/*
 * 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.ode.nonstiff;

import org.apache.commons.math.linear.Array2DRowRealMatrix;
import org.apache.commons.math.ode.DerivativeException;
import org.apache.commons.math.ode.FirstOrderDifferentialEquations;
import org.apache.commons.math.ode.IntegratorException;
import org.apache.commons.math.ode.events.CombinedEventsManager;
import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator;
import org.apache.commons.math.ode.sampling.StepHandler;

/**
 * This class implements explicit Adams-Bashforth integrators for Ordinary
 * Differential Equations.
 *
 * <p>
 * Adams-Bashforth methods (in fact due to Adams alone) are explicit multistep
 * ODE solvers. This implementation is a variation of the classical one: it uses
 * adaptive stepsize to implement error control, whereas classical
 * implementations are fixed step size. The value of state vector at step n+1 is
 * a simple combination of the value at step n and of the derivatives at steps
 * n, n-1, n-2 ... Depending on the number k of previous steps one wants to use
 * for computing the next value, different formulas are available:
 * </p>
 * <ul>
 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li>
 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h
 * (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li>
 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h
 * (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li>
 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h
 * (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li>
 * <li>...</li>
 * </ul>
 *
 * <p>
 * A k-steps Adams-Bashforth method is of order k.
 * </p>
 *
 * <h3>Implementation details</h3>
 *
 * <p>
 * We define scaled derivatives s<sub>i</sub>(n) at step n as:
 * 
 * <pre>
 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative
 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative
 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative
 * ...
 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative
 * </pre>
 * </p>
 *
 * <p>
 * The definitions above use the classical representation with several previous
 * first derivatives. Lets define
 * 
 * <pre>
 *   q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup>
 * </pre>
 * 
 * (we omit the k index in the notation for clarity). With these definitions,
 * Adams-Bashforth methods can be written:
 * <ul>
 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li>
 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n) + [ -1/2 ]
 * q<sub>n</sub></li>
 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n) + [
 * -16/12 5/12 ] q<sub>n</sub></li>
 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n) + [
 * -59/24 37/24 -9/24 ] q<sub>n</sub></li>
 * <li>...</li>
 * </ul>
 * </p>
 *
 * <p>
 * Instead of using the classical representation with first derivatives only
 * (y<sub>n</sub>, s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses
 * the Nordsieck vector with higher degrees scaled derivatives all taken at the
 * same step (y<sub>n</sub>, s<sub>1</sub>(n) and r<sub>n</sub>) where
 * r<sub>n</sub> is defined as:
 * 
 * <pre>
 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup>
 * </pre>
 * 
 * (here again we omit the k index in the notation for clarity)
 * </p>
 *
 * <p>
 * Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i)
 * can be computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n),
 * the formula being exact for degree k polynomials.
 * 
 * <pre>
 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + &sum;<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n)
 * </pre>
 * 
 * The previous formula can be used with several values for i to compute the
 * transform between classical representation and Nordsieck vector. The
 * transform between r<sub>n</sub> and q<sub>n</sub> resulting from the Taylor
 * series formulas above is:
 * 
 * <pre>
 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub>
 * </pre>
 * 
 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the
 * (k-1)&times;(k-1) matrix built with the j (-i)<sup>j-1</sup> terms:
 * 
 * <pre>
 *        [  -2   3   -4    5  ... ]
 *        [  -4  12  -32   80  ... ]
 *   P =  [  -6  27 -108  405  ... ]
 *        [  -8  48 -256 1280  ... ]
 *        [          ...           ]
 * </pre>
 * </p>
 * 
 * <p>
 * Using the Nordsieck vector has several advantages:
 * <ul>
 * <li>it greatly simplifies step interpolation as the interpolator mainly
 * applies Taylor series formulas,</li>
 * <li>it simplifies step changes that occur when discrete events that truncate
 * the step are triggered,</li>
 * <li>it allows to extend the methods in order to support adaptive
 * stepsize.</li>
 * </ul>
 * </p>
 * 
 * <p>
 * The Nordsieck vector at step n+1 is computed from the Nordsieck vector at
 * step n as follows:
 * <ul>
 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup>
 * r<sub>n</sub></li>
 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li>
 * <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup>
 * u + P<sup>-1</sup> A P r<sub>n</sub></li>
 * </ul>
 * where A is a rows shifting matrix (the lower left part is an identity
 * matrix):
 * 
 * <pre>
 *        [ 0 0   ...  0 0 | 0 ]
 *        [ ---------------+---]
 *        [ 1 0   ...  0 0 | 0 ]
 *    A = [ 0 1   ...  0 0 | 0 ]
 *        [       ...      | 0 ]
 *        [ 0 0   ...  1 0 | 0 ]
 *        [ 0 0   ...  0 1 | 0 ]
 * </pre>
 * </p>
 *
 * <p>
 * The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on
 * the state, they only depend on k and therefore are precomputed once for all.
 * </p>
 *
 * @version $Revision$ $Date$
 * @since 2.0
 */
public class AdamsBashforthIntegrator extends AdamsIntegrator {

