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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      https://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  
18  /*
19   * This is not the original file distributed by the Apache Software Foundation
20   * It has been modified by the Hipparchus project
21   */
22  package org.hipparchus.analysis.interpolation;
23  
24  import java.lang.reflect.Array;
25  
26  import org.hipparchus.Field;
27  import org.hipparchus.CalculusFieldElement;
28  import org.hipparchus.analysis.polynomials.FieldPolynomialFunction;
29  import org.hipparchus.analysis.polynomials.FieldPolynomialSplineFunction;
30  import org.hipparchus.analysis.polynomials.PolynomialFunction;
31  import org.hipparchus.analysis.polynomials.PolynomialSplineFunction;
32  import org.hipparchus.exception.LocalizedCoreFormats;
33  import org.hipparchus.exception.MathIllegalArgumentException;
34  import org.hipparchus.util.MathArrays;
35  import org.hipparchus.util.MathUtils;
36  
37  /**
38   * Computes a natural (also known as "free", "unclamped") cubic spline interpolation for the data set.
39   * <p>
40   * The {@link #interpolate(double[], double[])} method returns a {@link PolynomialSplineFunction}
41   * consisting of n cubic polynomials, defined over the subintervals determined by the x values,
42   * {@code x[0] < x[i] ... < x[n].}  The x values are referred to as "knot points."</p>
43   * <p>
44   * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest
45   * knot point and strictly less than the largest knot point is computed by finding the subinterval to which
46   * x belongs and computing the value of the corresponding polynomial at <code>x - x[i] </code> where
47   * <code>i</code> is the index of the subinterval.  See {@link PolynomialSplineFunction} for more details.
48   * </p>
49   * <p>
50   * The interpolating polynomials satisfy:
51   * </p>
52   * <ol>
53   * <li>The value of the PolynomialSplineFunction at each of the input x values equals the
54   *  corresponding y value.</li>
55   * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials
56   *  "match up" at the knot points, as do their first and second derivatives).</li>
57   * </ol>
58   * <p>
59   * The cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires,
60   * <u>Numerical Analysis</u>, 4th Ed., 1989, PWS-Kent, ISBN 0-53491-585-X, pp 126-131.
61   * </p>
62   *
63   */
64  public class SplineInterpolator implements UnivariateInterpolator, FieldUnivariateInterpolator {
65  
66      /** Empty constructor.
67       * <p>
68       * This constructor is not strictly necessary, but it prevents spurious
69       * javadoc warnings with JDK 18 and later.
70       * </p>
71       * @since 3.0
72       */
73      public SplineInterpolator() { // NOPMD - unnecessary constructor added intentionally to make javadoc happy
74          // nothing to do
75      }
76  
77      /**
78       * Computes an interpolating function for the data set.
79       * @param x the arguments for the interpolation points
80       * @param y the values for the interpolation points
81       * @return a function which interpolates the data set
82       * @throws MathIllegalArgumentException if {@code x} and {@code y}
83       * have different sizes.
84       * @throws MathIllegalArgumentException if {@code x} is not sorted in
85       * strict increasing order.
86       * @throws MathIllegalArgumentException if the size of {@code x} is smaller
87       * than 3.
88       */
89      @Override
90      public PolynomialSplineFunction interpolate(double x[], double y[])
91          throws MathIllegalArgumentException {
92  
93          MathUtils.checkNotNull(x);
94          MathUtils.checkNotNull(y);
95          MathArrays.checkEqualLength(x, y);
96          if (x.length < 3) {
97              throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_OF_POINTS,
98                                                     x.length, 3, true);
99          }
100 
101         // Number of intervals.  The number of data points is n + 1.
102         final int n = x.length - 1;
103 
104         MathArrays.checkOrder(x);
105 
106         // Differences between knot points
107         final double h[] = new double[n];
108         for (int i = 0; i < n; i++) {
109             h[i] = x[i + 1] - x[i];
110         }
111 
112         final double mu[] = new double[n];
113         final double z[] = new double[n + 1];
114         mu[0] = 0d;
115         z[0] = 0d;
116         double g;
117         for (int i = 1; i < n; i++) {
118             g = 2d * (x[i+1]  - x[i - 1]) - h[i - 1] * mu[i -1];
119             mu[i] = h[i] / g;
120             z[i] = (3d * (y[i + 1] * h[i - 1] - y[i] * (x[i + 1] - x[i - 1])+ y[i - 1] * h[i]) /
121                     (h[i - 1] * h[i]) - h[i - 1] * z[i - 1]) / g;
122         }
123 
124         // cubic spline coefficients --  b is linear, c quadratic, d is cubic (original y's are constants)
125         final double b[] = new double[n];
126         final double c[] = new double[n + 1];
127         final double d[] = new double[n];
128 
129         z[n] = 0d;
130         c[n] = 0d;
131 
132         for (int j = n -1; j >=0; j--) {
133             c[j] = z[j] - mu[j] * c[j + 1];
134             b[j] = (y[j + 1] - y[j]) / h[j] - h[j] * (c[j + 1] + 2d * c[j]) / 3d;
135             d[j] = (c[j + 1] - c[j]) / (3d * h[j]);
136         }
137 
138         final PolynomialFunction polynomials[] = new PolynomialFunction[n];
139         final double coefficients[] = new double[4];
140         for (int i = 0; i < n; i++) {
141             coefficients[0] = y[i];
142             coefficients[1] = b[i];
143             coefficients[2] = c[i];
144             coefficients[3] = d[i];
145             polynomials[i] = new PolynomialFunction(coefficients);
146         }
147 
148         return new PolynomialSplineFunction(x, polynomials);
149     }
150 
151     /**
152      * Computes an interpolating function for the data set.
