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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
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 * 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
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package org.apache.commons.math3.analysis.integration;

import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;

Implements the 
Legendre-Gauss quadrature formula.

Legendre-Gauss integrators are efficient integrators that can
accurately integrate functions with few function evaluations. A
Legendre-Gauss integrator using an n-points quadrature formula can
integrate 2n-1 degree polynomials exactly.


These integrators evaluate the function on n carefully chosen
abscissas in each step interval (mapped to the canonical [-1,1] interval).
The evaluation abscissas are not evenly spaced and none of them are
at the interval endpoints. This implies the function integrated can be
undefined at integration interval endpoints.


The evaluation abscissas xi are the roots of the degree n
Legendre polynomial. The weights ai of the quadrature formula
integrals from -1 to +1 ∫ Li2 where Li (x) =
∏ (x-xk)/(xi-xk) for k != i.


Since:
1.2
Deprecated:
As of 3.1 (to be removed in 4.0). Please use IterativeLegendreGaussIntegrator instead./**
 * Implements the <a href="http://mathworld.wolfram.com/Legendre-GaussQuadrature.html">
 * Legendre-Gauss</a> quadrature formula.
 * <p>
 * Legendre-Gauss integrators are efficient integrators that can
 * accurately integrate functions with few function evaluations. A
 * Legendre-Gauss integrator using an n-points quadrature formula can
 * integrate 2n-1 degree polynomials exactly.
 * </p>
 * <p>
 * These integrators evaluate the function on n carefully chosen
 * abscissas in each step interval (mapped to the canonical [-1,1] interval).
 * The evaluation abscissas are not evenly spaced and none of them are
 * at the interval endpoints. This implies the function integrated can be
 * undefined at integration interval endpoints.
 * </p>
 * <p>
 * The evaluation abscissas x<sub>i</sub> are the roots of the degree n
 * Legendre polynomial. The weights a<sub>i</sub> of the quadrature formula
 * integrals from -1 to +1 &int; Li<sup>2</sup> where Li (x) =
 * &prod; (x-x<sub>k</sub>)/(x<sub>i</sub>-x<sub>k</sub>) for k != i.
 * </p>
 * <p>
 * @since 1.2
 * @deprecated As of 3.1 (to be removed in 4.0). Please use
 * {@link IterativeLegendreGaussIntegrator} instead.
 */
@Deprecated
public class LegendreGaussIntegrator extends BaseAbstractUnivariateIntegrator {

    
Abscissas for the 2 points method. /** Abscissas for the 2 points method. */
    private static final double[] ABSCISSAS_2 = {
        -1.0 / FastMath.sqrt(3.0),
         1.0 / FastMath.sqrt(3.0)
    };

    
Weights for the 2 points method. /** Weights for the 2 points method. */
    private static final double[] WEIGHTS_2 = {
        1.0,
        1.0
    };

    
Abscissas for the 3 points method. /** Abscissas for the 3 points method. */
    private static final double[] ABSCISSAS_3 = {
        -FastMath.sqrt(0.6),
         0.0,
         FastMath.sqrt(0.6)
    };

    
Weights for the 3 points method. /** Weights for the 3 points method. */
    private static final double[] WEIGHTS_3 = {
        5.0 / 9.0,
        8.0 / 9.0,
        5.0 / 9.0
    };

    
Abscissas for the 4 points method. /** Abscissas for the 4 points method. */
    private static final double[] ABSCISSAS_4 = {
        -FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0),
        -FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0),
         FastMath.sqrt((15.0 - 2.0 * FastMath.sqrt(30.0)) / 35.0),
         FastMath.sqrt((15.0 + 2.0 * FastMath.sqrt(30.0)) / 35.0)
    };

    
Weights for the 4 points method. /** Weights for the 4 points method. */
    private static final double[] WEIGHTS_4 = {
        (90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0,
        (90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0,
        (90.0 + 5.0 * FastMath.sqrt(30.0)) / 180.0,
        (90.0 - 5.0 * FastMath.sqrt(30.0)) / 180.0
    };

    
Abscissas for the 5 points method. /** Abscissas for the 5 points method. */
    private static final double[] ABSCISSAS_5 = {
        -FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0),
        -FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0),
         0.0,
         FastMath.sqrt((35.0 - 2.0 * FastMath.sqrt(70.0)) / 63.0),
         FastMath.sqrt((35.0 + 2.0 * FastMath.sqrt(70.0)) / 63.0)
    };

    
Weights for the 5 points method. /** Weights for the 5 points method. */
    private static final double[] WEIGHTS_5 = {
        (322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0,
        (322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0,
        128.0 / 225.0,
        (322.0 + 13.0 * FastMath.sqrt(70.0)) / 900.0,
        (322.0 - 13.0 * FastMath.sqrt(70.0)) / 900.0
    };

    
Abscissas for the current method. /** Abscissas for the current method. */
    private final double[] abscissas;

    
Weights for the current method. /** Weights for the current method. */
    private final double[] weights;

