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| <div class="subTitle">org.apache.commons.math3.analysis.differentiation</div> |
| <h2 title="Class FiniteDifferencesDifferentiator" class="title">Class FiniteDifferencesDifferentiator</h2> |
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| <li>org.apache.commons.math3.analysis.differentiation.FiniteDifferencesDifferentiator</li> |
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| <pre>public class <span class="strong">FiniteDifferencesDifferentiator</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> |
| implements <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateFunctionDifferentiator.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateFunctionDifferentiator</a>, <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateVectorFunctionDifferentiator.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateVectorFunctionDifferentiator</a>, <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateMatrixFunctionDifferentiator.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateMatrixFunctionDifferentiator</a>, <a href="http://docs.oracle.com/javase/6/docs/api/java/io/Serializable.html?is-external=true" title="class or interface in java.io">Serializable</a></pre> |
| <div class="block">Univariate functions differentiator using finite differences. |
| <p> |
| This class creates some wrapper objects around regular |
| <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis"><code>univariate functions</code></a> (or <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateVectorFunction.html" title="interface in org.apache.commons.math3.analysis"><code>univariate vector functions</code></a> or <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateMatrixFunction.html" title="interface in org.apache.commons.math3.analysis"><code>univariate matrix functions</code></a>). These |
| wrapper objects compute derivatives in addition to function |
| value. |
| </p> |
| <p> |
| The wrapper objects work by calling the underlying function on |
| a sampling grid around the current point and performing polynomial |
| interpolation. A finite differences scheme with n points is |
| theoretically able to compute derivatives up to order n-1, but |
| it is generally better to have a slight margin. The step size must |
| also be small enough in order for the polynomial approximation to |
| be good in the current point neighborhood, but it should not be too |
| small because numerical instability appears quickly (there are several |
| differences of close points). Choosing the number of points and |
| the step size is highly problem dependent. |
| </p> |
| <p> |
| As an example of good and bad settings, lets consider the quintic |
| polynomial function <code>f(x) = (x-1)*(x-0.5)*x*(x+0.5)*(x+1)</code>. |
| Since it is a polynomial, finite differences with at least 6 points |
| should theoretically recover the exact same polynomial and hence |
| compute accurate derivatives for any order. However, due to numerical |
| errors, we get the following results for a 7 points finite differences |
| for abscissae in the [-10, 10] range: |
| <ul> |
| <li>step size = 0.25, second order derivative error about 9.97e-10</li> |
| <li>step size = 0.25, fourth order derivative error about 5.43e-8</li> |
| <li>step size = 1.0e-6, second order derivative error about 148</li> |
| <li>step size = 1.0e-6, fourth order derivative error about 6.35e+14</li> |
| </ul> |
| This example shows that the small step size is really bad, even simply |
| for second order derivative! |
| </p></div> |
| <dl><dt><span class="strong">Since:</span></dt> |
| <dd>3.1</dd> |
| <dt><span class="strong">Version:</span></dt> |
| <dd>$Id: FiniteDifferencesDifferentiator.java 1420666 2012-12-12 13:33:20Z erans $</dd> |
| <dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../../serialized-form.html#org.apache.commons.math3.analysis.differentiation.FiniteDifferencesDifferentiator">Serialized Form</a></dd></dl> |
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| <td class="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#FiniteDifferencesDifferentiator(int, double)">FiniteDifferencesDifferentiator</a></strong>(int nbPoints, |
| double stepSize)</code> |
| <div class="block">Build a differentiator with number of points and step size when independent variable is unbounded.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colOne"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#FiniteDifferencesDifferentiator(int, double, double, double)">FiniteDifferencesDifferentiator</a></strong>(int nbPoints, |
| double stepSize, |
| double tLower, |
| double tUpper)</code> |
| <div class="block">Build a differentiator with number of points and step size when independent variable is bounded.</div> |
| </td> |
| </tr> |
| </table> |
| </li> |
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| <tr class="altColor"> |
| <td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateDifferentiableFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#differentiate(org.apache.commons.math3.analysis.UnivariateFunction)">differentiate</a></strong>(<a href="../../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateFunction</a> function)</code> |
| <div class="block">Create an implementation of a <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation"><code>differential</code></a> from a regular <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis"><code>function</code></a>.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableMatrixFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateDifferentiableMatrixFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#differentiate(org.apache.commons.math3.analysis.UnivariateMatrixFunction)">differentiate</a></strong>(<a href="../../../../../../org/apache/commons/math3/analysis/UnivariateMatrixFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateMatrixFunction</a> function)</code> |
| <div class="block">Create an implementation of a <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableMatrixFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation"><code>differential</code></a> from a regular <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateMatrixFunction.html" title="interface in org.apache.commons.math3.analysis"><code>matrix function</code></a>.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableVectorFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateDifferentiableVectorFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#differentiate(org.apache.commons.math3.analysis.UnivariateVectorFunction)">differentiate</a></strong>(<a href="../../../../../../org/apache/commons/math3/analysis/UnivariateVectorFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateVectorFunction</a> function)</code> |
