001    /*
002     * Licensed to the Apache Software Foundation (ASF) under one or more
003     * contributor license agreements.  See the NOTICE file distributed with
004     * this work for additional information regarding copyright ownership.
005     * The ASF licenses this file to You under the Apache License, Version 2.0
006     * (the "License"); you may not use this file except in compliance with
007     * the License.  You may obtain a copy of the License at
008     *
009     *      http://www.apache.org/licenses/LICENSE-2.0
010     *
011     * Unless required by applicable law or agreed to in writing, software
012     * distributed under the License is distributed on an "AS IS" BASIS,
013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014     * See the License for the specific language governing permissions and
015     * limitations under the License.
016     */
017    
018    package org.apache.commons.math3.analysis.function;
019    
020    import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
021    import org.apache.commons.math3.analysis.FunctionUtils;
022    import org.apache.commons.math3.analysis.UnivariateFunction;
023    import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
024    import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
025    import org.apache.commons.math3.util.FastMath;
026    
027    /**
028     * <a href="http://en.wikipedia.org/wiki/Sinc_function">Sinc</a> function,
029     * defined by
030     * <pre><code>
031     *   sinc(x) = 1            if x = 0,
032     *             sin(x) / x   otherwise.
033     * </code></pre>
034     *
035     * @since 3.0
036     * @version $Id: Sinc.java 1383441 2012-09-11 14:56:39Z luc $
037     */
038    public class Sinc implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction {
039        /**
040         * Value below which the computations are done using Taylor series.
041         * <p>
042         * The Taylor series for sinc even order derivatives are:
043         * <pre>
044         * d^(2n)sinc/dx^(2n)     = Sum_(k>=0) (-1)^(n+k) / ((2k)!(2n+2k+1)) x^(2k)
045         *                        = (-1)^n     [ 1/(2n+1) - x^2/(4n+6) + x^4/(48n+120) - x^6/(1440n+5040) + O(x^8) ]
046         * </pre>
047         * </p>
048         * <p>
049         * The Taylor series for sinc odd order derivatives are:
050         * <pre>
051         * d^(2n+1)sinc/dx^(2n+1) = Sum_(k>=0) (-1)^(n+k+1) / ((2k+1)!(2n+2k+3)) x^(2k+1)
052         *                        = (-1)^(n+1) [ x/(2n+3) - x^3/(12n+30) + x^5/(240n+840) - x^7/(10080n+45360) + O(x^9) ]
053         * </pre>
054         * </p>
055         * <p>
056         * So the ratio of the fourth term with respect to the first term
057         * is always smaller than x^6/720, for all derivative orders.
058         * This implies that neglecting this term and using only the first three terms induces
059         * a relative error bounded by x^6/720. The SHORTCUT value is chosen such that this
060         * relative error is below double precision accuracy when |x| <= SHORTCUT.
061         * </p>
062         */
063        private static final double SHORTCUT = 6.0e-3;
064        /** For normalized sinc function. */
065        private final boolean normalized;
066    
067        /**
068         * The sinc function, {@code sin(x) / x}.
069         */
070        public Sinc() {
071            this(false);
072        }
073    
074        /**
075         * Instantiates the sinc function.
076         *
077         * @param normalized If {@code true}, the function is
078         * <code> sin(&pi;x) / &pi;x</code>, otherwise {@code sin(x) / x}.
