FieldBracketingNthOrderBrentSolver.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.solvers;
import org.apache.commons.math4.legacy.core.Field;
import org.apache.commons.math4.legacy.core.RealFieldElement;
import org.apache.commons.math4.legacy.analysis.RealFieldUnivariateFunction;
import org.apache.commons.math4.legacy.exception.MathInternalError;
import org.apache.commons.math4.legacy.exception.NoBracketingException;
import org.apache.commons.math4.legacy.exception.NullArgumentException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
import org.apache.commons.math4.legacy.core.IntegerSequence;
import org.apache.commons.math4.legacy.core.MathArrays;
import org.apache.commons.numbers.core.Precision;
/**
* This class implements a modification of the <a
* href="http://mathworld.wolfram.com/BrentsMethod.html"> Brent algorithm</a>.
* <p>
* The changes with respect to the original Brent algorithm are:
* <ul>
* <li>the returned value is chosen in the current interval according
* to user specified {@link AllowedSolution}</li>
* <li>the maximal order for the invert polynomial root search is
* user-specified instead of being invert quadratic only</li>
* </ul><p>
* The given interval must bracket the root.</p>
*
* @param <T> the type of the field elements
* @since 3.6
*/
public class FieldBracketingNthOrderBrentSolver<T extends RealFieldElement<T>>
implements BracketedRealFieldUnivariateSolver<T> {
/** Maximal aging triggering an attempt to balance the bracketing interval. */
private static final int MAXIMAL_AGING = 2;
/** Field to which the elements belong. */
private final Field<T> field;
/** Maximal order. */
private final int maximalOrder;
/** Function value accuracy. */
private final T functionValueAccuracy;
/** Absolute accuracy. */
private final T absoluteAccuracy;
/** Relative accuracy. */
private final T relativeAccuracy;
/** Evaluations counter. */
private IntegerSequence.Incrementor evaluations;
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
* @param functionValueAccuracy Function value accuracy.
* @param maximalOrder maximal order.
* @exception NumberIsTooSmallException if maximal order is lower than 2
*/
public FieldBracketingNthOrderBrentSolver(final T relativeAccuracy,
final T absoluteAccuracy,
final T functionValueAccuracy,
final int maximalOrder)
throws NumberIsTooSmallException {
if (maximalOrder < 2) {
throw new NumberIsTooSmallException(maximalOrder, 2, true);
}
this.field = relativeAccuracy.getField();
this.maximalOrder = maximalOrder;
this.absoluteAccuracy = absoluteAccuracy;
this.relativeAccuracy = relativeAccuracy;
this.functionValueAccuracy = functionValueAccuracy;
this.evaluations = IntegerSequence.Incrementor.create();
}
/** Get the maximal order.
* @return maximal order
*/
public int getMaximalOrder() {
return maximalOrder;
}
/**
* Get the maximal number of function evaluations.
*
* @return the maximal number of function evaluations.
*/
@Override
public int getMaxEvaluations() {
return evaluations.getMaximalCount();
}
/**
* Get the number of evaluations of the objective function.
* The number of evaluations corresponds to the last call to the
* {@code optimize} method. It is 0 if the method has not been
* called yet.
*
* @return the number of evaluations of the objective function.
*/
@Override
public int getEvaluations() {
return evaluations.getCount();
}
/**
* Get the absolute accuracy.
* @return absolute accuracy
*/
@Override
public T getAbsoluteAccuracy() {
return absoluteAccuracy;
}
/**
* Get the relative accuracy.
* @return relative accuracy
*/
@Override
public T getRelativeAccuracy() {
return relativeAccuracy;
}
/**
* Get the function accuracy.
* @return function accuracy
*/
@Override
public T getFunctionValueAccuracy() {
return functionValueAccuracy;
}
/**
* Solve for a zero in the given interval.
* A solver may require that the interval brackets a single zero root.
* Solvers that do require bracketing should be able to handle the case
* where one of the endpoints is itself a root.
*
* @param maxEval Maximum number of evaluations.
* @param f Function to solve.
* @param min Lower bound for the interval.
* @param max Upper bound for the interval.
* @param allowedSolution The kind of solutions that the root-finding algorithm may
* accept as solutions.
* @return a value where the function is zero.
* @exception NullArgumentException if f is null.