| <div class="block">Create an implementation of a <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableVectorFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation"><code>differential</code></a> from a regular <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateVectorFunction.html" title="interface in org.apache.commons.math3.analysis"><code>vector function</code></a>.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><code>int</code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#getNbPoints()">getNbPoints</a></strong>()</code> |
| <div class="block">Get the number of points to use.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><code>double</code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/FiniteDifferencesDifferentiator.html#getStepSize()">getStepSize</a></strong>()</code> |
| <div class="block">Get the step size.</div> |
| </td> |
| </tr> |
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| <code><a href="http://docs.oracle.com/javase/6/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/6/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/6/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/6/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/6/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/6/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/6/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/6/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/6/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/6/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/6/docs/api/java/lang/Object.html?is-external=true#wait(long, int)" title="class or interface in java.lang">wait</a></code></li> |
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| </a> |
| <h3>Constructor Detail</h3> |
| <a name="FiniteDifferencesDifferentiator(int, double)"> |
| <!-- --> |
| </a> |
| <ul class="blockList"> |
| <li class="blockList"> |
| <h4>FiniteDifferencesDifferentiator</h4> |
| <pre>public FiniteDifferencesDifferentiator(int nbPoints, |
| double stepSize) |
| throws <a href="../../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>, |
| <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></pre> |
| <div class="block">Build a differentiator with number of points and step size when independent variable is unbounded. |
| <p> |
| Beware that wrong settings for the finite differences differentiator |
| can lead to highly unstable and inaccurate results, especially for |
| high derivation orders. Using very small step sizes is often a |
| <em>bad</em> idea. |
| </p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>nbPoints</code> - number of points to use</dd><dd><code>stepSize</code> - step size (gap between each point)</dd> |
| <dt><span class="strong">Throws:</span></dt> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>stepsize <= 0</code> (note that |
| <a href="../../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception"><code>NotPositiveException</code></a> extends <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception"><code>NumberIsTooSmallException</code></a>)</dd> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></code> - <code>nbPoint <= 1</code></dd></dl> |
| </li> |
| </ul> |
| <a name="FiniteDifferencesDifferentiator(int, double, double, double)"> |
| <!-- --> |
| </a> |
| <ul class="blockListLast"> |
| <li class="blockList"> |
| <h4>FiniteDifferencesDifferentiator</h4> |
| <pre>public FiniteDifferencesDifferentiator(int nbPoints, |
| double stepSize, |
| double tLower, |
| double tUpper) |
| throws <a href="../../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a>, |
| <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a>, |
| <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></pre> |
| <div class="block">Build a differentiator with number of points and step size when independent variable is bounded. |
| <p> |
| When the independent variable is bounded (tLower < t < tUpper), the sampling |
| points used for differentiation will be adapted to ensure the constraint holds |
| even near the boundaries. This means the sample will not be centered anymore in |
| these cases. At an extreme case, computing derivatives exactly at the lower bound |
| will lead the sample to be entirely on the right side of the derivation point. |
| </p> |
| <p> |
| Note that the boundaries are considered to be excluded for function evaluation. |
| </p> |
| <p> |
| Beware that wrong settings for the finite differences differentiator |
| can lead to highly unstable and inaccurate results, especially for |
| high derivation orders. Using very small step sizes is often a |
| <em>bad</em> idea. |
| </p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>nbPoints</code> - number of points to use</dd><dd><code>stepSize</code> - step size (gap between each point)</dd><dd><code>tLower</code> - lower bound for independent variable (may be <code>Double.NEGATIVE_INFINITY</code> |
| if there are no lower bounds)</dd><dd><code>tUpper</code> - upper bound for independent variable (may be <code>Double.POSITIVE_INFINITY</code> |
| if there are no upper bounds)</dd> |
| <dt><span class="strong">Throws:</span></dt> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception">NotPositiveException</a></code> - if <code>stepsize <= 0</code> (note that |
| <a href="../../../../../../org/apache/commons/math3/exception/NotPositiveException.html" title="class in org.apache.commons.math3.exception"><code>NotPositiveException</code></a> extends <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception"><code>NumberIsTooSmallException</code></a>)</dd> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooSmallException.html" title="class in org.apache.commons.math3.exception">NumberIsTooSmallException</a></code> - <code>nbPoint <= 1</code></dd> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception">NumberIsTooLargeException</a></code> - <code>stepSize * (nbPoints - 1) >= tUpper - tLower</code></dd></dl> |
| </li> |
| </ul> |
| </li> |
| </ul> |
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| <h3>Method Detail</h3> |
| <a name="getNbPoints()"> |
