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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
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package org.apache.commons.math3.analysis.solvers;
import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.exception.MathInternalError;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.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>
*
*/
public class BracketingNthOrderBrentSolver
extends AbstractUnivariateSolver
implements BracketedUnivariateSolver<UnivariateFunction> {
/** Default absolute accuracy. */
private static final double DEFAULT_ABSOLUTE_ACCURACY = 1e-6;
/** Default maximal order. */
private static final int DEFAULT_MAXIMAL_ORDER = 5;
/** Maximal aging triggering an attempt to balance the bracketing interval. */
private static final int MAXIMAL_AGING = 2;
/** Reduction factor for attempts to balance the bracketing interval. */
private static final double REDUCTION_FACTOR = 1.0 / 16.0;
/** Maximal order. */
private final int maximalOrder;
/** The kinds of solutions that the algorithm may accept. */
private AllowedSolution allowed;
/**
* Construct a solver with default accuracy and maximal order (1e-6 and 5 respectively)
*/
public BracketingNthOrderBrentSolver() {
this(DEFAULT_ABSOLUTE_ACCURACY, DEFAULT_MAXIMAL_ORDER);
}
/**
* Construct a solver.
*
* @param absoluteAccuracy Absolute accuracy.
* @param maximalOrder maximal order.
* @exception NumberIsTooSmallException if maximal order is lower than 2
*/
public BracketingNthOrderBrentSolver(final double absoluteAccuracy,
final int maximalOrder)
throws NumberIsTooSmallException {
super(absoluteAccuracy);
if (maximalOrder < 2) {
throw new NumberIsTooSmallException(maximalOrder, 2, true);
}
this.maximalOrder = maximalOrder;
this.allowed = AllowedSolution.ANY_SIDE;
}
/**
* Construct a solver.
*
* @param relativeAccuracy Relative accuracy.
* @param absoluteAccuracy Absolute accuracy.
* @param maximalOrder maximal order.
* @exception NumberIsTooSmallException if maximal order is lower than 2
*/
public BracketingNthOrderBrentSolver(final double relativeAccuracy,
final double absoluteAccuracy,
final int maximalOrder)
throws NumberIsTooSmallException {
super(relativeAccuracy, absoluteAccuracy);
if (maximalOrder < 2) {
throw new NumberIsTooSmallException(maximalOrder, 2, true);
}
this.maximalOrder = maximalOrder;
this.allowed = AllowedSolution.ANY_SIDE;
}
/**
* 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 BracketingNthOrderBrentSolver(final double relativeAccuracy,
final double absoluteAccuracy,
final double functionValueAccuracy,
final int maximalOrder)
throws NumberIsTooSmallException {
super(relativeAccuracy, absoluteAccuracy, functionValueAccuracy);
if (maximalOrder < 2) {
throw new NumberIsTooSmallException(maximalOrder, 2, true);
}
this.maximalOrder = maximalOrder;
this.allowed = AllowedSolution.ANY_SIDE;
}
/** Get the maximal order.
* @return maximal order
*/
public int getMaximalOrder() {
return maximalOrder;
}
/**
* {@inheritDoc}
*/
@Override
protected double doSolve()
throws TooManyEvaluationsException,
NumberIsTooLargeException,
NoBracketingException {
// prepare arrays with the first points
final double[] x = new double[maximalOrder + 1];
final double[] y = new double[maximalOrder + 1];
x[0] = getMin();
x[1] = getStartValue();
x[2] = getMax();
verifySequence(x[0], x[1], x[2]);
// evaluate initial guess
y[1] = computeObjectiveValue(x[1]);
if (Precision.equals(y[1], 0.0, 1)) {
// return the initial guess if it is a perfect root.
return x[1];
}
// evaluate first endpoint
y[0] = computeObjectiveValue(x[0]);
if (Precision.equals(y[0], 0.0, 1)) {
// return the first endpoint if it is a perfect root.
return x[0];
}
int nbPoints;
int signChangeIndex;
if (y[0] * y[1] < 0) {
// reduce interval if it brackets the root
nbPoints = 2;
signChangeIndex = 1;
} else {
// evaluate second endpoint
y[2] = computeObjectiveValue(x[2]);
if (Precision.equals(y[2], 0.0, 1)) {
// return the second endpoint if it is a perfect root.