* @exception NoBracketingException if root cannot be bracketed
*/
@Override
public T solve(final int maxEval, final RealFieldUnivariateFunction<T> f,
final T min, final T max, final AllowedSolution allowedSolution)
throws NullArgumentException, NoBracketingException {
return solve(maxEval, f, min, max, min.add(max).divide(2), allowedSolution);
}
/**
* Solve for a zero in the given interval, start at {@code startValue}.
* A solver may require that the interval brackets a single zero root.
* Solvers that do require bracketing should be able to handle the case
* where one of the endpoints is itself a root.
*
* @param maxEval Maximum number of evaluations.
* @param f Function to solve.
* @param min Lower bound for the interval.
* @param max Upper bound for the interval.
* @param startValue Start value to use.
* @param allowedSolution The kind of solutions that the root-finding algorithm may
* accept as solutions.
* @return a value where the function is zero.
* @exception NullArgumentException if f is null.
* @exception NoBracketingException if root cannot be bracketed
*/
@Override
public T solve(final int maxEval, final RealFieldUnivariateFunction<T> f,
final T min, final T max, final T startValue,
final AllowedSolution allowedSolution)
throws NullArgumentException, NoBracketingException {
// Checks.
NullArgumentException.check(f);
// Reset.
evaluations = evaluations.withMaximalCount(maxEval).withStart(0);
T zero = field.getZero();
T nan = zero.add(Double.NaN);
// prepare arrays with the first points
final T[] x = MathArrays.buildArray(field, maximalOrder + 1);
final T[] y = MathArrays.buildArray(field, maximalOrder + 1);
x[0] = min;
x[1] = startValue;
x[2] = max;
// evaluate initial guess
evaluations.increment();
y[1] = f.value(x[1]);
if (Precision.equals(y[1].getReal(), 0.0, 1)) {
// return the initial guess if it is a perfect root.
return x[1];
}
// evaluate first endpoint
evaluations.increment();
y[0] = f.value(x[0]);
if (Precision.equals(y[0].getReal(), 0.0, 1)) {
// return the first endpoint if it is a perfect root.
return x[0];
}
int nbPoints;
int signChangeIndex;
if (y[0].multiply(y[1]).getReal() < 0) {
// reduce interval if it brackets the root
nbPoints = 2;
signChangeIndex = 1;
} else {
// evaluate second endpoint
evaluations.increment();
y[2] = f.value(x[2]);
if (Precision.equals(y[2].getReal(), 0.0, 1)) {
// return the second endpoint if it is a perfect root.
return x[2];
}
if (y[1].multiply(y[2]).getReal() < 0) {
// use all computed point as a start sampling array for solving
nbPoints = 3;
signChangeIndex = 2;
} else {
throw new NoBracketingException(x[0].getReal(), x[2].getReal(),
y[0].getReal(), y[2].getReal());
}
}
// prepare a work array for inverse polynomial interpolation
final T[] tmpX = MathArrays.buildArray(field, x.length);
// current tightest bracketing of the root
T xA = x[signChangeIndex - 1];
T yA = y[signChangeIndex - 1];
T absXA = xA.abs();
T absYA = yA.abs();
int agingA = 0;
T xB = x[signChangeIndex];
T yB = y[signChangeIndex];
T absXB = xB.abs();
T absYB = yB.abs();
int agingB = 0;
// search loop
while (true) {
// check convergence of bracketing interval
T maxX = absXA.subtract(absXB).getReal() < 0 ? absXB : absXA;
T maxY = absYA.subtract(absYB).getReal() < 0 ? absYB : absYA;
final T xTol = absoluteAccuracy.add(relativeAccuracy.multiply(maxX));
if (xB.subtract(xA).subtract(xTol).getReal() <= 0 ||
maxY.subtract(functionValueAccuracy).getReal() < 0) {
switch (allowedSolution) {
case ANY_SIDE :
return absYA.subtract(absYB).getReal() < 0 ? xA : xB;
case LEFT_SIDE :
return xA;
case RIGHT_SIDE :
return xB;
case BELOW_SIDE :
return yA.getReal() <= 0 ? xA : xB;
case ABOVE_SIDE :
return yA.getReal() < 0 ? xB : xA;
default :
// this should never happen
throw new MathInternalError(null);
}
}
// target for the next evaluation point
T targetY;
if (agingA >= MAXIMAL_AGING) {
// we keep updating the high bracket, try to compensate this
targetY = yB.divide(16).negate();
} else if (agingB >= MAXIMAL_AGING) {
// we keep updating the low bracket, try to compensate this
targetY = yA.divide(16).negate();
