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/*
* 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.interpolation;
import java.io.Serializable;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunctionLagrangeForm;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunctionNewtonForm;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.NonMonotonicSequenceException;
/**
* Implements the <a href=
* "http://mathworld.wolfram.com/NewtonsDividedDifferenceInterpolationFormula.html">
* Divided Difference Algorithm</a> for interpolation of real univariate
* functions. For reference, see <b>Introduction to Numerical Analysis</b>,
* ISBN 038795452X, chapter 2.
* <p>
* The actual code of Neville's evaluation is in PolynomialFunctionLagrangeForm,
* this class provides an easy-to-use interface to it.</p>
*
* @since 1.2
*/
public class DividedDifferenceInterpolator
implements UnivariateInterpolator, Serializable {
/** serializable version identifier */
private static final long serialVersionUID = 107049519551235069L;
/**
* Compute an interpolating function for the dataset.
*
* @param x Interpolating points array.
* @param y Interpolating values array.
* @return a function which interpolates the dataset.
* @throws DimensionMismatchException if the array lengths are different.
* @throws NumberIsTooSmallException if the number of points is less than 2.
* @throws NonMonotonicSequenceException if {@code x} is not sorted in
* strictly increasing order.
*/
public PolynomialFunctionNewtonForm interpolate(double x[], double y[])
throws DimensionMismatchException,
NumberIsTooSmallException,
NonMonotonicSequenceException {
/**
* a[] and c[] are defined in the general formula of Newton form:
* p(x) = a[0] + a[1](x-c[0]) + a[2](x-c[0])(x-c[1]) + ... +
* a[n](x-c[0])(x-c[1])...(x-c[n-1])
*/
PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);
/**
* When used for interpolation, the Newton form formula becomes
* p(x) = f[x0] + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ... +
* f[x0,x1,...,x[n-1]](x-x0)(x-x1)...(x-x[n-2])
* Therefore, a[k] = f[x0,x1,...,xk], c[k] = x[k].
* <p>
* Note x[], y[], a[] have the same length but c[]'s size is one less.</p>
*/
final double[] c = new double[x.length-1];
System.arraycopy(x, 0, c, 0, c.length);
final double[] a = computeDividedDifference(x, y);
return new PolynomialFunctionNewtonForm(a, c);
}
/**
* Return a copy of the divided difference array.
* <p>
* The divided difference array is defined recursively by <pre>
* f[x0] = f(x0)
* f[x0,x1,...,xk] = (f[x1,...,xk] - f[x0,...,x[k-1]]) / (xk - x0)
* </pre>
* <p>
* The computational complexity is \(O(n^2)\) where \(n\) is the common
* length of {@code x} and {@code y}.</p>
*
* @param x Interpolating points array.
* @param y Interpolating values array.
* @return a fresh copy of the divided difference array.
* @throws DimensionMismatchException if the array lengths are different.
* @throws NumberIsTooSmallException if the number of points is less than 2.
* @throws NonMonotonicSequenceException
* if {@code x} is not sorted in strictly increasing order.
*/
protected static double[] computeDividedDifference(final double x[], final double y[])
throws DimensionMismatchException,
NumberIsTooSmallException,
NonMonotonicSequenceException {
PolynomialFunctionLagrangeForm.verifyInterpolationArray(x, y, true);
final double[] divdiff = y.clone(); // initialization
final int n = x.length;
final double[] a = new double [n];
a[0] = divdiff[0];
for (int i = 1; i < n; i++) {
for (int j = 0; j < n-i; j++) {
final double denominator = x[j+i] - x[j];
divdiff[j] = (divdiff[j+1] - divdiff[j]) / denominator;
}
a[i] = divdiff[0];
}
return a;
}
}
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