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<div class="subTitle">org.apache.commons.math3.ode.nonstiff</div>
<h2 title="Class AdamsNordsieckTransformer" class="title">Class AdamsNordsieckTransformer</h2>
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<li>org.apache.commons.math3.ode.nonstiff.AdamsNordsieckTransformer</li>
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<pre>public class <span class="strong">AdamsNordsieckTransformer</span>
extends <a href="http://docs.oracle.com/javase/6/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) + &sum;<sub>j&gt;1</sub> j (-i)<sup>j-1</sup> s<sub>j</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)&times;(k-1) matrix built
with the j (-i)<sup>j-1</sup> terms:
<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 ... &plusmn;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>
<dt><span class="strong">Version:</span></dt>
<dd>$Id: AdamsNordsieckTransformer.java 1416643 2012-12-03 19:37:14Z tn $</dd></dl>
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<caption><span>Methods</span><span class="tabEnd">&nbsp;</span></caption>
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<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
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<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&nbsp;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">Get the number of steps of the method
(excluding the one being computed).</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, double[], double[][], double[][])">initializeHighOrderDerivatives</a></strong>(double&nbsp;h,
double[]&nbsp;t,
double[][]&nbsp;y,
double[][]&nbsp;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>&nbsp;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[], double[], org.apache.commons.math3.linear.Array2DRowRealMatrix)">updateHighOrderDerivativesPhase2</a></strong>(double[]&nbsp;start,
double[]&nbsp;end,
<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a>&nbsp;highOrder)</code>
<div class="block">Update the high order scaled derivatives Adams integrators (phase 2).</div>
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<h4>getInstance</h4>
<pre>public static&nbsp;<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckTransformer.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsNordsieckTransformer</a>&nbsp;getInstance(int&nbsp;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>
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<h4>getNSteps</h4>
<pre>public&nbsp;int&nbsp;getNSteps()</pre>
<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>
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<h4>initializeHighOrderDerivatives</h4>
<pre>public&nbsp;<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a>&nbsp;initializeHighOrderDerivatives(double&nbsp;h,
double[]&nbsp;t,
double[][]&nbsp;y,
double[][]&nbsp;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 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>
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<h4>updateHighOrderDerivativesPhase1</h4>
<pre>public&nbsp;<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a>&nbsp;updateHighOrderDerivativesPhase1(<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a>&nbsp;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[], double[], org.apache.commons.math3.linear.Array2DRowRealMatrix)"><code>updateHighOrderDerivativesPhase2(double[], double[], Array2DRowRealMatrix)</code></a></dd></dl>
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<h4>updateHighOrderDerivativesPhase2</h4>
<pre>public&nbsp;void&nbsp;updateHighOrderDerivativesPhase2(double[]&nbsp;start,
double[]&nbsp;end,
<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowRealMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowRealMatrix</a>&nbsp;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>
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