/*
 * 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.math.analysis.solvers;

import org.apache.commons.math.ConvergenceException;
import org.apache.commons.math.FunctionEvaluationException;
import org.apache.commons.math.MaxIterationsExceededException;
import org.apache.commons.math.analysis.UnivariateRealFunction;
import org.apache.commons.math.util.MathUtils;

/**
 * Implements the <a href="http://mathworld.wolfram.com/MullersMethod.html">
 * Muller's Method</a> for root finding of real univariate functions. For
 * reference, see <b>Elementary Numerical Analysis</b>, ISBN 0070124477, chapter
 * 3.
 * <p>
 * Muller's method applies to both real and complex functions, but here we
 * restrict ourselves to real functions. Methods solve() and solve2() find real
 * zeros, using different ways to bypass complex arithmetics.
 * </p>
 *
 * @version $Revision$ $Date$
 * @since 1.2
 */
public class MullerSolver extends UnivariateRealSolverImpl {

	/**
	 * Construct a solver for the given function.
	 * 
	 * @param f
	 *            function to solve
	 * @deprecated as of 2.0 the function to solve is passed as an argument to
	 *             the {@link #solve(UnivariateRealFunction, double, double)} or
	 *             {@link UnivariateRealSolverImpl#solve(UnivariateRealFunction, double, double, double)}
	 *             method.
	 */
	@Deprecated
	public MullerSolver(UnivariateRealFunction f) {
		super(f, 100, 1E-6);
	}

	/**
	 * Construct a solver.
	 */
	public MullerSolver() {
		super(100, 1E-6);
	}

	/** {@inheritDoc} */
	@Deprecated
	public double solve(final double min, final double max) throws ConvergenceException, FunctionEvaluationException {
		return solve(f, min, max);
	}

	/** {@inheritDoc} */
	@Deprecated
	public double solve(final double min, final double max, final double initial)
			throws ConvergenceException, FunctionEvaluationException {
		return solve(f, min, max, initial);
	}

	/**
	 * Find a real root in the given interval with initial value.
	 * <p>
	 * Requires bracketing condition.
	 * </p>
	 * 
	 * @param f
	 *            the function to solve
	 * @param min
	 *            the lower bound for the interval
	 * @param max
	 *            the upper bound for the interval
	 * @param initial
	 *            the start value to use
	 * @return the point at which the function value is zero
	 * @throws MaxIterationsExceededException
	 *             if the maximum iteration count is exceeded or the solver
	 *             detects convergence problems otherwise
	 * @throws FunctionEvaluationException
	 *             if an error occurs evaluating the function
	 * @throws IllegalArgumentException
	 *             if any parameters are invalid
	 */
	public double solve(final UnivariateRealFunction f, final double min, final double max, final double initial)
			throws MaxIterationsExceededException, FunctionEvaluationException {

		// check for zeros before verifying bracketing
		if (f.value(min) == 0.0) {
			return min;
		}
		if (f.value(max) == 0.0) {
			return max;
		}
		if (f.value(initial) == 0.0) {
			return initial;
		}

		verifyBracketing(min, max, f);
		verifySequence(min, initial, max);
		if (isBracketing(min, initial, f)) {
			return solve(f, min, initial);
		} else {
			return solve(f, initial, max);
		}
	}

	/**
	 * Find a real root in the given interval.
	 * <p>
	 * Original Muller's method would have function evaluation at complex point.
	 * Since our f(x) is real, we have to find ways to avoid that. Bracketing
	 * condition is one way to go: by requiring bracketing in every iteration,
	 * the newly computed approximation is guaranteed to be real.
	 * </p>
	 * <p>
	 * Normally Muller's method converges quadratically in the vicinity of a
	 * zero, however it may be very slow in regions far away from zeros. For
	 * example, f(x) = exp(x) - 1, min = -50, max = 100. In such case we use
	 * bisection as a safety backup if it performs very poorly.
	 * </p>
	 * <p>
	 * The formulas here use divided differences directly.
	 * </p>
	 * 
	 * @param f
	 *            the function to solve
	 * @param min
	 *            the lower bound for the interval
	 * @param max
	 *            the upper bound for the interval
	 * @return the point at which the function value is zero
	 * @throws MaxIterationsExceededException
	 *             if the maximum iteration count is exceeded or the solver
	 *             detects convergence problems otherwise
	 * @throws FunctionEvaluationException
	 *             if an error occurs evaluating the function
	 * @throws IllegalArgumentException
	 *             if any parameters are invalid
	 */
	public double solve(final UnivariateRealFunction f, final double min, final double max)
			throws MaxIterationsExceededException, FunctionEvaluationException {

