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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    *      http://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  package org.apache.commons.math.analysis;
18  
19  import java.io.Serializable;
20  import java.util.Arrays;
21  
22  import org.apache.commons.math.ArgumentOutsideDomainException;
23  
24  /**
25   * Represents a polynomial spline function.
26   * <p>
27   * A <strong>polynomial spline function</strong> consists of a set of 
28   * <i>interpolating polynomials</i> and an ascending array of domain 
29   * <i>knot points</i>, determining the intervals over which the spline function
30   * is defined by the constituent polynomials.  The polynomials are assumed to
31   * have been computed to match the values of another function at the knot
32   * points.  The value consistency constraints are not currently enforced by 
33   * <code>PolynomialSplineFunction</code> itself, but are assumed to hold among
34   * the polynomials and knot points passed to the constructor.</p>
35   * <p>
36   * N.B.:  The polynomials in the <code>polynomials</code> property must be
37   * centered on the knot points to compute the spline function values.  
38   * See below.</p>
39   * <p>
40   * The domain of the polynomial spline function is 
41   * <code>[smallest knot, largest knot]</code>.  Attempts to evaluate the
42   * function at values outside of this range generate IllegalArgumentExceptions.
43   * </p>
44   * <p>
45   * The value of the polynomial spline function for an argument <code>x</code>
46   * is computed as follows:
47   * <ol>
48   * <li>The knot array is searched to find the segment to which <code>x</code>
49   * belongs.  If <code>x</code> is less than the smallest knot point or greater
50   * than the largest one, an <code>IllegalArgumentException</code>
51   * is thrown.</li>
52   * <li> Let <code>j</code> be the index of the largest knot point that is less
53   * than or equal to <code>x</code>.  The value returned is <br>
54   * <code>polynomials[j](x - knot[j])</code></li></ol></p>
55   *
56   * @version $Revision: 615734 $ $Date: 2008-01-27 23:10:03 -0700 (Sun, 27 Jan 2008) $
57   */
58  public class PolynomialSplineFunction 
59      implements DifferentiableUnivariateRealFunction, Serializable {
60  
61      /** Serializable version identifier */
62      private static final long serialVersionUID = 1619940313389547244L;
63  
64      /** Spline segment interval delimiters (knots).   Size is n+1 for n segments. */
65      private double knots[];
66  
67      /**
68       * The polynomial functions that make up the spline.  The first element
69       * determines the value of the spline over the first subinterval, the
70       * second over the second, etc.   Spline function values are determined by
71       * evaluating these functions at <code>(x - knot[i])</code> where i is the
72       * knot segment to which x belongs.
73       */
74      private PolynomialFunction polynomials[] = null;
75      
76      /** 
77       * Number of spline segments = number of polynomials
78       *  = number of partition points - 1 
79       */
80      private int n = 0;
81      
82  
83      /**
84       * Construct a polynomial spline function with the given segment delimiters
85       * and interpolating polynomials.
86       * <p>
87       * The constructor copies both arrays and assigns the copies to the knots
88       * and polynomials properties, respectively.</p>
89       * 
90       * @param knots spline segment interval delimiters
91       * @param polynomials polynomial functions that make up the spline
92       * @throws NullPointerException if either of the input arrays is null
93       * @throws IllegalArgumentException if knots has length less than 2,  
94       * <code>polynomials.length != knots.length - 1 </code>, or the knots array
95       * is not strictly increasing.
96       * 
97       */
98      public PolynomialSplineFunction(double knots[], PolynomialFunction polynomials[]) {
99          if (knots.length < 2) {
100             throw new IllegalArgumentException
101                 ("Not enough knot values -- spline partition must have at least 2 points.");
102         }
103         if (knots.length - 1 != polynomials.length) {
104             throw new IllegalArgumentException 
105             ("Number of polynomial interpolants must match the number of segments.");
106         }
107         if (!isStrictlyIncreasing(knots)) {
108             throw new IllegalArgumentException 
109                 ("Knot values must be strictly increasing.");
110         }
111         
112         this.n = knots.length -1;
113         this.knots = new double[n + 1];
114         System.arraycopy(knots, 0, this.knots, 0, n + 1);
115         this.polynomials = new PolynomialFunction[n];
116         System.arraycopy(polynomials, 0, this.polynomials, 0, n);
117     }
118 
119     /**
120      * Compute the value for the function.
