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package org.apache.commons.math3.analysis.solvers;


import org.apache.commons.math3.Field;
import org.apache.commons.math3.RealFieldElement;
import org.apache.commons.math3.analysis.RealFieldUnivariateFunction;
import org.apache.commons.math3.exception.MathInternalError;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.util.IntegerSequence;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;
import org.apache.commons.math3.util.Precision;

This class implements a modification of the Brent algorithm.

The changes with respect to the original Brent algorithm are:

  • the returned value is chosen in the current interval according to user specified AllowedSolution
  • the maximal order for the invert polynomial root search is user-specified instead of being invert quadratic only

The given interval must bracket the root.

Type parameters:
  • <T> – the type of the field elements
Since:3.6
/** * 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.
/** Maximal aging triggering an attempt to balance the bracketing interval. */
private static final int MAXIMAL_AGING = 2;
Field to which the elements belong.
/** Field to which the elements belong. */
private final Field<T> field;
Maximal order.
/** Maximal order. */
private final int maximalOrder;
Function value accuracy.
/** Function value accuracy. */
private final T functionValueAccuracy;
Absolute accuracy.
/** Absolute accuracy. */
private final T absoluteAccuracy;
Relative accuracy.
/** Relative accuracy. */
private final T relativeAccuracy;
Evaluations counter.
/** Evaluations counter. */
private IntegerSequence.Incrementor evaluations;
Construct a solver.
Params:
  • relativeAccuracy – Relative accuracy.
  • absoluteAccuracy – Absolute accuracy.
  • functionValueAccuracy – Function value accuracy.
  • maximalOrder – maximal order.
Throws:
/** * 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.
Returns:maximal order
/** Get the maximal order. * @return maximal order */
public int getMaximalOrder() { return maximalOrder; }
Get the maximal number of function evaluations.
Returns:the maximal number of function evaluations.
/** * Get the maximal number of function evaluations. * * @return the maximal number of function evaluations. */
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 optimize method. It is 0 if the method has not been called yet.
Returns:the number of evaluations of the objective function.
/** * 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. */
public int getEvaluations() { return evaluations.getCount(); }
Get the absolute accuracy.
Returns:absolute accuracy
/** * Get the absolute accuracy. * @return absolute accuracy */
public T getAbsoluteAccuracy() { return absoluteAccuracy; }
Get the relative accuracy.
Returns:relative accuracy
/** * Get the relative accuracy. * @return relative accuracy */
public T getRelativeAccuracy() { return relativeAccuracy; }
Get the function accuracy.
Returns:function accuracy
/** * Get the function accuracy. * @return function accuracy */
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.
Params:
  • maxEval – Maximum number of evaluations.
  • f – Function to solve.
  • min – Lower bound for the interval.
  • max – Upper bound for the interval.
  • allowedSolution – The kind of solutions that the root-finding algorithm may accept as solutions.
Throws:
Returns:a value where the function is zero.
/** * 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 */
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 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.
Params:
  • maxEval – Maximum number of evaluations.
  • f – Function to solve.
  • min – Lower bound for the interval.
  • max – Upper bound for the interval.
  • startValue – Start value to use.
  • allowedSolution – The kind of solutions that the root-finding algorithm may accept as solutions.
Throws:
Returns:a value where the function is zero.
/** * 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 */
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. MathUtils.checkNotNull(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 nth order inverse polynomial interpolation.

The x value is guessed by evaluating polynomial Q(y) at y = targetY, where Q is built such that for all considered points (xi, yi), Q(yi) = xi.

Params:
  • targetY – target value for y
  • x – reference points abscissas for interpolation, note that this array is modified during computation
  • y – reference points ordinates for interpolation
  • start – start index of the points to consider (inclusive)
  • end – end index of the points to consider (exclusive)
Returns:guessed root (will be a NaN if two points share the same y)
/** 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; } }