    
Build a Legendre-Gauss integrator with given accuracies and iterations counts.
Params:
n – number of points desired (must be between 2 and 5 inclusive)
relativeAccuracy – relative accuracy of the result
absoluteAccuracy – absolute accuracy of the result
minimalIterationCount – minimum number of iterations
maximalIterationCount – maximum number of iterations
Throws:
MathIllegalArgumentException – if number of points is out of [2; 5]
NotStrictlyPositiveException – if minimal number of iterations
is not strictly positive
NumberIsTooSmallException – if maximal number of iterations
is lesser than or equal to the minimal number of iterations/**
     * Build a Legendre-Gauss integrator with given accuracies and iterations counts.
     * @param n number of points desired (must be between 2 and 5 inclusive)
     * @param relativeAccuracy relative accuracy of the result
     * @param absoluteAccuracy absolute accuracy of the result
     * @param minimalIterationCount minimum number of iterations
     * @param maximalIterationCount maximum number of iterations
     * @exception MathIllegalArgumentException if number of points is out of [2; 5]
     * @exception NotStrictlyPositiveException if minimal number of iterations
     * is not strictly positive
     * @exception NumberIsTooSmallException if maximal number of iterations
     * is lesser than or equal to the minimal number of iterations
     */
    public LegendreGaussIntegrator(final int n,
                                   final double relativeAccuracy,
                                   final double absoluteAccuracy,
                                   final int minimalIterationCount,
                                   final int maximalIterationCount)
        throws MathIllegalArgumentException, NotStrictlyPositiveException, NumberIsTooSmallException {
        super(relativeAccuracy, absoluteAccuracy, minimalIterationCount, maximalIterationCount);
        switch(n) {
        case 2 :
            abscissas = ABSCISSAS_2;
            weights   = WEIGHTS_2;
            break;
        case 3 :
            abscissas = ABSCISSAS_3;
            weights   = WEIGHTS_3;
            break;
        case 4 :
            abscissas = ABSCISSAS_4;
            weights   = WEIGHTS_4;
            break;
        case 5 :
            abscissas = ABSCISSAS_5;
            weights   = WEIGHTS_5;
            break;
        default :
            throw new MathIllegalArgumentException(
                    LocalizedFormats.N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED,
                    n, 2, 5);
        }

    }

    
Build a Legendre-Gauss integrator with given accuracies.
Params:
n – number of points desired (must be between 2 and 5 inclusive)
relativeAccuracy – relative accuracy of the result
absoluteAccuracy – absolute accuracy of the result
Throws:
MathIllegalArgumentException – if number of points is out of [2; 5]/**
     * Build a Legendre-Gauss integrator with given accuracies.
     * @param n number of points desired (must be between 2 and 5 inclusive)
     * @param relativeAccuracy relative accuracy of the result
     * @param absoluteAccuracy absolute accuracy of the result
     * @exception MathIllegalArgumentException if number of points is out of [2; 5]
     */
    public LegendreGaussIntegrator(final int n,
                                   final double relativeAccuracy,
                                   final double absoluteAccuracy)
        throws MathIllegalArgumentException {
        this(n, relativeAccuracy, absoluteAccuracy,
             DEFAULT_MIN_ITERATIONS_COUNT, DEFAULT_MAX_ITERATIONS_COUNT);
    }

    
Build a Legendre-Gauss integrator with given iteration counts.
Params:
n – number of points desired (must be between 2 and 5 inclusive)
minimalIterationCount – minimum number of iterations
maximalIterationCount – maximum number of iterations
Throws:
MathIllegalArgumentException – if number of points is out of [2; 5]
NotStrictlyPositiveException – if minimal number of iterations
is not strictly positive
NumberIsTooSmallException – if maximal number of iterations
is lesser than or equal to the minimal number of iterations/**
     * Build a Legendre-Gauss integrator with given iteration counts.
     * @param n number of points desired (must be between 2 and 5 inclusive)
     * @param minimalIterationCount minimum number of iterations
     * @param maximalIterationCount maximum number of iterations
     * @exception MathIllegalArgumentException if number of points is out of [2; 5]
     * @exception NotStrictlyPositiveException if minimal number of iterations
     * is not strictly positive
     * @exception NumberIsTooSmallException if maximal number of iterations
     * is lesser than or equal to the minimal number of iterations
     */
    public LegendreGaussIntegrator(final int n,
                                   final int minimalIterationCount,
                                   final int maximalIterationCount)
        throws MathIllegalArgumentException {
        this(n, DEFAULT_RELATIVE_ACCURACY, DEFAULT_ABSOLUTE_ACCURACY,
             minimalIterationCount, maximalIterationCount);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    protected double doIntegrate()
        throws MathIllegalArgumentException, TooManyEvaluationsException, MaxCountExceededException {

        // compute first estimate with a single step
        double oldt = stage(1);

        int n = 2;
        while (true) {

            // improve integral with a larger number of steps
            final double t = stage(n);

            // estimate error
            final double delta = FastMath.abs(t - oldt);
            final double limit =
                FastMath.max(getAbsoluteAccuracy(),
                             getRelativeAccuracy() * (FastMath.abs(oldt) + FastMath.abs(t)) * 0.5);

            // check convergence
            if ((getIterations() + 1 >= getMinimalIterationCount()) && (delta <= limit)) {
                return t;
            }

            // prepare next iteration
            double ratio = FastMath.min(4, FastMath.pow(delta / limit, 0.5 / abscissas.length));
            n = FastMath.max((int) (ratio * n), n + 1);
            oldt = t;
            incrementCount();

        }

    }

    
Compute the n-th stage integral.
Params:
n – number of steps
Throws:
TooManyEvaluationsException – if the maximum number of evaluations
is exceeded.
Returns:
the value of n-th stage integral/**
     * Compute the n-th stage integral.
     * @param n number of steps
     * @return the value of n-th stage integral
     * @throws TooManyEvaluationsException if the maximum number of evaluations
     * is exceeded.
     */
    private double stage(final int n)
        throws TooManyEvaluationsException {

        // set up the step for the current stage
        final double step     = (getMax() - getMin()) / n;
        final double halfStep = step / 2.0;

        // integrate over all elementary steps
        double midPoint = getMin() + halfStep;
        double sum = 0.0;
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < abscissas.length; ++j) {
                sum += weights[j] * computeObjectiveValue(midPoint + halfStep * abscissas[j]);
            }
            midPoint += step;
        }

        return halfStep * sum;

    }

}