		// [x0, x2] is the bracketing interval in each iteration
		// x1 is the last approximation and an interpolation point in (x0, x2)
		// x is the new root approximation and new x1 for next round
		// d01, d12, d012 are divided differences
		double x0, x1, x2, x, oldx, y0, y1, y2, y;
		double d01, d12, d012, c1, delta, xplus, xminus, tolerance;

		x0 = min;
		y0 = f.value(x0);
		x2 = max;
		y2 = f.value(x2);
		x1 = 0.5 * (x0 + x2);
		y1 = f.value(x1);

		// check for zeros before verifying bracketing
		if (y0 == 0.0) {
			return min;
		}
		if (y2 == 0.0) {
			return max;
		}
		verifyBracketing(min, max, f);

		int i = 1;
		oldx = Double.POSITIVE_INFINITY;
		while (i <= maximalIterationCount) {
			// Muller's method employs quadratic interpolation through
			// x0, x1, x2 and x is the zero of the interpolating parabola.
			// Due to bracketing condition, this parabola must have two
			// real roots and we choose one in [x0, x2] to be x.
			d01 = (y1 - y0) / (x1 - x0);
			d12 = (y2 - y1) / (x2 - x1);
			d012 = (d12 - d01) / (x2 - x0);
			c1 = d01 + (x1 - x0) * d012;
			delta = c1 * c1 - 4 * y1 * d012;
			xplus = x1 + (-2.0 * y1) / (c1 + Math.sqrt(delta));
			xminus = x1 + (-2.0 * y1) / (c1 - Math.sqrt(delta));
			// xplus and xminus are two roots of parabola and at least
			// one of them should lie in (x0, x2)
			x = isSequence(x0, xplus, x2) ? xplus : xminus;
			y = f.value(x);

			// check for convergence
			tolerance = Math.max(relativeAccuracy * Math.abs(x), absoluteAccuracy);
			if (Math.abs(x - oldx) <= tolerance) {
				setResult(x, i);
				return result;
			}
			if (Math.abs(y) <= functionValueAccuracy) {
				setResult(x, i);
				return result;
			}

			// Bisect if convergence is too slow. Bisection would waste
			// our calculation of x, hopefully it won't happen often.
			// the real number equality test x == x1 is intentional and
			// completes the proximity tests above it
			boolean bisect = (x < x1 && (x1 - x0) > 0.95 * (x2 - x0)) || (x > x1 && (x2 - x1) > 0.95 * (x2 - x0))
					|| (x == x1);
			// prepare the new bracketing interval for next iteration
			if (!bisect) {
				x0 = x < x1 ? x0 : x1;
				y0 = x < x1 ? y0 : y1;
				x2 = x > x1 ? x2 : x1;
				y2 = x > x1 ? y2 : y1;
				x1 = x;
				y1 = y;
				oldx = x;
			} else {
				double xm = 0.5 * (x0 + x2);
				double ym = f.value(xm);
				if (MathUtils.sign(y0) + MathUtils.sign(ym) == 0.0) {
					x2 = xm;
					y2 = ym;
				} else {
					x0 = xm;
					y0 = ym;
				}
				x1 = 0.5 * (x0 + x2);
				y1 = f.value(x1);
				oldx = Double.POSITIVE_INFINITY;
			}
			i++;
		}
		throw new MaxIterationsExceededException(maximalIterationCount);
	}

	/**
	 * Find a real root in the given interval.
	 * <p>
	 * solve2() differs from solve() in the way it avoids complex operations.
	 * Except for the initial [min, max], solve2() does not require bracketing
	 * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex
	 * number arises in the computation, we simply use its modulus as real
	 * approximation.
	 * </p>
	 * <p>
	 * Because the interval may not be bracketing, bisection alternative is not
	 * applicable here. However in practice our treatment usually works well,
	 * especially near real zeros where the imaginary part of complex
	 * approximation is often negligible.
	 * </p>
	 * <p>
	 * The formulas here do not use divided differences directly.
	 * </p>
	 * 
	 * @param min
	 *            the lower bound for the interval
	 * @param max
	 *            the upper bound for the interval
	 * @return the point at which the function value is zero
	 * @throws MaxIterationsExceededException
	 *             if the maximum iteration count is exceeded or the solver
	 *             detects convergence problems otherwise
	 * @throws FunctionEvaluationException
	 *             if an error occurs evaluating the function
	 * @throws IllegalArgumentException
	 *             if any parameters are invalid
	 * @deprecated replaced by
	 *             {@link #solve2(UnivariateRealFunction, double, double)} since
	 *             2.0
	 */
	@Deprecated
	public double solve2(final double min, final double max)
			throws MaxIterationsExceededException, FunctionEvaluationException {
		return solve2(f, min, max);
	}

