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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.polynomials;
import java.io.Serializable;
import java.util.Arrays;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.analysis.DifferentiableUnivariateFunction;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.analysis.ParametricUnivariateFunction;
import org.apache.commons.math3.analysis.differentiation.DerivativeStructure;
import org.apache.commons.math3.analysis.differentiation.UnivariateDifferentiableFunction;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;
/**
* Immutable representation of a real polynomial function with real coefficients.
*
* Horner's Method
* is used to evaluate the function.
*
*/
public class PolynomialFunction implements UnivariateDifferentiableFunction, DifferentiableUnivariateFunction, Serializable {
/**
* Serialization identifier
*/
private static final long serialVersionUID = -7726511984200295583L;
/**
* The coefficients of the polynomial, ordered by degree -- i.e.,
* coefficients[0] is the constant term and coefficients[n] is the
* coefficient of x^n where n is the degree of the polynomial.
*/
private final double coefficients[];
/**
* Construct a polynomial with the given coefficients. The first element
* of the coefficients array is the constant term. Higher degree
* coefficients follow in sequence. The degree of the resulting polynomial
* is the index of the last non-null element of the array, or 0 if all elements
* are null.
*
* The constructor makes a copy of the input array and assigns the copy to
* the coefficients property.
*
* @param c Polynomial coefficients.
* @throws NullArgumentException if {@code c} is {@code null}.
* @throws NoDataException if {@code c} is empty.
*/
public PolynomialFunction(double c[])
throws NullArgumentException, NoDataException {
super();
MathUtils.checkNotNull(c);
int n = c.length;
if (n == 0) {
throw new NoDataException(LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);
}
while ((n > 1) && (c[n - 1] == 0)) {
--n;
}
this.coefficients = new double[n];
System.arraycopy(c, 0, this.coefficients, 0, n);
}
/**
* Compute the value of the function for the given argument.
*
* The value returned is
* coefficients[n] * x^n + ... + coefficients[1] * x + coefficients[0]
*
*
* @param x Argument for which the function value should be computed.
* @return the value of the polynomial at the given point.
* @see UnivariateFunction#value(double)
*/
public double value(double x) {
return evaluate(coefficients, x);
}
/**
* Returns the degree of the polynomial.
*
* @return the degree of the polynomial.
*/
public int degree() {
return coefficients.length - 1;
}
/**
* Returns a copy of the coefficients array.
*
* Changes made to the returned copy will not affect the coefficients of
* the polynomial.
*
* @return a fresh copy of the coefficients array.
*/
public double[] getCoefficients() {
return coefficients.clone();
}
/**
* Uses Horner's Method to evaluate the polynomial with the given coefficients at
* the argument.
*
* @param coefficients Coefficients of the polynomial to evaluate.
* @param argument Input value.
* @return the value of the polynomial.
* @throws NoDataException if {@code coefficients} is empty.
* @throws NullArgumentException if {@code coefficients} is {@code null}.
*/
protected static double evaluate(double[] coefficients, double argument)
throws NullArgumentException, NoDataException {
MathUtils.checkNotNull(coefficients);
int n = coefficients.length;
if (n == 0) {
throw new NoDataException(LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);
}
double result = coefficients[n - 1];
for (int j = n - 2; j >= 0; j--) {
result = argument * result + coefficients[j];
}
return result;
}
/** {@inheritDoc}
* @since 3.1
* @throws NoDataException if {@code coefficients} is empty.
* @throws NullArgumentException if {@code coefficients} is {@code null}.
*/
public DerivativeStructure value(final DerivativeStructure t)
throws NullArgumentException, NoDataException {
MathUtils.checkNotNull(coefficients);
int n = coefficients.length;
if (n == 0) {
throw new NoDataException(LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);
}
DerivativeStructure result =
new DerivativeStructure(t.getFreeParameters(), t.getOrder(), coefficients[n - 1]);
for (int j = n - 2; j >= 0; j--) {
result = result.multiply(t).add(coefficients[j]);
}
return result;
}
/**
* Add a polynomial to the instance.
*
* @param p Polynomial to add.
* @return a new polynomial which is the sum of the instance and {@code p}.
*/
public PolynomialFunction add(final PolynomialFunction p) {
// identify the lowest degree polynomial
final int lowLength = FastMath.min(coefficients.length, p.coefficients.length);
final int highLength = FastMath.max(coefficients.length, p.coefficients.length);
// build the coefficients array
double[] newCoefficients = new double[highLength];
for (int i = 0; i < lowLength; ++i) {
newCoefficients[i] = coefficients[i] + p.coefficients[i];
}
System.arraycopy((coefficients.length < p.coefficients.length) ?
p.coefficients : coefficients,
lowLength,
newCoefficients, lowLength,
highLength - lowLength);
return new PolynomialFunction(newCoefficients);
}
/**
* Subtract a polynomial from the instance.
*
* @param p Polynomial to subtract.
* @return a new polynomial which is the difference the instance minus {@code p}.