153      * @param x the arguments for the interpolation points
154      * @param y the values for the interpolation points
155      * @param <T> the type of the field elements
156      * @return a function which interpolates the data set
157      * @throws MathIllegalArgumentException if {@code x} and {@code y}
158      * have different sizes.
159      * @throws MathIllegalArgumentException if {@code x} is not sorted in
160      * strict increasing order.
161      * @throws MathIllegalArgumentException if the size of {@code x} is smaller
162      * than 3.
163      * @since 1.5
164      */
165     @Override
166     public <T extends CalculusFieldElement<T>> FieldPolynomialSplineFunction<T> interpolate(T x[], T y[])
167         throws MathIllegalArgumentException {
168 
169         MathUtils.checkNotNull(x);
170         MathUtils.checkNotNull(y);
171         MathArrays.checkEqualLength(x, y);
172         if (x.length < 3) {
173             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_OF_POINTS,
174                                                    x.length, 3, true);
175         }
176 
177         // Number of intervals.  The number of data points is n + 1.
178         final int n = x.length - 1;
179 
180         MathArrays.checkOrder(x);
181 
182         // Differences between knot points
183         final Field<T> field = x[0].getField();
184         final T h[] = MathArrays.buildArray(field, n);
185         for (int i = 0; i < n; i++) {
186             h[i] = x[i + 1].subtract(x[i]);
187         }
188 
189         final T mu[] = MathArrays.buildArray(field, n);
190         final T z[]  = MathArrays.buildArray(field, n + 1);
191         mu[0] = field.getZero();
192         z[0]  = field.getZero();
193         for (int i = 1; i < n; i++) {
194             final T g = x[i+1].subtract(x[i - 1]).multiply(2).subtract(h[i - 1].multiply(mu[i -1]));
195             mu[i] = h[i].divide(g);
196             z[i] =          y[i + 1].multiply(h[i - 1]).
197                    subtract(y[i].multiply(x[i + 1].subtract(x[i - 1]))).
198                         add(y[i - 1].multiply(h[i])).
199                    multiply(3).
200                    divide(h[i - 1].multiply(h[i])).
201                    subtract(h[i - 1].multiply(z[i - 1])).
202                    divide(g);
203         }
204 
205         // cubic spline coefficients --  b is linear, c quadratic, d is cubic (original y's are constants)
206         final T b[] = MathArrays.buildArray(field, n);
207         final T c[] = MathArrays.buildArray(field, n + 1);
208         final T d[] = MathArrays.buildArray(field, n);
209 
210         z[n] = field.getZero();
211         c[n] = field.getZero();
212 
213         for (int j = n -1; j >=0; j--) {
214             c[j] = z[j].subtract(mu[j].multiply(c[j + 1]));
215             b[j] = y[j + 1].subtract(y[j]).divide(h[j]).
216                    subtract(h[j].multiply(c[j + 1].add(c[j]).add(c[j])).divide(3));
217             d[j] = c[j + 1].subtract(c[j]).divide(h[j].multiply(3));
218         }
219 
220         @SuppressWarnings("unchecked")
221         final FieldPolynomialFunction<T> polynomials[] =
222                         (FieldPolynomialFunction<T>[]) Array.newInstance(FieldPolynomialFunction.class, n);
223         final T coefficients[] = MathArrays.buildArray(field, 4);
224         for (int i = 0; i < n; i++) {
225             coefficients[0] = y[i];
226             coefficients[1] = b[i];
227             coefficients[2] = c[i];
228             coefficients[3] = d[i];
229             polynomials[i] = new FieldPolynomialFunction<>(coefficients);
230         }
231 
232         return new FieldPolynomialSplineFunction<>(x, polynomials);
233     }
234 
235 }