	/**
	 * Find a real root in the given interval.
	 * <p>
	 * solve2() differs from solve() in the way it avoids complex operations.
	 * Except for the initial [min, max], solve2() does not require bracketing
	 * condition, e.g. f(x0), f(x1), f(x2) can have the same sign. If complex
	 * number arises in the computation, we simply use its modulus as real
	 * approximation.
	 * </p>
	 * <p>
	 * Because the interval may not be bracketing, bisection alternative is not
	 * applicable here. However in practice our treatment usually works well,
	 * especially near real zeros where the imaginary part of complex
	 * approximation is often negligible.
	 * </p>
	 * <p>
	 * The formulas here do not use divided differences directly.
	 * </p>
	 * 
	 * @param f
	 *            the function to solve
	 * @param min
	 *            the lower bound for the interval
	 * @param max
	 *            the upper bound for the interval
	 * @return the point at which the function value is zero
	 * @throws MaxIterationsExceededException
	 *             if the maximum iteration count is exceeded or the solver
	 *             detects convergence problems otherwise
	 * @throws FunctionEvaluationException
	 *             if an error occurs evaluating the function
	 * @throws IllegalArgumentException
	 *             if any parameters are invalid
	 */
	public double solve2(final UnivariateRealFunction f, final double min, final double max)
			throws MaxIterationsExceededException, FunctionEvaluationException {

		// x2 is the last root approximation
		// x is the new approximation and new x2 for next round
		// x0 < x1 < x2 does not hold here
		double x0, x1, x2, x, oldx, y0, y1, y2, y;
		double q, A, B, C, delta, denominator, tolerance;

		x0 = min;
		y0 = f.value(x0);
		x1 = max;
		y1 = f.value(x1);
		x2 = 0.5 * (x0 + x1);
		y2 = f.value(x2);

		// check for zeros before verifying bracketing
		if (y0 == 0.0) {
			return min;
		}
		if (y1 == 0.0) {
			return max;
		}
		verifyBracketing(min, max, f);

		int i = 1;
		oldx = Double.POSITIVE_INFINITY;
		while (i <= maximalIterationCount) {
			// quadratic interpolation through x0, x1, x2
			q = (x2 - x1) / (x1 - x0);
			A = q * (y2 - (1 + q) * y1 + q * y0);
			B = (2 * q + 1) * y2 - (1 + q) * (1 + q) * y1 + q * q * y0;
			C = (1 + q) * y2;
			delta = B * B - 4 * A * C;
			if (delta >= 0.0) {
				// choose a denominator larger in magnitude
				double dplus = B + Math.sqrt(delta);
				double dminus = B - Math.sqrt(delta);
				denominator = Math.abs(dplus) > Math.abs(dminus) ? dplus : dminus;
			} else {
				// take the modulus of (B +/- Math.sqrt(delta))
				denominator = Math.sqrt(B * B - delta);
			}
			if (denominator != 0) {
				x = x2 - 2.0 * C * (x2 - x1) / denominator;
				// perturb x if it exactly coincides with x1 or x2
				// the equality tests here are intentional
				while (x == x1 || x == x2) {
					x += absoluteAccuracy;
				}
			} else {
				// extremely rare case, get a random number to skip it
				x = min + Math.random() * (max - min);
				oldx = Double.POSITIVE_INFINITY;
			}
			y = f.value(x);

			// check for convergence
			tolerance = Math.max(relativeAccuracy * Math.abs(x), absoluteAccuracy);
			if (Math.abs(x - oldx) <= tolerance) {
				setResult(x, i);
				return result;
			}
			if (Math.abs(y) <= functionValueAccuracy) {
				setResult(x, i);
				return result;
			}

			// prepare the next iteration
			x0 = x1;
			y0 = y1;
			x1 = x2;
			y1 = y2;
			x2 = x;
			y2 = y;
			oldx = x;
			i++;
		}
		throw new MaxIterationsExceededException(maximalIterationCount);
	}
}
