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<div class="subTitle">org.apache.commons.math3.ode.nonstiff</div>
<h2 title="Class AdamsNordsieckFieldTransformer" class="title">Class AdamsNordsieckFieldTransformer<T extends <a href="../../../../../../org/apache/commons/math3/RealFieldElement.html" title="interface in org.apache.commons.math3">RealFieldElement</a><T>></h2>
</div>
<div class="contentContainer">
<ul class="inheritance">
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<li>org.apache.commons.math3.ode.nonstiff.AdamsNordsieckFieldTransformer<T></li>
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<div class="description">
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<dl><dt><span class="strong">Type Parameters:</span></dt><dd><code>T</code> - the type of the field elements</dd></dl>
<hr>
<br>
<pre>public class <span class="strong">AdamsNordsieckFieldTransformer<T extends <a href="../../../../../../org/apache/commons/math3/RealFieldElement.html" title="interface in org.apache.commons.math3">RealFieldElement</a><T>></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>3.6</dd></dl>
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<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 <T extends <a href="../../../../../../org/apache/commons/math3/RealFieldElement.html" title="interface in org.apache.commons.math3">RealFieldElement</a><T>> <br><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsNordsieckFieldTransformer</a><T></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html#getInstance(org.apache.commons.math3.Field,%20int)">getInstance</a></strong>(<a href="../../../../../../org/apache/commons/math3/Field.html" title="interface in org.apache.commons.math3">Field</a><T> field,
int nSteps)</code>
<div class="block">Get the Nordsieck transformer for a given field and number of steps.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html#initializeHighOrderDerivatives(T,%20T[],%20T[][],%20T[][])">initializeHighOrderDerivatives</a></strong>(<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a> h,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[] t,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[][] y,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[][] yDot)</code>
<div class="block">Initialize the high order scaled derivatives at step start.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html#updateHighOrderDerivativesPhase1(org.apache.commons.math3.linear.Array2DRowFieldMatrix)">updateHighOrderDerivativesPhase1</a></strong>(<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>> highOrder)</code>
<div class="block">Update the high order scaled derivatives for Adams integrators (phase 1).</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>void</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html#updateHighOrderDerivativesPhase2(T[],%20T[],%20org.apache.commons.math3.linear.Array2DRowFieldMatrix)">updateHighOrderDerivativesPhase2</a></strong>(<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[] start,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[] end,
<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>> highOrder)</code>
<div class="block">Update the high order scaled derivatives Adams integrators (phase 2).</div>
</td>
</tr>
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<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>
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<h4>getInstance</h4>
<pre>public static <T extends <a href="../../../../../../org/apache/commons/math3/RealFieldElement.html" title="interface in org.apache.commons.math3">RealFieldElement</a><T>> <a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="class in org.apache.commons.math3.ode.nonstiff">AdamsNordsieckFieldTransformer</a><T> getInstance(<a href="../../../../../../org/apache/commons/math3/Field.html" title="interface in org.apache.commons.math3">Field</a><T> field,
int nSteps)</pre>
<div class="block">Get the Nordsieck transformer for a given field and number of steps.</div>
<dl><dt><span class="strong">Type Parameters:</span></dt><dd><code>T</code> - the type of the field elements</dd><dt><span class="strong">Parameters:</span></dt><dd><code>field</code> - field to which the time and state vector elements belong</dd><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 field and number of steps</dd></dl>
</li>
</ul>
<a name="initializeHighOrderDerivatives(org.apache.commons.math3.RealFieldElement,org.apache.commons.math3.RealFieldElement[],org.apache.commons.math3.RealFieldElement[][],org.apache.commons.math3.RealFieldElement[][])">
<!-- -->
</a><a name="initializeHighOrderDerivatives(T, T[], T[][], T[][])">
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<h4>initializeHighOrderDerivatives</h4>
<pre>public <a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>> initializeHighOrderDerivatives(<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a> h,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[] t,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[][] y,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[][] 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>
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<h4>updateHighOrderDerivativesPhase1</h4>
<pre>public <a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>> updateHighOrderDerivativesPhase1(<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</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/AdamsNordsieckFieldTransformer.html#updateHighOrderDerivativesPhase2(T[],%20T[],%20org.apache.commons.math3.linear.Array2DRowFieldMatrix)"><code>updateHighOrderDerivativesPhase2(RealFieldElement[], RealFieldElement[], Array2DRowFieldMatrix)</code></a></dd></dl>
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<h4>updateHighOrderDerivativesPhase2</h4>
<pre>public void updateHighOrderDerivativesPhase2(<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[] start,
<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</a>[] end,
<a href="../../../../../../org/apache/commons/math3/linear/Array2DRowFieldMatrix.html" title="class in org.apache.commons.math3.linear">Array2DRowFieldMatrix</a><<a href="../../../../../../org/apache/commons/math3/ode/nonstiff/AdamsNordsieckFieldTransformer.html" title="type parameter in AdamsNordsieckFieldTransformer">T</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/AdamsNordsieckFieldTransformer.html#updateHighOrderDerivativesPhase1(org.apache.commons.math3.linear.Array2DRowFieldMatrix)"><code>updateHighOrderDerivativesPhase1(Array2DRowFieldMatrix)</code></a></dd></dl>
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