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

import org.apache.commons.math3.analysis.UnivariateFunction;
import org.apache.commons.math3.exception.NoBracketingException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;

/**
 * Utility routines for {@link UnivariateSolver} objects.
 *
 */
public class UnivariateSolverUtils {
    /**
     * Class contains only static methods.
     */
    private UnivariateSolverUtils() {}

    /**
     * Convenience method to find a zero of a univariate real function.  A default
     * solver is used.
     *
     * @param function Function.
     * @param x0 Lower bound for the interval.
     * @param x1 Upper bound for the interval.
     * @return a value where the function is zero.
     * @throws NoBracketingException if the function has the same sign at the
     * endpoints.
     * @throws NullArgumentException if {@code function} is {@code null}.
     */
    public static double solve(UnivariateFunction function, double x0, double x1)
        throws NullArgumentException,
               NoBracketingException {
        if (function == null) {
            throw new NullArgumentException(LocalizedFormats.FUNCTION);
        }
        final UnivariateSolver solver = new BrentSolver();
        return solver.solve(Integer.MAX_VALUE, function, x0, x1);
    }

    /**
     * Convenience method to find a zero of a univariate real function.  A default
     * solver is used.
     *
     * @param function Function.
     * @param x0 Lower bound for the interval.
     * @param x1 Upper bound for the interval.
     * @param absoluteAccuracy Accuracy to be used by the solver.
     * @return a value where the function is zero.
     * @throws NoBracketingException if the function has the same sign at the
     * endpoints.
     * @throws NullArgumentException if {@code function} is {@code null}.
     */
    public static double solve(UnivariateFunction function,
                               double x0, double x1,
                               double absoluteAccuracy)
        throws NullArgumentException,
               NoBracketingException {
        if (function == null) {
            throw new NullArgumentException(LocalizedFormats.FUNCTION);
        }
        final UnivariateSolver solver = new BrentSolver(absoluteAccuracy);
        return solver.solve(Integer.MAX_VALUE, function, x0, x1);
    }

    /**
     * Force a root found by a non-bracketing solver to lie on a specified side,
     * as if the solver were a bracketing one.
     *
     * @param maxEval maximal number of new evaluations of the function
     * (evaluations already done for finding the root should have already been subtracted
     * from this number)
     * @param f function to solve
     * @param bracketing bracketing solver to use for shifting the root
     * @param baseRoot original root found by a previous non-bracketing solver
     * @param min minimal bound of the search interval
     * @param max maximal bound of the search interval
     * @param allowedSolution the kind of solutions that the root-finding algorithm may
     * accept as solutions.
     * @return a root approximation, on the specified side of the exact root
     * @throws NoBracketingException if the function has the same sign at the
     * endpoints.
     */
    public static double forceSide(final int maxEval, final UnivariateFunction f,
                                   final BracketedUnivariateSolver bracketing,
                                   final double baseRoot, final double min, final double max,
                                   final AllowedSolution allowedSolution)
        throws NoBracketingException {

        if (allowedSolution == AllowedSolution.ANY_SIDE) {
            // no further bracketing required
            return baseRoot;
        }

        // find a very small interval bracketing the root
        final double step = FastMath.max(bracketing.getAbsoluteAccuracy(),
                                         FastMath.abs(baseRoot * bracketing.getRelativeAccuracy()));
        double xLo        = FastMath.max(min, baseRoot - step);
        double fLo        = f.value(xLo);
        double xHi        = FastMath.min(max, baseRoot + step);
        double fHi        = f.value(xHi);
        int remainingEval = maxEval - 2;
        while (remainingEval > 0) {

            if ((fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0)) {
                // compute the root on the selected side
                return bracketing.solve(remainingEval, f, xLo, xHi, baseRoot, allowedSolution);
            }

            // try increasing the interval
            boolean changeLo = false;
            boolean changeHi = false;
            if (fLo < fHi) {
                // increasing function
                if (fLo >= 0) {
                    changeLo = true;
                } else {
                    changeHi = true;
                }
            } else if (fLo > fHi) {
                // decreasing function
                if (fLo <= 0) {
                    changeLo = true;
                } else {
                    changeHi = true;
                }
            } else {
                // unknown variation
                changeLo = true;
                changeHi = true;
            }

            // update the lower bound
            if (changeLo) {
                xLo = FastMath.max(min, xLo - step);
                fLo  = f.value(xLo);
                remainingEval--;
            }

            // update the higher bound
            if (changeHi) {
                xHi = FastMath.min(max, xHi + step);
                fHi  = f.value(xHi);
                remainingEval--;
            }

        }

        throw new NoBracketingException(LocalizedFormats.FAILED_BRACKETING,
                                        xLo, xHi, fLo, fHi,
                                        maxEval - remainingEval, maxEval, baseRoot,
                                        min, max);

    }

    /**
     * This method simply calls {@link #bracket(UnivariateFunction, double, double, double,
     * double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}
     * with {@code q} and {@code r} set to 1.0 and {@code maximumIterations} set to {@code Integer.MAX_VALUE}.
     * 

* Note: this method can take {@code Integer.MAX_VALUE} * iterations to throw a {@code ConvergenceException.} Unless you are * confident that there is a root between {@code lowerBound} and * {@code upperBound} near {@code initial}, it is better to use * {@link #bracket(UnivariateFunction, double, double, double, double,double, int) * bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)}, * explicitly specifying the maximum number of iterations.

