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| <div class="subTitle">org.apache.commons.math3.analysis.polynomials</div> |
| <h2 title="Class PolynomialsUtils" class="title">Class PolynomialsUtils</h2> |
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| <li>org.apache.commons.math3.analysis.polynomials.PolynomialsUtils</li> |
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| <pre>public class <span class="strong">PolynomialsUtils</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">A collection of static methods that operate on or return polynomials.</div> |
| <dl><dt><span class="strong">Since:</span></dt> |
| <dd>2.0</dd> |
| <dt><span class="strong">Version:</span></dt> |
| <dd>$Id: PolynomialsUtils.java 1517203 2013-08-24 21:55:35Z psteitz $</dd></dl> |
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| <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> |
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| <td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createChebyshevPolynomial(int)">createChebyshevPolynomial</a></strong>(int degree)</code> |
| <div class="block">Create a Chebyshev polynomial of the first kind.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createHermitePolynomial(int)">createHermitePolynomial</a></strong>(int degree)</code> |
| <div class="block">Create a Hermite polynomial.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createJacobiPolynomial(int, int, int)">createJacobiPolynomial</a></strong>(int degree, |
| int v, |
| int w)</code> |
| <div class="block">Create a Jacobi polynomial.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createLaguerrePolynomial(int)">createLaguerrePolynomial</a></strong>(int degree)</code> |
| <div class="block">Create a Laguerre polynomial.</div> |
| </td> |
| </tr> |
| <tr class="altColor"> |
| <td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createLegendrePolynomial(int)">createLegendrePolynomial</a></strong>(int degree)</code> |
| <div class="block">Create a Legendre polynomial.</div> |
| </td> |
| </tr> |
| <tr class="rowColor"> |
| <td class="colFirst"><code>static double[]</code></td> |
| <td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#shift(double[], double)">shift</a></strong>(double[] coefficients, |
| double shift)</code> |
| <div class="block">Compute the coefficients of the polynomial <code>P<sub>s</sub>(x)</code> |
| whose values at point <code>x</code> will be the same as the those from the |
| original polynomial <code>P(x)</code> when computed at <code>x + shift</code>.</div> |
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| <a name="createChebyshevPolynomial(int)"> |
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| <h4>createChebyshevPolynomial</h4> |
| <pre>public static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a> createChebyshevPolynomial(int degree)</pre> |
| <div class="block">Create a Chebyshev polynomial of the first kind. |
| <p><a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev |
| polynomials of the first kind</a> are orthogonal polynomials. |
| They can be defined by the following recurrence relations: |
| <pre> |
| T<sub>0</sub>(X) = 1 |
| T<sub>1</sub>(X) = X |
| T<sub>k+1</sub>(X) = 2X T<sub>k</sub>(X) - T<sub>k-1</sub>(X) |
| </pre></p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>Chebyshev polynomial of specified degree</dd></dl> |
| </li> |
| </ul> |
| <a name="createHermitePolynomial(int)"> |
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| </a> |
| <ul class="blockList"> |
| <li class="blockList"> |
| <h4>createHermitePolynomial</h4> |
| <pre>public static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a> createHermitePolynomial(int degree)</pre> |
| <div class="block">Create a Hermite polynomial. |
| <p><a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite |
| polynomials</a> are orthogonal polynomials. |
| They can be defined by the following recurrence relations: |
| <pre> |
| H<sub>0</sub>(X) = 1 |
| H<sub>1</sub>(X) = 2X |
| H<sub>k+1</sub>(X) = 2X H<sub>k</sub>(X) - 2k H<sub>k-1</sub>(X) |