079         */
080        public Sinc(boolean normalized) {
081            this.normalized = normalized;
082        }
083    
084        /** {@inheritDoc} */
085        public double value(final double x) {
086            final double scaledX = normalized ? FastMath.PI * x : x;
087            if (FastMath.abs(scaledX) <= SHORTCUT) {
088                // use Taylor series
089                final double scaledX2 = scaledX * scaledX;
090                return ((scaledX2 - 20) * scaledX2 + 120) / 120;
091            } else {
092                // use definition expression
093                return FastMath.sin(scaledX) / scaledX;
094            }
095        }
096    
097        /** {@inheritDoc}
098         * @deprecated as of 3.1, replaced by {@link #value(DerivativeStructure)}
099         */
100        @Deprecated
101        public UnivariateFunction derivative() {
102            return FunctionUtils.toDifferentiableUnivariateFunction(this).derivative();
103        }
104    
105        /** {@inheritDoc}
106         * @since 3.1
107         */
108        public DerivativeStructure value(final DerivativeStructure t) {
109    
110            final double scaledX  = (normalized ? FastMath.PI : 1) * t.getValue();
111            final double scaledX2 = scaledX * scaledX;
112    
113            double[] f = new double[t.getOrder() + 1];
114    
115            if (FastMath.abs(scaledX) <= SHORTCUT) {
116    
117                for (int i = 0; i < f.length; ++i) {
118                    final int k = i / 2;
119                    if ((i & 0x1) == 0) {
120                        // even derivation order
121                        f[i] = (((k & 0x1) == 0) ? 1 : -1) *
122                               (1.0 / (i + 1) - scaledX2 * (1.0 / (2 * i + 6) - scaledX2 / (24 * i + 120)));
123                    } else {
124                        // odd derivation order
125                        f[i] = (((k & 0x1) == 0) ? -scaledX : scaledX) *
126                               (1.0 / (i + 2) - scaledX2 * (1.0 / (6 * i + 24) - scaledX2 / (120 * i + 720)));
127                    }
128                }
129    
130            } else {
131    
132                final double inv = 1 / scaledX;
133                final double cos = FastMath.cos(scaledX);
134                final double sin = FastMath.sin(scaledX);
135    
136                f[0] = inv * sin;
137    
138                // the nth order derivative of sinc has the form:
139                // dn(sinc(x)/dxn = [S_n(x) sin(x) + C_n(x) cos(x)] / x^(n+1)
140                // where S_n(x) is an even polynomial with degree n-1 or n (depending on parity)
141                // and C_n(x) is an odd polynomial with degree n-1 or n (depending on parity)
142                // S_0(x) = 1, S_1(x) = -1, S_2(x) = -x^2 + 2, S_3(x) = 3x^2 - 6...
143                // C_0(x) = 0, C_1(x) = x, C_2(x) = -2x, C_3(x) = -x^3 + 6x...
144                // the general recurrence relations for S_n and C_n are:
145                // S_n(x) = x S_(n-1)'(x) - n S_(n-1)(x) - x C_(n-1)(x)
146                // C_n(x) = x C_(n-1)'(x) - n C_(n-1)(x) + x S_(n-1)(x)
147                // as per polynomials parity, we can store both S_n and C_n in the same array
148                final double[] sc = new double[f.length];
149                sc[0] = 1;
150    
151                double coeff = inv;
152                for (int n = 1; n < f.length; ++n) {
153    
154                    double s = 0;
155                    double c = 0;
156    
157                    // update and evaluate polynomials S_n(x) and C_n(x)
158                    final int kStart;
159                    if ((n & 0x1) == 0) {
160                        // even derivation order, S_n is degree n and C_n is degree n-1
161                        sc[n] = 0;
162                        kStart = n;
163                    } else {
164                        // odd derivation order, S_n is degree n-1 and C_n is degree n
165                        sc[n] = sc[n - 1];
166                        c = sc[n];
167                        kStart = n - 1;
168                    }
169    
170                    // in this loop, k is always even
171                    for (int k = kStart; k > 1; k -= 2) {
172    
173                        // sine part
174                        sc[k]     = (k - n) * sc[k] - sc[k - 1];
175                        s         = s * scaledX2 + sc[k];
176    
177                        // cosine part
178                        sc[k - 1] = (k - 1 - n) * sc[k - 1] + sc[k -2];
179                        c         = c * scaledX2 + sc[k - 1];
180    
181                    }
182                    sc[0] *= -n;
183                    s      = s * scaledX2 + sc[0];
184    
185                    coeff *= inv;
186                    f[n]   = coeff * (s * sin + c * scaledX * cos);
187    
188                }
189    
190            }
191    
192            if (normalized) {
193                double scale = FastMath.PI;
194                for (int i = 1; i < f.length; ++i) {
195                    f[i]  *= scale;
196                    scale *= FastMath.PI;
197                }
198            }
199    
200            return t.compose(f);
201    
202        }
203    
204    }