| /* |
| * 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.math4.legacy.analysis.solvers; |
| |
| |
| import org.apache.commons.math4.legacy.analysis.UnivariateFunction; |
| import org.apache.commons.math4.legacy.exception.MathInternalError; |
| import org.apache.commons.math4.legacy.exception.NoBracketingException; |
| import org.apache.commons.math4.legacy.exception.NumberIsTooLargeException; |
| import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException; |
| import org.apache.commons.math4.legacy.exception.TooManyEvaluationsException; |
| import org.apache.commons.math4.core.jdkmath.JdkMath; |
| 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> |
| * |
| */ |
| 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 = JdkMath.abs(yA); |
| int agingA = 0; |
| double xB = x[signChangeIndex]; |
| double yB = y[signChangeIndex]; |
| double absYB = JdkMath.abs(yB); |
| int agingB = 0; |
| |
| // search loop |
| while (true) { |
| |
| // check convergence of bracketing interval |
| final double xTol = getAbsoluteAccuracy() + |
| getRelativeAccuracy() * JdkMath.max(JdkMath.abs(xA), JdkMath.abs(xB)); |
| if (((xB - xA) <= xTol) || (JdkMath.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 = JdkMath.abs(yB); |
| ++agingA; |
| agingB = 0; |
| } else { |
| // the sign change occurs after the inserted point |
| xA = nextX; |
| yA = nextY; |
| absYA = JdkMath.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} */ |
| @Override |
| 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} */ |
| @Override |
| 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); |
| } |
| |
| } |