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| </a> |
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| <li class="blockList"> |
| <h4>getNbPoints</h4> |
| <pre>public int getNbPoints()</pre> |
| <div class="block">Get the number of points to use.</div> |
| <dl><dt><span class="strong">Returns:</span></dt><dd>number of points to use</dd></dl> |
| </li> |
| </ul> |
| <a name="getStepSize()"> |
| <!-- --> |
| </a> |
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| <li class="blockList"> |
| <h4>getStepSize</h4> |
| <pre>public double getStepSize()</pre> |
| <div class="block">Get the step size.</div> |
| <dl><dt><span class="strong">Returns:</span></dt><dd>step size</dd></dl> |
| </li> |
| </ul> |
| <a name="differentiate(org.apache.commons.math3.analysis.UnivariateFunction)"> |
| <!-- --> |
| </a> |
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| <h4>differentiate</h4> |
| <pre>public <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateDifferentiableFunction</a> differentiate(<a href="../../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateFunction</a> function)</pre> |
| <div class="block">Create an implementation of a <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation"><code>differential</code></a> from a regular <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateFunction.html" title="interface in org.apache.commons.math3.analysis"><code>function</code></a>. |
| <p>The returned object cannot compute derivatives to arbitrary orders. The |
| value function will throw a <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception"><code>NumberIsTooLargeException</code></a> if the requested |
| derivation order is larger or equal to the number of points. |
| </p></div> |
| <dl> |
| <dt><strong>Specified by:</strong></dt> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateFunctionDifferentiator.html#differentiate(org.apache.commons.math3.analysis.UnivariateFunction)">differentiate</a></code> in interface <code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateFunctionDifferentiator.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateFunctionDifferentiator</a></code></dd> |
| <dt><span class="strong">Parameters:</span></dt><dd><code>function</code> - function to differentiate</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>differential function</dd></dl> |
| </li> |
| </ul> |
| <a name="differentiate(org.apache.commons.math3.analysis.UnivariateVectorFunction)"> |
| <!-- --> |
| </a> |
| <ul class="blockList"> |
| <li class="blockList"> |
| <h4>differentiate</h4> |
| <pre>public <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableVectorFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateDifferentiableVectorFunction</a> differentiate(<a href="../../../../../../org/apache/commons/math3/analysis/UnivariateVectorFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateVectorFunction</a> function)</pre> |
| <div class="block">Create an implementation of a <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableVectorFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation"><code>differential</code></a> from a regular <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateVectorFunction.html" title="interface in org.apache.commons.math3.analysis"><code>vector function</code></a>. |
| <p>The returned object cannot compute derivatives to arbitrary orders. The |
| value function will throw a <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception"><code>NumberIsTooLargeException</code></a> if the requested |
| derivation order is larger or equal to the number of points. |
| </p></div> |
| <dl> |
| <dt><strong>Specified by:</strong></dt> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateVectorFunctionDifferentiator.html#differentiate(org.apache.commons.math3.analysis.UnivariateVectorFunction)">differentiate</a></code> in interface <code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateVectorFunctionDifferentiator.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateVectorFunctionDifferentiator</a></code></dd> |
| <dt><span class="strong">Parameters:</span></dt><dd><code>function</code> - function to differentiate</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>differential function</dd></dl> |
| </li> |
| </ul> |
| <a name="differentiate(org.apache.commons.math3.analysis.UnivariateMatrixFunction)"> |
| <!-- --> |
| </a> |
| <ul class="blockListLast"> |
| <li class="blockList"> |
| <h4>differentiate</h4> |
| <pre>public <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableMatrixFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateDifferentiableMatrixFunction</a> differentiate(<a href="../../../../../../org/apache/commons/math3/analysis/UnivariateMatrixFunction.html" title="interface in org.apache.commons.math3.analysis">UnivariateMatrixFunction</a> function)</pre> |
| <div class="block">Create an implementation of a <a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateDifferentiableMatrixFunction.html" title="interface in org.apache.commons.math3.analysis.differentiation"><code>differential</code></a> from a regular <a href="../../../../../../org/apache/commons/math3/analysis/UnivariateMatrixFunction.html" title="interface in org.apache.commons.math3.analysis"><code>matrix function</code></a>. |
| <p>The returned object cannot compute derivatives to arbitrary orders. The |
| value function will throw a <a href="../../../../../../org/apache/commons/math3/exception/NumberIsTooLargeException.html" title="class in org.apache.commons.math3.exception"><code>NumberIsTooLargeException</code></a> if the requested |
| derivation order is larger or equal to the number of points. |
| </p></div> |
| <dl> |
| <dt><strong>Specified by:</strong></dt> |
| <dd><code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateMatrixFunctionDifferentiator.html#differentiate(org.apache.commons.math3.analysis.UnivariateMatrixFunction)">differentiate</a></code> in interface <code><a href="../../../../../../org/apache/commons/math3/analysis/differentiation/UnivariateMatrixFunctionDifferentiator.html" title="interface in org.apache.commons.math3.analysis.differentiation">UnivariateMatrixFunctionDifferentiator</a></code></dd> |
| <dt><span class="strong">Parameters:</span></dt><dd><code>function</code> - function to differentiate</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>differential function</dd></dl> |
| </li> |
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