} else {
// bracketing is balanced, try to find the root itself
targetY = zero;
}
// make a few attempts to guess a root,
T nextX;
int start = 0;
int end = nbPoints;
do {
// guess a value for current target, using inverse polynomial interpolation
System.arraycopy(x, start, tmpX, start, end - start);
nextX = guessX(targetY, tmpX, y, start, end);
if (!(nextX.subtract(xA).getReal() > 0 && nextX.subtract(xB).getReal() < 0)) {
// the guessed root is not strictly inside of the tightest bracketing interval
// the guessed root is either not strictly inside the interval or it
// is a NaN (which occurs when some sampling points share the same y)
// we try again with a lower interpolation order
if (signChangeIndex - start >= end - signChangeIndex) {
// we have more points before the sign change, drop the lowest point
++start;
} else {
// we have more points after sign change, drop the highest point
--end;
}
// we need to do one more attempt
nextX = nan;
}
} while (Double.isNaN(nextX.getReal()) && end - start > 1);
if (Double.isNaN(nextX.getReal())) {
// fall back to bisection
nextX = xA.add(xB.subtract(xA).divide(2));
start = signChangeIndex - 1;
end = signChangeIndex;
}
// evaluate the function at the guessed root
evaluations.increment();
final T nextY = f.value(nextX);
if (Precision.equals(nextY.getReal(), 0.0, 1)) {
// we have found an exact root, since it is not an approximation
// we don't need to bother about the allowed solutions setting
return nextX;
}
if (nbPoints > 2 && end - start != nbPoints) {
// we have been forced to ignore some points to keep bracketing,
// they are probably too far from the root, drop them from now on
nbPoints = end - start;
System.arraycopy(x, start, x, 0, nbPoints);
System.arraycopy(y, start, y, 0, nbPoints);
signChangeIndex -= start;
} else if (nbPoints == x.length) {
// we have to drop one point in order to insert the new one
nbPoints--;
// keep the tightest bracketing interval as centered as possible
if (signChangeIndex >= (x.length + 1) / 2) {
// we drop the lowest point, we have to shift the arrays and the index
System.arraycopy(x, 1, x, 0, nbPoints);
System.arraycopy(y, 1, y, 0, nbPoints);
--signChangeIndex;
}
}
// insert the last computed point
//(by construction, we know it lies inside the tightest bracketing interval)
System.arraycopy(x, signChangeIndex, x, signChangeIndex + 1, nbPoints - signChangeIndex);
x[signChangeIndex] = nextX;
System.arraycopy(y, signChangeIndex, y, signChangeIndex + 1, nbPoints - signChangeIndex);
y[signChangeIndex] = nextY;
++nbPoints;
// update the bracketing interval
if (nextY.multiply(yA).getReal() <= 0) {
// the sign change occurs before the inserted point
xB = nextX;
yB = nextY;
absYB = yB.abs();
++agingA;
agingB = 0;
} else {
// the sign change occurs after the inserted point
xA = nextX;
yA = nextY;
absYA = yA.abs();
agingA = 0;
++agingB;
// update the sign change index
signChangeIndex++;
}
}
}
/** Guess an x value by n<sup>th</sup> order inverse polynomial interpolation.
* <p>
* The x value is guessed by evaluating polynomial Q(y) at y = targetY, where Q
* is built such that for all considered points (x<sub>i</sub>, y<sub>i</sub>),
* Q(y<sub>i</sub>) = x<sub>i</sub>.
* </p>
* @param targetY target value for y
* @param x reference points abscissas for interpolation,
* note that this array <em>is</em> modified during computation
* @param y reference points ordinates for interpolation
* @param start start index of the points to consider (inclusive)
* @param end end index of the points to consider (exclusive)
* @return guessed root (will be a NaN if two points share the same y)
*/
private T guessX(final T targetY, final T[] x, final T[] y,
final int start, final int end) {
// compute Q Newton coefficients by divided differences
for (int i = start; i < end - 1; ++i) {
final int delta = i + 1 - start;
for (int j = end - 1; j > i; --j) {
x[j] = x[j].subtract(x[j-1]).divide(y[j].subtract(y[j - delta]));
}
}
// evaluate Q(targetY)
T x0 = field.getZero();
for (int j = end - 1; j >= start; --j) {
x0 = x[j].add(x0.multiply(targetY.subtract(y[j])));
}
return x0;
}
}