*/
public PolynomialFunction subtract(final PolynomialFunction p) {
// identify the lowest degree polynomial
int lowLength = FastMath.min(coefficients.length, p.coefficients.length);
int highLength = FastMath.max(coefficients.length, p.coefficients.length);
// build the coefficients array
double[] newCoefficients = new double[highLength];
for (int i = 0; i < lowLength; ++i) {
newCoefficients[i] = coefficients[i] - p.coefficients[i];
}
if (coefficients.length < p.coefficients.length) {
for (int i = lowLength; i < highLength; ++i) {
newCoefficients[i] = -p.coefficients[i];
}
} else {
System.arraycopy(coefficients, lowLength, newCoefficients, lowLength,
highLength - lowLength);
}
return new PolynomialFunction(newCoefficients);
}
/**
* Negate the instance.
*
* @return a new polynomial.
*/
public PolynomialFunction negate() {
double[] newCoefficients = new double[coefficients.length];
for (int i = 0; i < coefficients.length; ++i) {
newCoefficients[i] = -coefficients[i];
}
return new PolynomialFunction(newCoefficients);
}
/**
* Multiply the instance by a polynomial.
*
* @param p Polynomial to multiply by.
* @return a new polynomial.
*/
public PolynomialFunction multiply(final PolynomialFunction p) {
double[] newCoefficients = new double[coefficients.length + p.coefficients.length - 1];
for (int i = 0; i < newCoefficients.length; ++i) {
newCoefficients[i] = 0.0;
for (int j = FastMath.max(0, i + 1 - p.coefficients.length);
j < FastMath.min(coefficients.length, i + 1);
++j) {
newCoefficients[i] += coefficients[j] * p.coefficients[i-j];
}
}
return new PolynomialFunction(newCoefficients);
}
/**
* Returns the coefficients of the derivative of the polynomial with the given coefficients.
*
* @param coefficients Coefficients of the polynomial to differentiate.
* @return the coefficients of the derivative or {@code null} if coefficients has length 1.
* @throws NoDataException if {@code coefficients} is empty.
* @throws NullArgumentException if {@code coefficients} is {@code null}.
*/
protected static double[] differentiate(double[] coefficients)
throws NullArgumentException, NoDataException {
MathUtils.checkNotNull(coefficients);
int n = coefficients.length;
if (n == 0) {
throw new NoDataException(LocalizedFormats.EMPTY_POLYNOMIALS_COEFFICIENTS_ARRAY);
}
if (n == 1) {
return new double[]{0};
}
double[] result = new double[n - 1];
for (int i = n - 1; i > 0; i--) {
result[i - 1] = i * coefficients[i];
}
return result;
}
/**
* Returns the derivative as a {@link PolynomialFunction}.
*
* @return the derivative polynomial.
*/
public PolynomialFunction polynomialDerivative() {
return new PolynomialFunction(differentiate(coefficients));
}
/**
* Returns the derivative as a {@link UnivariateFunction}.
*
* @return the derivative function.
*/
public UnivariateFunction derivative() {
return polynomialDerivative();
}
/**
* Returns a string representation of the polynomial.
*
* The representation is user oriented. Terms are displayed lowest
* degrees first. The multiplications signs, coefficients equals to
* one and null terms are not displayed (except if the polynomial is 0,
* in which case the 0 constant term is displayed). Addition of terms
* with negative coefficients are replaced by subtraction of terms
* with positive coefficients except for the first displayed term
* (i.e. we display -3 for a constant negative polynomial,
* but 1 - 3 x + x^2 if the negative coefficient is not
* the first one displayed).
*
* @return a string representation of the polynomial.
*/
@Override
public String toString() {
StringBuilder s = new StringBuilder();
if (coefficients[0] == 0.0) {
if (coefficients.length == 1) {
return "0";
}
} else {
s.append(toString(coefficients[0]));
}
for (int i = 1; i < coefficients.length; ++i) {
if (coefficients[i] != 0) {
if (s.length() > 0) {
if (coefficients[i] < 0) {
s.append(" - ");
} else {
s.append(" + ");
}
} else {
if (coefficients[i] < 0) {
s.append("-");
}
}
double absAi = FastMath.abs(coefficients[i]);
if ((absAi - 1) != 0) {
s.append(toString(absAi));
s.append(' ');
}
s.append("x");
if (i > 1) {
s.append('^');
s.append(Integer.toString(i));
}
}
}
return s.toString();
}
/**
* Creates a string representing a coefficient, removing ".0" endings.
*
* @param coeff Coefficient.
* @return a string representation of {@code coeff}.
*/
private static String toString(double coeff) {
final String c = Double.toString(coeff);
if (c.endsWith(".0")) {
return c.substring(0, c.length() - 2);
} else {
return c;
}
}
/** {@inheritDoc} */
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + Arrays.hashCode(coefficients);
return result;
}
/** {@inheritDoc} */
@Override
public boolean equals(Object obj) {
if (this == obj) {
return true;
}
if (!(obj instanceof PolynomialFunction)) {
return false;
}
PolynomialFunction other = (PolynomialFunction) obj;
if (!Arrays.equals(coefficients, other.coefficients)) {
return false;
}
return true;
}
/**
* Dedicated parametric polynomial class.
*
* @since 3.0
*/
public static class Parametric implements ParametricUnivariateFunction {
/** {@inheritDoc} */
public double[] gradient(double x, double ... parameters) {
final double[] gradient = new double[parameters.length];
double xn = 1.0;
for (int i = 0; i < parameters.length; ++i) {
gradient[i] = xn;
xn *= x;
}
return gradient;
}
/** {@inheritDoc} */
public double value(final double x, final double ... parameters)
throws NoDataException {
return PolynomialFunction.evaluate(parameters, x);
}
}
}