121      * <p>
122      * Throws FunctionEvaluationException if v is outside of the domain of the
123      * function.  The domain is [smallest knot, largest knot].</p>
124      * <p>
125      * See {@link PolynomialSplineFunction} for details on the algorithm for
126      * computing the value of the function.</p>
127      * 
128      * @param v the point for which the function value should be computed
129      * @return the value
130      * @throws ArgumentOutsideDomainException if v is outside of the domain of
131      * of the spline function (less than the smallest knot point or greater
132      * than the largest knot point)
133      */
134     public double value(double v) throws ArgumentOutsideDomainException {
135         if (v < knots[0] || v > knots[n]) {
136             throw new ArgumentOutsideDomainException(v, knots[0], knots[n]);
137         }
138         int i = Arrays.binarySearch(knots, v);
139         if (i < 0) {
140             i = -i - 2;
141         }
142         //This will handle the case where v is the last knot value
143         //There are only n-1 polynomials, so if v is the last knot
144         //then we will use the last polynomial to calculate the value.
145         if ( i >= polynomials.length ) {
146             i--;
147         }
148         return polynomials[i].value(v - knots[i]);
149     }
150     
151     /**
152      * Returns the derivative of the polynomial spline function as a UnivariateRealFunction
153      * @return  the derivative function
154      */
155     public UnivariateRealFunction derivative() {
156         return polynomialSplineDerivative();
157     }
158     
159     /**
160      * Returns the derivative of the polynomial spline function as a PolynomialSplineFunction
161      * 
162      * @return  the derivative function
163      */
164     public PolynomialSplineFunction polynomialSplineDerivative() {
165         PolynomialFunction derivativePolynomials[] = new PolynomialFunction[n];
166         for (int i = 0; i < n; i++) {
167             derivativePolynomials[i] = polynomials[i].polynomialDerivative();
168         }
169         return new PolynomialSplineFunction(knots, derivativePolynomials);
170     }
171 
172     /**
173      * Returns the number of spline segments = the number of polynomials 
174      * = the number of knot points - 1.
175      * 
176      * @return the number of spline segments
177      */
178     public int getN() {
179         return n;
180     }
181 
182     /**
183      * Returns a copy of the interpolating polynomials array.
184      * <p>
185      * Returns a fresh copy of the array. Changes made to the copy will
186      * not affect the polynomials property.</p>
187      * 
188      * @return the interpolating polynomials
189      */
190     public PolynomialFunction[] getPolynomials() {
191         PolynomialFunction p[] = new PolynomialFunction[n];
192         System.arraycopy(polynomials, 0, p, 0, n);
193         return p;
194     }
195 
196     /**
197      * Returns an array copy of the knot points.
198      * <p>
199      * Returns a fresh copy of the array. Changes made to the copy
200      * will not affect the knots property.</p>
201      * 
202      * @return the knot points
203      */
204     public double[] getKnots() {
205         double out[] = new double[n + 1];
206         System.arraycopy(knots, 0, out, 0, n + 1);
207         return out;  
208     }
209 
210     /**
211      * Determines if the given array is ordered in a strictly increasing
212      * fashion.
213      * 
214      * @param x the array to examine.
215      * @return <code>true</code> if the elements in <code>x</code> are ordered
216      * in a stricly increasing manner.  <code>false</code>, otherwise.
217      */
218     private static boolean isStrictlyIncreasing(double[] x) {
219         for (int i = 1; i < x.length; ++i) {
220             if (x[i - 1] >= x[i]) {
221                 return false;
222             }
223         }
224         return true;
225     }
226 }