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    1   /*
    2    * Copyright 2003-2004 The Apache Software Foundation.
    3    *
    4    * Licensed under the Apache License, Version 2.0 (the "License");
    5    * you may not use this file except in compliance with the License.
    6    * You may obtain a copy of the License at
    7    *
    8    *      http://www.apache.org/licenses/LICENSE-2.0
    9    *
    10   * Unless required by applicable law or agreed to in writing, software
    11   * distributed under the License is distributed on an "AS IS" BASIS,
    12   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    13   * See the License for the specific language governing permissions and
    14   * limitations under the License.
    15   */
    16  package org.apache.commons.math.analysis;
    17
    18
    19  import org.apache.commons.math.ConvergenceException;
    20  import org.apache.commons.math.FunctionEvaluationException;
    21
    22  /***
    23   * Implements the <a href="http://mathworld.wolfram.com/BrentsMethod.html">
    24   * Brent algorithm</a> for  finding zeros of real univariate functions.
    25   * <p>
    26   * The function should be continuous but not necessarily smooth.
    27   *
    28   * @version $Revision: 1.18 $ $Date: 2004/11/07 02:17:56 $
    29   */
    30  public class BrentSolver extends UnivariateRealSolverImpl {
    31
    32      /*** Serializable version identifier */
    33      static final long serialVersionUID = 3350616277306882875L;
    34
    35      /***
    36       * Construct a solver for the given function.
    37       *
    38       * @param f function to solve.
    39       */
    40      public BrentSolver(UnivariateRealFunction f) {
    41          super(f, 100, 1E-6);
    42      }
    43
    44      /***
    45       * Find a zero in the given interval.
    46       * <p>
    47       * Throws <code>ConvergenceException</code> if the values of the function
    48       * at the endpoints of the interval have the same sign.
    49       *
    50       * @param min the lower bound for the interval.
    51       * @param max the upper bound for the interval.
    52       * @param initial the start value to use (ignored).
    53       * @return the value where the function is zero
    54       * @throws ConvergenceException the maximum iteration count is exceeded
    55       * @throws FunctionEvaluationException if an error occurs evaluating
    56       *  the function
    57       * @throws IllegalArgumentException if initial is not between min and max
    58       */
    59      public double solve(double min, double max, double initial)
    60          throws ConvergenceException, FunctionEvaluationException {
    61
    62          return solve(min, max);
    63      }
    64
    65      /***
    66       * Find a zero in the given interval.
    67       * <p>
    68       * Requires that the values of the function at the endpoints have opposite
    69       * signs. An <code>IllegalArgumentException</code> is thrown if this is not
    70       * the case.
    71       *
    72       * @param min the lower bound for the interval.
    73       * @param max the upper bound for the interval.
    74       * @return the value where the function is zero
    75       * @throws ConvergenceException if the maximum iteration count is exceeded
    76       * @throws FunctionEvaluationException if an error occurs evaluating the
    77       * function
    78       * @throws IllegalArgumentException if min is not less than max or the
    79       * signs of the values of the function at the endpoints are not opposites
    80       */
    81      public double solve(double min, double max) throws ConvergenceException,
    82          FunctionEvaluationException {
    83
    84          clearResult();
    85          verifyBracketing(min, max, f);
    86
    87          // Index 0 is the old approximation for the root.
    88          // Index 1 is the last calculated approximation  for the root.
    89          // Index 2 is a bracket for the root with respect to x1.
    90          double x0 = min;
    91          double x1 = max;
    92          double y0;
    93          double y1;
    94          y0 = f.value(x0);
    95          y1 = f.value(x1);
    96
    97          double x2 = x0;
    98          double y2 = y0;
    99          double delta = x1 - x0;
    100         double oldDelta = delta;
    101
    102         int i = 0;
    103         while (i < maximalIterationCount) {
    104             if (Math.abs(y2) < Math.abs(y1)) {
    105                 x0 = x1;
    106                 x1 = x2;
    107                 x2 = x0;
    108                 y0 = y1;
    109                 y1 = y2;
    110                 y2 = y0;
    111             }
    112             if (Math.abs(y1) <= functionValueAccuracy) {
    113                 // Avoid division by very small values. Assume
    114                 // the iteration has converged (the problem may
    115                 // still be ill conditioned)
    116                 setResult(x1, i);
    117                 return result;
    118             }
    119             double dx = (x2 - x1);
    120             double tolerance =
    121                 Math.max(relativeAccuracy * Math.abs(x1), absoluteAccuracy);
    122             if (Math.abs(dx) <= tolerance) {
    123                 setResult(x1, i);
    124                 return result;
    125             }
    126             if ((Math.abs(oldDelta) < tolerance) ||
    127                     (Math.abs(y0) <= Math.abs(y1))) {
    128                 // Force bisection.
    129                 delta = 0.5 * dx;
    130                 oldDelta = delta;
    131             } else {
    132                 double r3 = y1 / y0;
    133                 double p;
    134                 double p1;
    135                 if (x0 == x2) {
    136                     // Linear interpolation.
    137                     p = dx * r3;
    138                     p1 = 1.0 - r3;
    139                 } else {
    140                     // Inverse quadratic interpolation.
    141                     double r1 = y0 / y2;
    142                     double r2 = y1 / y2;
    143                     p = r3 * (dx * r1 * (r1 - r2) - (x1 - x0) * (r2 - 1.0));
    144                     p1 = (r1 - 1.0) * (r2 - 1.0) * (r3 - 1.0);
    145                 }
    146                 if (p > 0.0) {
    147                     p1 = -p1;
    148                 } else {
    149                     p = -p;
    150                 }
    151                 if (2.0 * p >= 1.5 * dx * p1 - Math.abs(tolerance * p1) ||
    152                         p >= Math.abs(0.5 * oldDelta * p1)) {
    153                     // Inverse quadratic interpolation gives a value
    154                     // in the wrong direction, or progress is slow.
    155                     // Fall back to bisection.
    156                     delta = 0.5 * dx;
    157                     oldDelta = delta;
    158                 } else {
    159                     oldDelta = delta;
    160                     delta = p / p1;
    161                 }
    162             }
    163             // Save old X1, Y1
    164             x0 = x1;
    165             y0 = y1;
    166             // Compute new X1, Y1
    167             if (Math.abs(delta) > tolerance) {
    168                 x1 = x1 + delta;
    169             } else if (dx > 0.0) {
    170                 x1 = x1 + 0.5 * tolerance;
    171             } else if (dx <= 0.0) {
    172                 x1 = x1 - 0.5 * tolerance;
    173             }
    174             y1 = f.value(x1);
    175             if ((y1 > 0) == (y2 > 0)) {
    176                 x2 = x0;
    177                 y2 = y0;
    178                 delta = x1 - x0;
    179                 oldDelta = delta;
    180             }
    181             i++;
    182         }
    183         throw new ConvergenceException("Maximum number of iterations exceeded.");
    184     }
    185 }