</div> +<dl><dt><span class="strong">Parameters:</span></dt><dd><code>h</code> - step size to use for scaling</dd><dd><code>t</code> - first steps times</dd><dd><code>y</code> - first steps states</dd><dd><code>yDot</code> - first steps derivatives</dd> +<dt><span class="strong">Returns:</span></dt><dd>Nordieck vector at start of first step (h<sup>2</sup>/2 y''<sub>n</sub>, + h<sup>3</sup>/6 y'''<sub>n</sub> ... h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub>)</dd></dl> +</li> +</ul> +<a name="updateHighOrderDerivativesPhase1(org.apache.commons.math3.linear.Array2DRowRealMatrix)"> +<!-- --> +</a> +<ul class="blockList"> +<li class="blockList"> +<h4>updateHighOrderDerivativesPhase1</h4> +<pre>public <a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a> updateHighOrderDerivativesPhase1(<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a> highOrder)</pre> +<div class="block">Update the high order scaled derivatives for Adams integrators (phase 1). + <p>The complete update of high order derivatives has a form similar to: + <pre> + r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub> + </pre> + this method computes the P<sup>-1</sup> A P r<sub>n</sub> part.</p></div> +<dl><dt><span class="strong">Parameters:</span></dt><dd><code>highOrder</code> - high order scaled derivatives + (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))</dd> +<dt><span class="strong">Returns:</span></dt><dd>updated high order derivatives</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#updateHighOrderDerivativesPhase2(double[],%20double[],%20org.apache.commons.math3.linear.Array2DRowRealMatrix)"><code>updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)</code></a></dd></dl> +</li> +</ul> +<a name="updateHighOrderDerivativesPhase2(double[], double[], org.apache.commons.math3.linear.Array2DRowRealMatrix)"> +<!-- --> +</a> +<ul class="blockListLast"> +<li class="blockList"> +<h4>updateHighOrderDerivativesPhase2</h4> +<pre>public void updateHighOrderDerivativesPhase2(double[] start, + double[] end, + <a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a> highOrder)</pre> +<div class="block">Update the high order scaled derivatives Adams integrators (phase 2). + <p>The complete update of high order derivatives has a form similar to: + <pre> + r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub> + </pre> + this method computes the (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u part.</p> + <p>Phase 1 of the update must already have been performed.</p></div> +<dl><dt><span class="strong">Parameters:</span></dt><dd><code>start</code> - first order scaled derivatives at step start</dd><dd><code>end</code> - first order scaled derivatives at step end</dd><dd><code>highOrder</code> - high order scaled derivatives, will be modified + (h<sup>2</sup>/2 y'', ... h<sup>k</sup>/k! y(k))</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html#updateHighOrderDerivativesPhase1(org.apache.commons.math3.linear.Array2DRowRealMatrix)"><code>updateHighOrderDerivativesPhase1(Array2DRowRealMatrix)</code></a></dd></dl> +</li> +</ul> +</li> +</ul> +</li> +</ul> +</div> +</div> +<!-- 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