1dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond/* 2dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Licensed to the Apache Software Foundation (ASF) under one or more 3dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * contributor license agreements. See the NOTICE file distributed with 4dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * this work for additional information regarding copyright ownership. 5dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * The ASF licenses this file to You under the Apache License, Version 2.0 6dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * (the "License"); you may not use this file except in compliance with 7dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * the License. You may obtain a copy of the License at 8dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * 9dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * http://www.apache.org/licenses/LICENSE-2.0 10dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * 11dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Unless required by applicable law or agreed to in writing, software 12dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * distributed under the License is distributed on an "AS IS" BASIS, 13dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 14dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * See the License for the specific language governing permissions and 15dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * limitations under the License. 16dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 17dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondpackage org.apache.commons.math.analysis.polynomials; 18dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 19dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport java.util.ArrayList; 20dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 21dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.fraction.BigFraction; 22dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondimport org.apache.commons.math.util.FastMath; 23dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 24dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond/** 25dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * A collection of static methods that operate on or return polynomials. 26dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * 27dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ 28dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @since 2.0 29dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 30dee0849a9704d532af0b550146cbafbaa6ee1d19Raymondpublic class PolynomialsUtils { 31dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 32dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Coefficients for Chebyshev polynomials. */ 33dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final ArrayList<BigFraction> CHEBYSHEV_COEFFICIENTS; 34dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 35dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Coefficients for Hermite polynomials. */ 36dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final ArrayList<BigFraction> HERMITE_COEFFICIENTS; 37dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 38dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Coefficients for Laguerre polynomials. */ 39dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final ArrayList<BigFraction> LAGUERRE_COEFFICIENTS; 40dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 41dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Coefficients for Legendre polynomials. */ 42dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static final ArrayList<BigFraction> LEGENDRE_COEFFICIENTS; 43dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 44dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond static { 45dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 46dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // initialize recurrence for Chebyshev polynomials 47dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // T0(X) = 1, T1(X) = 0 + 1 * X 48dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond CHEBYSHEV_COEFFICIENTS = new ArrayList<BigFraction>(); 49dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE); 50dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond CHEBYSHEV_COEFFICIENTS.add(BigFraction.ZERO); 51dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE); 52dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 53dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // initialize recurrence for Hermite polynomials 54dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // H0(X) = 1, H1(X) = 0 + 2 * X 55dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond HERMITE_COEFFICIENTS = new ArrayList<BigFraction>(); 56dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond HERMITE_COEFFICIENTS.add(BigFraction.ONE); 57dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond HERMITE_COEFFICIENTS.add(BigFraction.ZERO); 58dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond HERMITE_COEFFICIENTS.add(BigFraction.TWO); 59dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 60dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // initialize recurrence for Laguerre polynomials 61dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // L0(X) = 1, L1(X) = 1 - 1 * X 62dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LAGUERRE_COEFFICIENTS = new ArrayList<BigFraction>(); 63dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LAGUERRE_COEFFICIENTS.add(BigFraction.ONE); 64dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LAGUERRE_COEFFICIENTS.add(BigFraction.ONE); 65dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LAGUERRE_COEFFICIENTS.add(BigFraction.MINUS_ONE); 66dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 67dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // initialize recurrence for Legendre polynomials 68dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // P0(X) = 1, P1(X) = 0 + 1 * X 69dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LEGENDRE_COEFFICIENTS = new ArrayList<BigFraction>(); 70dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LEGENDRE_COEFFICIENTS.add(BigFraction.ONE); 71dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LEGENDRE_COEFFICIENTS.add(BigFraction.ZERO); 72dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond LEGENDRE_COEFFICIENTS.add(BigFraction.ONE); 73dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 74dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 75dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 76dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 77dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Private constructor, to prevent instantiation. 78dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 79dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private PolynomialsUtils() { 80dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 81dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 82dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 83dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Create a Chebyshev polynomial of the first kind. 84dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p><a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev 85dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * polynomials of the first kind</a> are orthogonal polynomials. 86dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * They can be defined by the following recurrence relations: 87dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <pre> 88dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * T<sub>0</sub>(X) = 1 89dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * T<sub>1</sub>(X) = X 90dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * T<sub>k+1</sub>(X) = 2X T<sub>k</sub>(X) - T<sub>k-1</sub>(X) 91dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </pre></p> 92dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param degree degree of the polynomial 93dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return Chebyshev polynomial of specified degree 94dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 95dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public static PolynomialFunction createChebyshevPolynomial(final int degree) { 96dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return buildPolynomial(degree, CHEBYSHEV_COEFFICIENTS, 97dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new RecurrenceCoefficientsGenerator() { 98dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private final BigFraction[] coeffs = { BigFraction.ZERO, BigFraction.TWO, BigFraction.ONE }; 99dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** {@inheritDoc} */ 100dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public BigFraction[] generate(int k) { 101dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return coeffs; 102dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 103dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }); 104dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 105dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 106dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 107dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Create a Hermite polynomial. 108dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p><a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite 109dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * polynomials</a> are orthogonal polynomials. 110dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * They can be defined by the following recurrence relations: 111dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <pre> 112dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * H<sub>0</sub>(X) = 1 113dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * H<sub>1</sub>(X) = 2X 114dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * H<sub>k+1</sub>(X) = 2X H<sub>k</sub>(X) - 2k H<sub>k-1</sub>(X) 115dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </pre></p> 116dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 117dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param degree degree of the polynomial 118dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return Hermite polynomial of specified degree 119dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 120dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public static PolynomialFunction createHermitePolynomial(final int degree) { 121dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return buildPolynomial(degree, HERMITE_COEFFICIENTS, 122dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new RecurrenceCoefficientsGenerator() { 123dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** {@inheritDoc} */ 124dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public BigFraction[] generate(int k) { 125dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return new BigFraction[] { 126dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction.ZERO, 127dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction.TWO, 128dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new BigFraction(2 * k)}; 129dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 130dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }); 131dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 132dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 133dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 134dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Create a Laguerre polynomial. 135dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p><a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre 136dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * polynomials</a> are orthogonal polynomials. 137dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * They can be defined by the following recurrence relations: 138dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <pre> 139dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * L<sub>0</sub>(X) = 1 140dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * L<sub>1</sub>(X) = 1 - X 141dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * (k+1) L<sub>k+1</sub>(X) = (2k + 1 - X) L<sub>k</sub>(X) - k L<sub>k-1</sub>(X) 142dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </pre></p> 143dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param degree degree of the polynomial 144dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return Laguerre polynomial of specified degree 145dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 146dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public static PolynomialFunction createLaguerrePolynomial(final int degree) { 147dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return buildPolynomial(degree, LAGUERRE_COEFFICIENTS, 148dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new RecurrenceCoefficientsGenerator() { 149dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** {@inheritDoc} */ 150dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public BigFraction[] generate(int k) { 151dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final int kP1 = k + 1; 152dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return new BigFraction[] { 153dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new BigFraction(2 * k + 1, kP1), 154dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new BigFraction(-1, kP1), 155dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new BigFraction(k, kP1)}; 156dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 157dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }); 158dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 159dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 160dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 161dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Create a Legendre polynomial. 162dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <p><a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre 163dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * polynomials</a> are orthogonal polynomials. 164dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * They can be defined by the following recurrence relations: 165dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * <pre> 166dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * P<sub>0</sub>(X) = 1 167dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * P<sub>1</sub>(X) = X 168dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * (k+1) P<sub>k+1</sub>(X) = (2k+1) X P<sub>k</sub>(X) - k P<sub>k-1</sub>(X) 169dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * </pre></p> 170dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param degree degree of the polynomial 171dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return Legendre polynomial of specified degree 172dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 173dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public static PolynomialFunction createLegendrePolynomial(final int degree) { 174dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return buildPolynomial(degree, LEGENDRE_COEFFICIENTS, 175dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new RecurrenceCoefficientsGenerator() { 176dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** {@inheritDoc} */ 177dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond public BigFraction[] generate(int k) { 178dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final int kP1 = k + 1; 179dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return new BigFraction[] { 180dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction.ZERO, 181dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new BigFraction(k + kP1, kP1), 182dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond new BigFraction(k, kP1)}; 183dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 184dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond }); 185dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 186dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 187dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Get the coefficients array for a given degree. 188dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param degree degree of the polynomial 189dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param coefficients list where the computed coefficients are stored 190dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param generator recurrence coefficients generator 191dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return coefficients array 192dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 193dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static PolynomialFunction buildPolynomial(final int degree, 194dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final ArrayList<BigFraction> coefficients, 195dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final RecurrenceCoefficientsGenerator generator) { 196dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 197dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final int maxDegree = (int) FastMath.floor(FastMath.sqrt(2 * coefficients.size())) - 1; 198dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond synchronized (PolynomialsUtils.class) { 199dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond if (degree > maxDegree) { 200dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond computeUpToDegree(degree, maxDegree, generator, coefficients); 201dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 202dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 203dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 204dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficient for polynomial 0 is l [0] 205dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficients for polynomial 1 are l [1] ... l [2] (degrees 0 ... 1) 206dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficients for polynomial 2 are l [3] ... l [5] (degrees 0 ... 2) 207dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficients for polynomial 3 are l [6] ... l [9] (degrees 0 ... 3) 208dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficients for polynomial 4 are l[10] ... l[14] (degrees 0 ... 4) 209dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficients for polynomial 5 are l[15] ... l[20] (degrees 0 ... 5) 210dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // coefficients for polynomial 6 are l[21] ... l[27] (degrees 0 ... 6) 211dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // ... 212dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final int start = degree * (degree + 1) / 2; 213dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 214dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final double[] a = new double[degree + 1]; 215dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond for (int i = 0; i <= degree; ++i) { 216dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond a[i] = coefficients.get(start + i).doubleValue(); 217dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 218dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 219dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // build the polynomial 220dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond return new PolynomialFunction(a); 221dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 222dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 223dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 224dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Compute polynomial coefficients up to a given degree. 225dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param degree maximal degree 226dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param maxDegree current maximal degree 227dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param generator recurrence coefficients generator 228dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param coefficients list where the computed coefficients should be appended 229dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 230dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static void computeUpToDegree(final int degree, final int maxDegree, 231dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final RecurrenceCoefficientsGenerator generator, 232dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final ArrayList<BigFraction> coefficients) { 233dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 234dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond int startK = (maxDegree - 1) * maxDegree / 2; 235dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond for (int k = maxDegree; k < degree; ++k) { 236dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 237dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // start indices of two previous polynomials Pk(X) and Pk-1(X) 238dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond int startKm1 = startK; 239dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond startK += k; 240dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 241dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // Pk+1(X) = (a[0] + a[1] X) Pk(X) - a[2] Pk-1(X) 242dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction[] ai = generator.generate(k); 243dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 244dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction ck = coefficients.get(startK); 245dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction ckm1 = coefficients.get(startKm1); 246dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 247dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // degree 0 coefficient 248dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond coefficients.add(ck.multiply(ai[0]).subtract(ckm1.multiply(ai[2]))); 249dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 250dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // degree 1 to degree k-1 coefficients 251dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond for (int i = 1; i < k; ++i) { 252dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final BigFraction ckPrev = ck; 253dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond ck = coefficients.get(startK + i); 254dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond ckm1 = coefficients.get(startKm1 + i); 255dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])).subtract(ckm1.multiply(ai[2]))); 256dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 257dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 258dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // degree k coefficient 259dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond final BigFraction ckPrev = ck; 260dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond ck = coefficients.get(startK + k); 261dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1]))); 262dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 263dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond // degree k+1 coefficient 264dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond coefficients.add(ck.multiply(ai[1])); 265dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 266dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 267dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 268dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 269dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 270dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** Interface for recurrence coefficients generation. */ 271dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond private static interface RecurrenceCoefficientsGenerator { 272dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond /** 273dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * Generate recurrence coefficients. 274dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @param k highest degree of the polynomials used in the recurrence 275dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * @return an array of three coefficients such that 276dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond * P<sub>k+1</sub>(X) = (a[0] + a[1] X) P<sub>k</sub>(X) - a[2] P<sub>k-1</sub>(X) 277dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond */ 278dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond BigFraction[] generate(int k); 279dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond } 280dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond 281dee0849a9704d532af0b550146cbafbaa6ee1d19Raymond} 282