| </pre></p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>Hermite polynomial of specified degree</dd></dl> |
| </li> |
| </ul> |
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| <h4>createLaguerrePolynomial</h4> |
| <pre>public static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a> createLaguerrePolynomial(int degree)</pre> |
| <div class="block">Create a Laguerre polynomial. |
| <p><a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre |
| polynomials</a> are orthogonal polynomials. |
| They can be defined by the following recurrence relations: |
| <pre> |
| L<sub>0</sub>(X) = 1 |
| L<sub>1</sub>(X) = 1 - X |
| (k+1) L<sub>k+1</sub>(X) = (2k + 1 - X) L<sub>k</sub>(X) - k L<sub>k-1</sub>(X) |
| </pre></p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>Laguerre polynomial of specified degree</dd></dl> |
| </li> |
| </ul> |
| <a name="createLegendrePolynomial(int)"> |
| <!-- --> |
| </a> |
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| <h4>createLegendrePolynomial</h4> |
| <pre>public static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a> createLegendrePolynomial(int degree)</pre> |
| <div class="block">Create a Legendre polynomial. |
| <p><a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre |
| polynomials</a> are orthogonal polynomials. |
| They can be defined by the following recurrence relations: |
| <pre> |
| P<sub>0</sub>(X) = 1 |
| P<sub>1</sub>(X) = X |
| (k+1) P<sub>k+1</sub>(X) = (2k+1) X P<sub>k</sub>(X) - k P<sub>k-1</sub>(X) |
| </pre></p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>Legendre polynomial of specified degree</dd></dl> |
| </li> |
| </ul> |
| <a name="createJacobiPolynomial(int, int, int)"> |
| <!-- --> |
| </a> |
| <ul class="blockList"> |
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| <h4>createJacobiPolynomial</h4> |
| <pre>public static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a> createJacobiPolynomial(int degree, |
| int v, |
| int w)</pre> |
| <div class="block">Create a Jacobi polynomial. |
| <p><a href="http://mathworld.wolfram.com/JacobiPolynomial.html">Jacobi |
| polynomials</a> are orthogonal polynomials. |
| They can be defined by the following recurrence relations: |
| <pre> |
| P<sub>0</sub><sup>vw</sup>(X) = 1 |
| P<sub>-1</sub><sup>vw</sup>(X) = 0 |
| 2k(k + v + w)(2k + v + w - 2) P<sub>k</sub><sup>vw</sup>(X) = |
| (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) X + v<sup>2</sup> - w<sup>2</sup>] P<sub>k-1</sub><sup>vw</sup>(X) |
| - 2(k + v - 1)(k + w - 1)(2k + v + w) P<sub>k-2</sub><sup>vw</sup>(X) |
| </pre></p></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd><dd><code>v</code> - first exponent</dd><dd><code>w</code> - second exponent</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>Jacobi polynomial of specified degree</dd></dl> |
| </li> |
| </ul> |
| <a name="shift(double[], double)"> |
| <!-- --> |
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| <h4>shift</h4> |
| <pre>public static double[] shift(double[] coefficients, |
| double shift)</pre> |
| <div class="block">Compute the coefficients of the polynomial <code>P<sub>s</sub>(x)</code> |
| whose values at point <code>x</code> will be the same as the those from the |
| original polynomial <code>P(x)</code> when computed at <code>x + shift</code>. |
| Thus, if <code>P(x) = Σ<sub>i</sub> a<sub>i</sub> x<sup>i</sup></code>, |
| then |
| <pre> |
| <table> |
| <tr> |
| <td><code>P<sub>s</sub>(x)</td> |
| <td>= Σ<sub>i</sub> b<sub>i</sub> x<sup>i</sup></code></td> |
| </tr> |
| <tr> |
| <td></td> |
| <td>= Σ<sub>i</sub> a<sub>i</sub> (x + shift)<sup>i</sup></code></td> |
| </tr> |
| </table> |
| </pre></div> |
| <dl><dt><span class="strong">Parameters:</span></dt><dd><code>coefficients</code> - Coefficients of the original polynomial.</dd><dd><code>shift</code> - Shift value.</dd> |
| <dt><span class="strong">Returns:</span></dt><dd>the coefficients <code>b<sub>i</sub></code> of the shifted |
| polynomial.</dd></dl> |
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