FieldHermiteInterpolator.java
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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.
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package org.apache.commons.math4.legacy.analysis.interpolation;
import java.util.ArrayList;
import java.util.List;
import org.apache.commons.math4.legacy.core.FieldElement;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.MathArithmeticException;
import org.apache.commons.math4.legacy.exception.NoDataException;
import org.apache.commons.math4.legacy.exception.NullArgumentException;
import org.apache.commons.math4.legacy.exception.ZeroException;
import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
import org.apache.commons.math4.legacy.core.MathArrays;
/** 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>
*
* @param <T> Type of the field elements.
*
* @since 3.2
*/
public class FieldHermiteInterpolator<T extends FieldElement<T>> {
/** Sample abscissae. */
private final List<T> abscissae;
/** Top diagonal of the divided differences array. */
private final List<T[]> topDiagonal;
/** Bottom diagonal of the divided differences array. */
private final List<T[]> bottomDiagonal;
/** Create an empty interpolator.
*/
public FieldHermiteInterpolator() {
this.abscissae = new ArrayList<>();
this.topDiagonal = new ArrayList<>();
this.bottomDiagonal = new ArrayList<>();
}
/** 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
* @throws DimensionMismatchException if derivative structures are inconsistent
* @throws NullArgumentException if x is null
*/
@SafeVarargs
public final void addSamplePoint(final T x, final T[] ... value)
throws ZeroException, MathArithmeticException,
DimensionMismatchException, NullArgumentException {
NullArgumentException.check(x);
T factorial = x.getField().getOne();
for (int i = 0; i < value.length; ++i) {
final T[] y = value[i].clone();
if (i > 1) {
factorial = factorial.multiply(i);
final T inv = factorial.reciprocal();
for (int j = 0; j < y.length; ++j) {
y[j] = y[j].multiply(inv);
}
}
// update the bottom diagonal of the divided differences array
final int n = abscissae.size();
bottomDiagonal.add(n - i, y);
T[] bottom0 = y;
for (int j = i; j < n; ++j) {
final T[] bottom1 = bottomDiagonal.get(n - (j + 1));
if (x.equals(abscissae.get(n - (j + 1)))) {
throw new ZeroException(LocalizedFormats.DUPLICATED_ABSCISSA_DIVISION_BY_ZERO, x);
}
final T inv = x.subtract(abscissae.get(n - (j + 1))).reciprocal();
for (int k = 0; k < y.length; ++k) {
bottom1[k] = inv.multiply(bottom0[k].subtract(bottom1[k]));
}
bottom0 = bottom1;
}
// update the top diagonal of the divided differences array
topDiagonal.add(bottom0.clone());
// update the abscissae array
abscissae.add(x);
}
}
/** Interpolate value at a specified abscissa.
* @param x interpolation abscissa
* @return interpolated value
* @exception NoDataException if sample is empty
* @throws NullArgumentException if x is null
*/
public T[] value(T x) throws NoDataException, NullArgumentException {
// safety check
NullArgumentException.check(x);
if (abscissae.isEmpty()) {
throw new NoDataException(LocalizedFormats.EMPTY_INTERPOLATION_SAMPLE);
}
final T[] value = MathArrays.buildArray(x.getField(), topDiagonal.get(0).length);
T valueCoeff = x.getField().getOne();
for (int i = 0; i < topDiagonal.size(); ++i) {
T[] dividedDifference = topDiagonal.get(i);
for (int k = 0; k < value.length; ++k) {
value[k] = value[k].add(dividedDifference[k].multiply(valueCoeff));
}
final T deltaX = x.subtract(abscissae.get(i));
valueCoeff = valueCoeff.multiply(deltaX);
}
return value;
}
/** Interpolate value and first derivatives at a specified abscissa.
* @param x interpolation abscissa
* @param order maximum derivation order
* @return interpolated value and derivatives (value in row 0,
* 1<sup>st</sup> derivative in row 1, ... n<sup>th</sup> derivative in row n)
* @exception NoDataException if sample is empty
* @throws NullArgumentException if x is null
*/
public T[][] derivatives(T x, int order) throws NoDataException, NullArgumentException {
// safety check
NullArgumentException.check(x);
if (abscissae.isEmpty()) {
throw new NoDataException(LocalizedFormats.EMPTY_INTERPOLATION_SAMPLE);
}
final T zero = x.getField().getZero();
final T one = x.getField().getOne();
final T[] tj = MathArrays.buildArray(x.getField(), order + 1);
tj[0] = zero;
for (int i = 0; i < order; ++i) {
tj[i + 1] = tj[i].add(one);
}
final T[][] derivatives =
MathArrays.buildArray(x.getField(), order + 1, topDiagonal.get(0).length);
final T[] valueCoeff = MathArrays.buildArray(x.getField(), order + 1);
valueCoeff[0] = x.getField().getOne();
for (int i = 0; i < topDiagonal.size(); ++i) {
T[] dividedDifference = topDiagonal.get(i);
final T deltaX = x.subtract(abscissae.get(i));
for (int j = order; j >= 0; --j) {
for (int k = 0; k < derivatives[j].length; ++k) {
derivatives[j][k] =
derivatives[j][k].add(dividedDifference[k].multiply(valueCoeff[j]));
}
valueCoeff[j] = valueCoeff[j].multiply(deltaX);
if (j > 0) {
valueCoeff[j] = valueCoeff[j].add(tj[j].multiply(valueCoeff[j - 1]));
}
}
}
return derivatives;
}
}