* * @param function Function. * @param initial Initial midpoint of interval being expanded to * bracket a root. * @param lowerBound Lower bound (a is never lower than this value) * @param upperBound Upper bound (b never is greater than this * value). * @return a two-element array holding a and b. * @throws NoBracketingException if a root cannot be bracketted. * @throws NotStrictlyPositiveException if {@code maximumIterations <= 0}. * @throws NullArgumentException if {@code function} is {@code null}. */ public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound) throws NullArgumentException, NotStrictlyPositiveException, NoBracketingException { return bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, Integer.MAX_VALUE); } /** * This method simply calls {@link #bracket(UnivariateFunction, double, double, double, * double, double, int) bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)} * with {@code q} and {@code r} set to 1.0. * @param function Function. * @param initial Initial midpoint of interval being expanded to * bracket a root. * @param lowerBound Lower bound (a is never lower than this value). * @param upperBound Upper bound (b never is greater than this * value). * @param maximumIterations Maximum number of iterations to perform * @return a two element array holding a and b. * @throws NoBracketingException if the algorithm fails to find a and b * satisfying the desired conditions. * @throws NotStrictlyPositiveException if {@code maximumIterations <= 0}. * @throws NullArgumentException if {@code function} is {@code null}. */ public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, int maximumIterations) throws NullArgumentException, NotStrictlyPositiveException, NoBracketingException { return bracket(function, initial, lowerBound, upperBound, 1.0, 1.0, maximumIterations); } /** * This method attempts to find two values a and b satisfying
    *
  • {@code lowerBound <= a < initial < b <= upperBound}
  • *
  • {@code f(a) * f(b) <= 0}
  • *
* If {@code f} is continuous on {@code [a,b]}, this means that {@code a} * and {@code b} bracket a root of {@code f}. *

* The algorithm checks the sign of \( f(l_k) \) and \( f(u_k) \) for increasing * values of k, where \( l_k = max(lower, initial - \delta_k) \), * \( u_k = min(upper, initial + \delta_k) \), using recurrence * \( \delta_{k+1} = r \delta_k + q, \delta_0 = 0\) and starting search with \( k=1 \). * The algorithm stops when one of the following happens:

    *
  • at least one positive and one negative value have been found -- success!
  • *
  • both endpoints have reached their respective limits -- NoBracketingException
  • *
  • {@code maximumIterations} iterations elapse -- NoBracketingException
*

* If different signs are found at first iteration ({@code k=1}), then the returned * interval will be \( [a, b] = [l_1, u_1] \). If different signs are found at a later * iteration {@code k>1}, then the returned interval will be either * \( [a, b] = [l_{k+1}, l_{k}] \) or \( [a, b] = [u_{k}, u_{k+1}] \). A root solver called * with these parameters will therefore start with the smallest bracketing interval known * at this step. *

*

* Interval expansion rate is tuned by changing the recurrence parameters {@code r} and * {@code q}. When the multiplicative factor {@code r} is set to 1, the sequence is a * simple arithmetic sequence with linear increase. When the multiplicative factor {@code r} * is larger than 1, the sequence has an asymptotically exponential rate. Note than the * additive parameter {@code q} should never be set to zero, otherwise the interval would * degenerate to the single initial point for all values of {@code k}. *

*

* As a rule of thumb, when the location of the root is expected to be approximately known * within some error margin, {@code r} should be set to 1 and {@code q} should be set to the * order of magnitude of the error margin. When the location of the root is really a wild guess, * then {@code r} should be set to a value larger than 1 (typically 2 to double the interval * length at each iteration) and {@code q} should be set according to half the initial * search interval length. *

*

* As an example, if we consider the trivial function {@code f(x) = 1 - x} and use * {@code initial = 4}, {@code r = 1}, {@code q = 2}, the algorithm will compute * {@code f(4-2) = f(2) = -1} and {@code f(4+2) = f(6) = -5} for {@code k = 1}, then * {@code f(4-4) = f(0) = +1} and {@code f(4+4) = f(8) = -7} for {@code k = 2}. Then it will * return the interval {@code [0, 2]} as the smallest one known to be bracketing the root. * As shown by this example, the initial value (here {@code 4}) may lie outside of the returned * bracketing interval. *

