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Java example source code file (Sinc.java)

This example Java source code file (Sinc.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

Learn more about this Java project at its project page.

Java - Java tags/keywords

deprecated, derivativestructure, differentiableunivariatefunction, dimensionmismatchexception, shortcut, sinc, univariatedifferentiablefunction, univariatefunction

The Sinc.java Java example source code

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.apache.commons.math3.analysis.function;

import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
import org.apache.commons.math3.analysis.FunctionUtils;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.FastMath;

/**
 * <a href="http://en.wikipedia.org/wiki/Sinc_function">Sinc function,
 * defined by
 * <pre>
 *   sinc(x) = 1            if x = 0,
 *             sin(x) / x   otherwise.
 * </code>
* * @since 3.0 */ public class Sinc implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction { /** * Value below which the computations are done using Taylor series. * <p> * The Taylor series for sinc even order derivatives are: * <pre> * d^(2n)sinc/dx^(2n) = Sum_(k>=0) (-1)^(n+k) / ((2k)!(2n+2k+1)) x^(2k) * = (-1)^n [ 1/(2n+1) - x^2/(4n+6) + x^4/(48n+120) - x^6/(1440n+5040) + O(x^8) ] * </pre> * </p> * <p> * The Taylor series for sinc odd order derivatives are: * <pre> * d^(2n+1)sinc/dx^(2n+1) = Sum_(k>=0) (-1)^(n+k+1) / ((2k+1)!(2n+2k+3)) x^(2k+1) * = (-1)^(n+1) [ x/(2n+3) - x^3/(12n+30) + x^5/(240n+840) - x^7/(10080n+45360) + O(x^9) ] * </pre> * </p> * <p> * So the ratio of the fourth term with respect to the first term * is always smaller than x^6/720, for all derivative orders. * This implies that neglecting this term and using only the first three terms induces * a relative error bounded by x^6/720. The SHORTCUT value is chosen such that this * relative error is below double precision accuracy when |x| <= SHORTCUT. * </p> */ private static final double SHORTCUT = 6.0e-3; /** For normalized sinc function. */ private final boolean normalized; /** * The sinc function, {@code sin(x) / x}. */ public Sinc() { this(false); } /** * Instantiates the sinc function. * * @param normalized If {@code true}, the function is * <code> sin(πx) / πx, otherwise {@code sin(x) / x}. */ public Sinc(boolean normalized) { this.normalized = normalized; } /** {@inheritDoc} */ public double value(final double x) { final double scaledX = normalized ? FastMath.PI * x : x; if (FastMath.abs(scaledX) <= SHORTCUT) { // use Taylor series final double scaledX2 = scaledX * scaledX; return ((scaledX2 - 20) * scaledX2 + 120) / 120; } else { // use definition expression return FastMath.sin(scaledX) / scaledX; } } /** {@inheritDoc} * @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)} */ @Deprecated public UnivariateFunction derivative() { return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative(); } /** {@inheritDoc} * @since 3.1 */ public DerivativeStructure value(final DerivativeStructure t) throws DimensionMismatchException { final double scaledX = (normalized ? FastMath.PI : 1) * t.getValue(); final double scaledX2 = scaledX * scaledX; double[] f = new double[t.getOrder() + 1]; if (FastMath.abs(scaledX) <= SHORTCUT) { for (int i = 0; i < f.length; ++i) { final int k = i / 2; if ((i & 0x1) == 0) { // even derivation order f[i] = (((k & 0x1) == 0) ? 1 : -1) * (1.0 / (i + 1) - scaledX2 * (1.0 / (2 * i + 6) - scaledX2 / (24 * i + 120))); } else { // odd derivation order f[i] = (((k & 0x1) == 0) ? -scaledX : scaledX) * (1.0 / (i + 2) - scaledX2 * (1.0 / (6 * i + 24) - scaledX2 / (120 * i + 720))); } } } else { final double inv = 1 / scaledX; final double cos = FastMath.cos(scaledX); final double sin = FastMath.sin(scaledX); f[0] = inv * sin; // the nth order derivative of sinc has the form: // dn(sinc(x)/dxn = [S_n(x) sin(x) + C_n(x) cos(x)] / x^(n+1) // where S_n(x) is an even polynomial with degree n-1 or n (depending on parity) // and C_n(x) is an odd polynomial with degree n-1 or n (depending on parity) // S_0(x) = 1, S_1(x) = -1, S_2(x) = -x^2 + 2, S_3(x) = 3x^2 - 6... // C_0(x) = 0, C_1(x) = x, C_2(x) = -2x, C_3(x) = -x^3 + 6x... // the general recurrence relations for S_n and C_n are: // S_n(x) = x S_(n-1)'(x) - n S_(n-1)(x) - x C_(n-1)(x) // C_n(x) = x C_(n-1)'(x) - n C_(n-1)(x) + x S_(n-1)(x) // as per polynomials parity, we can store both S_n and C_n in the same array final double[] sc = new double[f.length]; sc[0] = 1; double coeff = inv; for (int n = 1; n < f.length; ++n) { double s = 0; double c = 0; // update and evaluate polynomials S_n(x) and C_n(x) final int kStart; if ((n & 0x1) == 0) { // even derivation order, S_n is degree n and C_n is degree n-1 sc[n] = 0; kStart = n; } else { // odd derivation order, S_n is degree n-1 and C_n is degree n sc[n] = sc[n - 1]; c = sc[n]; kStart = n - 1; } // in this loop, k is always even for (int k = kStart; k > 1; k -= 2) { // sine part sc[k] = (k - n) * sc[k] - sc[k - 1]; s = s * scaledX2 + sc[k]; // cosine part sc[k - 1] = (k - 1 - n) * sc[k - 1] + sc[k -2]; c = c * scaledX2 + sc[k - 1]; } sc[0] *= -n; s = s * scaledX2 + sc[0]; coeff *= inv; f[n] = coeff * (s * sin + c * scaledX * cos); } } if (normalized) { double scale = FastMath.PI; for (int i = 1; i < f.length; ++i) { f[i] *= scale; scale *= FastMath.PI; } } return t.compose(f); } }

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