return x[2];
}
if (y[1] * y[2] < 0) {
// use all computed point as a start sampling array for solving
nbPoints = 3;
signChangeIndex = 2;
} else {
throw new NoBracketingException(x[0], x[2], y[0], y[2]);
}
}
// prepare a work array for inverse polynomial interpolation
final double[] tmpX = new double[x.length];
// current tightest bracketing of the root
double xA = x[signChangeIndex - 1];
double yA = y[signChangeIndex - 1];
double absYA = Math.abs(yA);
int agingA = 0;
double xB = x[signChangeIndex];
double yB = y[signChangeIndex];
double absYB = Math.abs(yB);
int agingB = 0;
// search loop
while (true) {
// check convergence of bracketing interval
final double xTol = getAbsoluteAccuracy() +
getRelativeAccuracy() * Math.max(Math.abs(xA), Math.abs(xB));
if (((xB - xA) <= xTol) || (Math.max(absYA, absYB) < getFunctionValueAccuracy())) {
switch (allowed) {
case ANY_SIDE :
return absYA < absYB ? xA : xB;
case LEFT_SIDE :
return xA;
case RIGHT_SIDE :
return xB;
case BELOW_SIDE :
return (yA <= 0) ? xA : xB;
case ABOVE_SIDE :
return (yA < 0) ? xB : xA;
default :
// this should never happen
throw new MathInternalError();
}
}
// target for the next evaluation point
double targetY;
if (agingA >= MAXIMAL_AGING) {
// we keep updating the high bracket, try to compensate this
final int p = agingA - MAXIMAL_AGING;
final double weightA = (1 << p) - 1;
final double weightB = p + 1;
targetY = (weightA * yA - weightB * REDUCTION_FACTOR * yB) / (weightA + weightB);
} else if (agingB >= MAXIMAL_AGING) {
// we keep updating the low bracket, try to compensate this
final int p = agingB - MAXIMAL_AGING;
final double weightA = p + 1;
final double weightB = (1 << p) - 1;
targetY = (weightB * yB - weightA * REDUCTION_FACTOR * yA) / (weightA + weightB);
} else {
// bracketing is balanced, try to find the root itself
targetY = 0;
}
// make a few attempts to guess a root,
double 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 > xA) && (nextX < xB))) {
// 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 = Double.NaN;
}
} while (Double.isNaN(nextX) && (end - start > 1));
if (Double.isNaN(nextX)) {
// fall back to bisection
nextX = xA + 0.5 * (xB - xA);
start = signChangeIndex - 1;
end = signChangeIndex;
}
// evaluate the function at the guessed root
final double nextY = computeObjectiveValue(nextX);
if (Precision.equals(nextY, 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 * yA <= 0) {
// the sign change occurs before the inserted point
xB = nextX;
yB = nextY;
absYB = Math.abs(yB);
++agingA;
agingB = 0;
} else {
// the sign change occurs after the inserted point
xA = nextX;
yA = nextY;
absYA = Math.abs(yA);
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 double guessX(final double targetY, final double[] x, final double[] 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] - x[j-1]) / (y[j] - y[j - delta]);
}
}
// evaluate Q(targetY)
double x0 = 0;
for (int j = end - 1; j >= start; --j) {
x0 = x[j] + x0 * (targetY - y[j]);
}
return x0;
}
/** {@inheritDoc} */
public double solve(int maxEval, UnivariateFunction f, double min,
double max, AllowedSolution allowedSolution)
throws TooManyEvaluationsException,
NumberIsTooLargeException,
NoBracketingException {
this.allowed = allowedSolution;
return super.solve(maxEval, f, min, max);
}
/** {@inheritDoc} */
public double solve(int maxEval, UnivariateFunction f, double min,
double max, double startValue,
AllowedSolution allowedSolution)
throws TooManyEvaluationsException,
NumberIsTooLargeException,
NoBracketingException {
this.allowed = allowedSolution;
return super.solve(maxEval, f, min, max, startValue);
}
}