	/**
	 * Build an Adams-Bashforth integrator with the given order and step control
	 * parameters.
	 * 
	 * @param nSteps
	 *            number of steps of the method excluding the one being computed
	 * @param minStep
	 *            minimal step (must be positive even for backward integration),
	 *            the last step can be smaller than this
	 * @param maxStep
	 *            maximal step (must be positive even for backward integration)
	 * @param scalAbsoluteTolerance
	 *            allowed absolute error
	 * @param scalRelativeTolerance
	 *            allowed relative error
	 * @exception IllegalArgumentException
	 *                if order is 1 or less
	 */
	public AdamsBashforthIntegrator(final int nSteps, final double minStep, final double maxStep,
			final double scalAbsoluteTolerance, final double scalRelativeTolerance) throws IllegalArgumentException {
		super("Adams-Bashforth", nSteps, nSteps, minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
	}

	/**
	 * Build an Adams-Bashforth integrator with the given order and step control
	 * parameters.
	 * 
	 * @param nSteps
	 *            number of steps of the method excluding the one being computed
	 * @param minStep
	 *            minimal step (must be positive even for backward integration),
	 *            the last step can be smaller than this
	 * @param maxStep
	 *            maximal step (must be positive even for backward integration)
	 * @param vecAbsoluteTolerance
	 *            allowed absolute error
	 * @param vecRelativeTolerance
	 *            allowed relative error
	 * @exception IllegalArgumentException
	 *                if order is 1 or less
	 */
	public AdamsBashforthIntegrator(final int nSteps, final double minStep, final double maxStep,
			final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance) throws IllegalArgumentException {
		super("Adams-Bashforth", nSteps, nSteps, minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
	}

	/** {@inheritDoc} */
	@Override
	public double integrate(final FirstOrderDifferentialEquations equations, final double t0, final double[] y0,
			final double t, final double[] y) throws DerivativeException, IntegratorException {

		final int n = y0.length;
		sanityChecks(equations, t0, y0, t, y);
		setEquations(equations);
		resetEvaluations();
		final boolean forward = (t > t0);

		// initialize working arrays
		if (y != y0) {
			System.arraycopy(y0, 0, y, 0, n);
		}
		final double[] yDot = new double[n];
		final double[] yTmp = new double[y0.length];

		// set up an interpolator sharing the integrator arrays
		final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator();
		interpolator.reinitialize(y, forward);
		final NordsieckStepInterpolator interpolatorTmp = new NordsieckStepInterpolator();
		interpolatorTmp.reinitialize(yTmp, forward);

		// set up integration control objects
		for (StepHandler handler : stepHandlers) {
			handler.reset();
		}
		CombinedEventsManager manager = addEndTimeChecker(t0, t, eventsHandlersManager);

