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 package org.apache.commons.math.analysis.polynomials;
018
019 import org.apache.commons.math.MathRuntimeException;
020 import org.apache.commons.math.analysis.UnivariateRealFunction;
021 import org.apache.commons.math.FunctionEvaluationException;
022 import org.apache.commons.math.exception.util.LocalizedFormats;
023
024 /**
025 * Implements the representation of a real polynomial function in
026 * Newton Form. For reference, see <b>Elementary Numerical Analysis</b>,
027 * ISBN 0070124477, chapter 2.
028 * <p>
029 * The formula of polynomial in Newton form is
030 * p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
031 * a[n](x-c[0])(x-c[1])...(x-c[n-1])
032 * Note that the length of a[] is one more than the length of c[]</p>
033 *
034 * @version $Revision: 1073498 $ $Date: 2011-02-22 21:57:26 +0100 (mar. 22 f??vr. 2011) $
035 * @since 1.2
036 */
037 public class PolynomialFunctionNewtonForm implements UnivariateRealFunction {
038
039 /**
040 * The coefficients of the polynomial, ordered by degree -- i.e.
041 * coefficients[0] is the constant term and coefficients[n] is the
042 * coefficient of x^n where n is the degree of the polynomial.
043 */
044 private double coefficients[];
045
046 /**
047 * Centers of the Newton polynomial.
048 */
049 private final double c[];
050
051 /**
052 * When all c[i] = 0, a[] becomes normal polynomial coefficients,
053 * i.e. a[i] = coefficients[i].
054 */
055 private final double a[];
056
057 /**
058 * Whether the polynomial coefficients are available.
059 */
060 private boolean coefficientsComputed;
061
062 /**
063 * Construct a Newton polynomial with the given a[] and c[]. The order of
064 * centers are important in that if c[] shuffle, then values of a[] would
065 * completely change, not just a permutation of old a[].
066 * <p>
067 * The constructor makes copy of the input arrays and assigns them.</p>
068 *
069 * @param a the coefficients in Newton form formula
070 * @param c the centers
071 * @throws IllegalArgumentException if input arrays are not valid
072 */
073 public PolynomialFunctionNewtonForm(double a[], double c[])
074 throws IllegalArgumentException {
075
076 verifyInputArray(a, c);
077 this.a = new double[a.length];
078 this.c = new double[c.length];
079 System.arraycopy(a, 0, this.a, 0, a.length);
080 System.arraycopy(c, 0, this.c, 0, c.length);
081 coefficientsComputed = false;
082 }
083
084 /**
085 * Calculate the function value at the given point.
086 *
087 * @param z the point at which the function value is to be computed
088 * @return the function value
089 * @throws FunctionEvaluationException if a runtime error occurs
090 * @see UnivariateRealFunction#value(double)
091 */
092 public double value(double z) throws FunctionEvaluationException {
093 return evaluate(a, c, z);
094 }
095
096 /**
097 * Returns the degree of the polynomial.
098 *
099 * @return the degree of the polynomial
100 */
101 public int degree() {
102 return c.length;
103 }
104
105 /**
106 * Returns a copy of coefficients in Newton form formula.
107 * <p>
108 * Changes made to the returned copy will not affect the polynomial.</p>
109 *
110 * @return a fresh copy of coefficients in Newton form formula
111 */
112 public double[] getNewtonCoefficients() {
113 double[] out = new double[a.length];
114 System.arraycopy(a, 0, out, 0, a.length);
115 return out;
116 }
117
118 /**
119 * Returns a copy of the centers array.
120 * <p>
121 * Changes made to the returned copy will not affect the polynomial.</p>
122 *
123 * @return a fresh copy of the centers array
124 */
125 public double[] getCenters() {
126 double[] out = new double[c.length];
127 System.arraycopy(c, 0, out, 0, c.length);
128 return out;
129 }
130
131 /**
132 * Returns a copy of the coefficients array.
133 * <p>
134 * Changes made to the returned copy will not affect the polynomial.</p>
135 *
136 * @return a fresh copy of the coefficients array
137 */
138 public double[] getCoefficients() {
139 if (!coefficientsComputed) {
140 computeCoefficients();
141 }
142 double[] out = new double[coefficients.length];
143 System.arraycopy(coefficients, 0, out, 0, coefficients.length);
144 return out;
145 }
146
147 /**
148 * Evaluate the Newton polynomial using nested multiplication. It is
149 * also called <a href="http://mathworld.wolfram.com/HornersRule.html">
150 * Horner's Rule</a> and takes O(N) time.
151 *
152 * @param a the coefficients in Newton form formula
153 * @param c the centers
154 * @param z the point at which the function value is to be computed
155 * @return the function value
156 * @throws FunctionEvaluationException if a runtime error occurs
157 * @throws IllegalArgumentException if inputs are not valid
158 */
159 public static double evaluate(double a[], double c[], double z)
160 throws FunctionEvaluationException, IllegalArgumentException {
161
162 verifyInputArray(a, c);
163
164 int n = c.length;
165 double value = a[n];
166 for (int i = n-1; i >= 0; i--) {
167 value = a[i] + (z - c[i]) * value;
168 }
169
170 return value;
171 }
172
173 /**
174 * Calculate the normal polynomial coefficients given the Newton form.
175 * It also uses nested multiplication but takes O(N^2) time.
176 */
177 protected void computeCoefficients() {
178 final int n = degree();
179
180 coefficients = new double[n+1];
181 for (int i = 0; i <= n; i++) {
182 coefficients[i] = 0.0;
183 }
184
185 coefficients[0] = a[n];
186 for (int i = n-1; i >= 0; i--) {
187 for (int j = n-i; j > 0; j--) {
188 coefficients[j] = coefficients[j-1] - c[i] * coefficients[j];
189 }
190 coefficients[0] = a[i] - c[i] * coefficients[0];
191 }
192
193 coefficientsComputed = true;
194 }
195
196 /**
197 * Verifies that the input arrays are valid.
198 * <p>
199 * The centers must be distinct for interpolation purposes, but not
200 * for general use. Thus it is not verified here.</p>
201 *
202 * @param a the coefficients in Newton form formula
203 * @param c the centers
204 * @throws IllegalArgumentException if not valid
205 * @see org.apache.commons.math.analysis.interpolation.DividedDifferenceInterpolator#computeDividedDifference(double[],
206 * double[])
207 */
208 protected static void verifyInputArray(double a[], double c[]) throws
209 IllegalArgumentException {
210
211 if (a.length < 1 || c.length < 1) {
212 throw MathRuntimeException.createIllegalArgumentException(
213 LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);
214 }
215 if (a.length != c.length + 1) {
216 throw MathRuntimeException.createIllegalArgumentException(
217 LocalizedFormats.ARRAY_SIZES_SHOULD_HAVE_DIFFERENCE_1,
218 a.length, c.length);
219 }
220 }
221 }