* @param function function to check * @param initial Initial midpoint of interval being expanded to * bracket a root. * @param lowerBound Lower bound (a is never lower than this value). * @param upperBound Upper bound (b never is greater than this * value). * @param q additive offset used to compute bounds sequence (must be strictly positive) * @param r multiplicative factor used to compute bounds sequence * @param maximumIterations Maximum number of iterations to perform * @return a two element array holding the bracketing values. * @exception NoBracketingException if function cannot be bracketed in the search interval */ public static double[] bracket(final UnivariateFunction function, final double initial, final double lowerBound, final double upperBound, final double q, final double r, final int maximumIterations) throws NoBracketingException { if (function == null) { throw new NullArgumentException(LocalizedFormats.FUNCTION); } if (q <= 0) { throw new NotStrictlyPositiveException(q); } if (maximumIterations <= 0) { throw new NotStrictlyPositiveException(LocalizedFormats.INVALID_MAX_ITERATIONS, maximumIterations); } verifySequence(lowerBound, initial, upperBound); // initialize the recurrence double a = initial; double b = initial; double fa = Double.NaN; double fb = Double.NaN; double delta = 0; for (int numIterations = 0; (numIterations < maximumIterations) && (a > lowerBound || b < upperBound); ++numIterations) { final double previousA = a; final double previousFa = fa; final double previousB = b; final double previousFb = fb; delta = r * delta + q; a = FastMath.max(initial - delta, lowerBound); b = FastMath.min(initial + delta, upperBound); fa = function.value(a); fb = function.value(b); if (numIterations == 0) { // at first iteration, we don't have a previous interval // we simply compare both sides of the initial interval if (fa * fb <= 0) { // the first interval already brackets a root return new double[] { a, b }; } } else { // we have a previous interval with constant sign and expand it, // we expect sign changes to occur at boundaries if (fa * previousFa <= 0) { // sign change detected at near lower bound return new double[] { a, previousA }; } else if (fb * previousFb <= 0) { // sign change detected at near upper bound return new double[] { previousB, b }; } } } // no bracketing found throw new NoBracketingException(a, b, fa, fb); } /** * Compute the midpoint of two values. * * @param a first value. * @param b second value. * @return the midpoint. */ public static double midpoint(double a, double b) { return (a + b) * 0.5; } /** * Check whether the interval bounds bracket a root. That is, if the * values at the endpoints are not equal to zero, then the function takes * opposite signs at the endpoints. * * @param function Function. * @param lower Lower endpoint. * @param upper Upper endpoint. * @return {@code true} if the function values have opposite signs at the * given points. * @throws NullArgumentException if {@code function} is {@code null}. */ public static boolean isBracketing(UnivariateFunction function, final double lower, final double upper) throws NullArgumentException { if (function == null) { throw new NullArgumentException(LocalizedFormats.FUNCTION); } final double fLo = function.value(lower); final double fHi = function.value(upper); return (fLo >= 0 && fHi <= 0) || (fLo <= 0 && fHi >= 0); } /** * Check whether the arguments form a (strictly) increasing sequence. * * @param start First number. * @param mid Second number. * @param end Third number. * @return {@code true} if the arguments form an increasing sequence. */ public static boolean isSequence(final double start, final double mid, final double end) { return (start < mid) && (mid < end); } /** * Check that the endpoints specify an interval. * * @param lower Lower endpoint. * @param upper Upper endpoint. * @throws NumberIsTooLargeException if {@code lower >= upper}. */ public static void verifyInterval(final double lower, final double upper) throws NumberIsTooLargeException { if (lower >= upper) { throw new NumberIsTooLargeException(LocalizedFormats.ENDPOINTS_NOT_AN_INTERVAL, lower, upper, false); } } /** * Check that {@code lower < initial < upper}. * * @param lower Lower endpoint. * @param initial Initial value. * @param upper Upper endpoint. * @throws NumberIsTooLargeException if {@code lower >= initial} or * {@code initial >= upper}. */ public static void verifySequence(final double lower, final double initial, final double upper) throws NumberIsTooLargeException { verifyInterval(lower, initial); verifyInterval(initial, upper); } /** * Check that the endpoints specify an interval and the end points * bracket a root. * * @param function Function. * @param lower Lower endpoint. * @param upper Upper endpoint. * @throws NoBracketingException if the function has the same sign at the * endpoints. * @throws NullArgumentException if {@code function} is {@code null}. */ public static void verifyBracketing(UnivariateFunction function, final double lower, final double upper) throws NullArgumentException, NoBracketingException { if (function == null) { throw new NullArgumentException(LocalizedFormats.FUNCTION); } verifyInterval(lower, upper); if (!isBracketing(function, lower, upper)) { throw new NoBracketingException(lower, upper, function.value(lower), function.value(upper)); } } }




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