		// compute the initial Nordsieck vector using the configured starter
		// integrator
		start(t0, y, t);
		interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);
		interpolator.storeTime(stepStart);
		final int lastRow = nordsieck.getRowDimension() - 1;

		// reuse the step that was chosen by the starter integrator
		double hNew = stepSize;
		interpolator.rescale(hNew);

		boolean lastStep = false;
		while (!lastStep) {

			// shift all data
			interpolator.shift();

			double error = 0;
			for (boolean loop = true; loop;) {

				stepSize = hNew;

				// evaluate error using the last term of the Taylor expansion
				error = 0;
				for (int i = 0; i < y0.length; ++i) {
					final double yScale = Math.abs(y[i]);
					final double tol = (vecAbsoluteTolerance == null)
							? (scalAbsoluteTolerance + scalRelativeTolerance * yScale)
							: (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale);
					final double ratio = nordsieck.getEntry(lastRow, i) / tol;
					error += ratio * ratio;
				}
				error = Math.sqrt(error / y0.length);

				if (error <= 1.0) {

					// predict a first estimate of the state at step end
					final double stepEnd = stepStart + stepSize;
					interpolator.setInterpolatedTime(stepEnd);
					System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length);

					// evaluate the derivative
					computeDerivatives(stepEnd, yTmp, yDot);

					// update Nordsieck vector
					final double[] predictedScaled = new double[y0.length];
					for (int j = 0; j < y0.length; ++j) {
						predictedScaled[j] = stepSize * yDot[j];
					}
					final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck);
					updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp);

					// discrete events handling
					interpolatorTmp.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp);
					interpolatorTmp.storeTime(stepStart);
					interpolatorTmp.shift();
					interpolatorTmp.storeTime(stepEnd);
					if (manager.evaluateStep(interpolatorTmp)) {
						final double dt = manager.getEventTime() - stepStart;
						if (Math.abs(dt) <= Math.ulp(stepStart)) {
							// rejecting the step would lead to a too small next
							// step, we accept it
							loop = false;
						} else {
							// reject the step to match exactly the next switch
							// time
							hNew = dt;
							interpolator.rescale(hNew);
						}
					} else {
						// accept the step
						scaled = predictedScaled;
						nordsieck = nordsieckTmp;
						interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck);
						loop = false;
					}

				} else {
					// reject the step and attempt to reduce error by stepsize
					// control
					final double factor = computeStepGrowShrinkFactor(error);
					hNew = filterStep(stepSize * factor, forward, false);
					interpolator.rescale(hNew);
				}

			}

			// the step has been accepted (may have been truncated)
			final double nextStep = stepStart + stepSize;
			System.arraycopy(yTmp, 0, y, 0, n);
			interpolator.storeTime(nextStep);
			manager.stepAccepted(nextStep, y);
			lastStep = manager.stop();

			// provide the step data to the step handler
			for (StepHandler handler : stepHandlers) {
				interpolator.setInterpolatedTime(nextStep);
				handler.handleStep(interpolator, lastStep);
			}
			stepStart = nextStep;

			if (!lastStep && manager.reset(stepStart, y)) {

				// some events handler has triggered changes that
				// invalidate the derivatives, we need to restart from scratch
				start(stepStart, y, t);
				interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck);

			}

			if (!lastStep) {
				// in some rare cases we may get here with stepSize = 0, for
				// example
				// when an event occurs at integration start, reducing the first
				// step
				// to zero; we have to reset the step to some safe non zero
				// value
				stepSize = filterStep(stepSize, forward, true);

				// stepsize control for next step
				final double factor = computeStepGrowShrinkFactor(error);
				final double scaledH = stepSize * factor;
				final double nextT = stepStart + scaledH;
				final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
				hNew = filterStep(scaledH, forward, nextIsLast);
				interpolator.rescale(hNew);
			}

		}

		final double stopTime = stepStart;
		stepStart = Double.NaN;
		stepSize = Double.NaN;
		return stopTime;

	}

}
