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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.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableVectorFunction;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.ZeroException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.CombinatoricsUtils;
/** Polynomial interpolator using both sample values and sample derivatives.
* <p>
* The interpolation polynomials match all sample points, including both values
* and provided derivatives. There is one polynomial for each component of
* the values vector. All polynomials have the same degree. The degree of the
* polynomials depends on the number of points and number of derivatives at each
* point. For example the interpolation polynomials for n sample points without
* any derivatives all have degree n-1. The interpolation polynomials for n
* sample points with the two extreme points having value and first derivative
* and the remaining points having value only all have degree n+1. The
* interpolation polynomial for n sample points with value, first and second
* derivative for all points all have degree 3n-1.
* </p>
*
* @since 3.1
*/
public class HermiteInterpolator implements UnivariateDifferentiableVectorFunction {
/** Sample abscissae. */
private final List<Double> abscissae;
/** Top diagonal of the divided differences array. */
private final List<double[]> topDiagonal;
/** Bottom diagonal of the divided differences array. */
private final List<double[]> bottomDiagonal;
/** Create an empty interpolator.
*/
public HermiteInterpolator() {
this.abscissae = new ArrayList<Double>();
this.topDiagonal = new ArrayList<double[]>();
this.bottomDiagonal = new ArrayList<double[]>();
}
/** Add a sample point.
* <p>
* This method must be called once for each sample point. It is allowed to
* mix some calls with values only with calls with values and first
* derivatives.
* </p>
* <p>
* The point abscissae for all calls <em>must</em> be different.
* </p>
* @param x abscissa of the sample point
* @param value value and derivatives of the sample point
* (if only one row is passed, it is the value, if two rows are
* passed the first one is the value and the second the derivative
* and so on)
* @exception ZeroException if the abscissa difference between added point
* and a previous point is zero (i.e. the two points are at same abscissa)
* @exception MathArithmeticException if the number of derivatives is larger
* than 20, which prevents computation of a factorial
*/
public void addSamplePoint(final double x, final double[] ... value)
throws ZeroException, MathArithmeticException {
for (int i = 0; i < value.length; ++i) {
final double[] y = value[i].clone();
if (i > 1) {
double inv = 1.0 / CombinatoricsUtils.factorial(i);
for (int j = 0; j < y.length; ++j) {
y[j] *= inv;
}
}
// update the bottom diagonal of the divided differences array
final int n = abscissae.size();
bottomDiagonal.add(n - i, y);
double[] bottom0 = y;
for (int j = i; j < n; ++j) {
final double[] bottom1 = bottomDiagonal.get(n - (j + 1));
final double inv = 1.0 / (x - abscissae.get(n - (j + 1)));
if (Double.isInfinite(inv)) {
throw new ZeroException(LocalizedFormats.DUPLICATED_ABSCISSA_DIVISION_BY_ZERO, x);
}
for (int k = 0; k < y.length; ++k) {
bottom1[k] = inv * (bottom0[k] - bottom1[k]);
}
bottom0 = bottom1;
}
// update the top diagonal of the divided differences array
topDiagonal.add(bottom0.clone());
// update the abscissae array
abscissae.add(x);
}
}
/** Compute the interpolation polynomials.
* @return interpolation polynomials array
* @exception NoDataException if sample is empty
*/
public PolynomialFunction[] getPolynomials()
throws NoDataException {
// safety check
checkInterpolation();
// iteration initialization
final PolynomialFunction zero = polynomial(0);
PolynomialFunction[] polynomials = new PolynomialFunction[topDiagonal.get(0).length];
for (int i = 0; i < polynomials.length; ++i) {
polynomials[i] = zero;
}
PolynomialFunction coeff = polynomial(1);
// build the polynomials by iterating on the top diagonal of the divided differences array
for (int i = 0; i < topDiagonal.size(); ++i) {
double[] tdi = topDiagonal.get(i);
for (int k = 0; k < polynomials.length; ++k) {
polynomials[k] = polynomials[k].add(coeff.multiply(polynomial(tdi[k])));
}
coeff = coeff.multiply(polynomial(-abscissae.get(i), 1.0));
}
return polynomials;
}
/** Interpolate value at a specified abscissa.
* <p>
* Calling this method is equivalent to call the {@link PolynomialFunction#value(double)
* value} methods of all polynomials returned by {@link #getPolynomials() getPolynomials},
* except it does not build the intermediate polynomials, so this method is faster and
* numerically more stable.
* </p>
* @param x interpolation abscissa
* @return interpolated value
* @exception NoDataException if sample is empty
*/
public double[] value(double x)
throws NoDataException {
// safety check
checkInterpolation();
final double[] value = new double[topDiagonal.get(0).length];
double valueCoeff = 1;
for (int i = 0; i < topDiagonal.size(); ++i) {
double[] dividedDifference = topDiagonal.get(i);
for (int k = 0; k < value.length; ++k) {
value[k] += dividedDifference[k] * valueCoeff;
}
final double deltaX = x - abscissae.get(i);
valueCoeff *= deltaX;
}
return value;
}
/** Interpolate value at a specified abscissa.
* <p>
* Calling this method is equivalent to call the {@link
* PolynomialFunction#value(DerivativeStructure) value} methods of all polynomials
* returned by {@link #getPolynomials() getPolynomials}, except it does not build the
* intermediate polynomials, so this method is faster and numerically more stable.
* </p>
* @param x interpolation abscissa
* @return interpolated value
* @exception NoDataException if sample is empty
*/
public DerivativeStructure[] value(final DerivativeStructure x)
throws NoDataException {
// safety check
checkInterpolation();
final DerivativeStructure[] value = new DerivativeStructure[topDiagonal.get(0).length];
Arrays.fill(value, x.getField().getZero());
DerivativeStructure valueCoeff = x.getField().getOne();
for (int i = 0; i < topDiagonal.size(); ++i) {
double[] dividedDifference = topDiagonal.get(i);
for (int k = 0; k < value.length; ++k) {
value[k] = value[k].add(valueCoeff.multiply(dividedDifference[k]));
}
final DerivativeStructure deltaX = x.subtract(abscissae.get(i));
valueCoeff = valueCoeff.multiply(deltaX);
}
return value;
}
/** Check interpolation can be performed.
* @exception NoDataException if interpolation cannot be performed
* because sample is empty
*/
private void checkInterpolation() throws NoDataException {
if (abscissae.isEmpty()) {
throw new NoDataException(LocalizedFormats.EMPTY_INTERPOLATION_SAMPLE);
}
}
/** Create a polynomial from its coefficients.
* @param c polynomials coefficients
* @return polynomial
*/
private PolynomialFunction polynomial(double ... c) {
return new PolynomialFunction(c);
}
}