/*
    * 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.math4.legacy.analysis.integration.gauss;
  
  import java.math.BigDecimal;
  import java.math.MathContext;
  
  import org.apache.commons.math4.legacy.core.Pair;
  
  /**
   * Factory that creates Gauss-type quadrature rule using Legendre polynomials.
   * In this implementation, the lower and upper bounds of the natural interval
   * of integration are -1 and 1, respectively.
   * The Legendre polynomials are evaluated using the recurrence relation
   * presented in <a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun">
   * Abramowitz and Stegun, 1964</a>.
   *
   * @since 3.1
   */
  public class LegendreHighPrecisionRuleFactory extends BaseRuleFactory<BigDecimal> {
      /** Settings for enhanced precision computations. */
      private final MathContext mContext;
      /** The number {@code 2}. */
      private final BigDecimal two;
      /** The number {@code -1}. */
      private final BigDecimal minusOne;
      /** The number {@code 0.5}. */
      private final BigDecimal oneHalf;
  
      /**
       * Default precision is {@link MathContext#DECIMAL128 DECIMAL128}.
       */
      public LegendreHighPrecisionRuleFactory() {
          this(MathContext.DECIMAL128);
      }
  
      /**
       * @param mContext Precision setting for computing the quadrature rules.
       */
      public LegendreHighPrecisionRuleFactory(MathContext mContext) {
          this.mContext = mContext;
          two = new BigDecimal("2", mContext);
          minusOne = new BigDecimal("-1", mContext);
          oneHalf = new BigDecimal("0.5", mContext);
      }
  
      /** {@inheritDoc} */
      @Override
      protected Pair<BigDecimal[], BigDecimal[]> computeRule(int numberOfPoints) {
          if (numberOfPoints == 1) {
              // Break recursion.
              return new Pair<>(new BigDecimal[] { BigDecimal.ZERO },
                                                          new BigDecimal[] { two });
          }
  
          // Get previous rule.
          // If it has not been computed yet it will trigger a recursive call
          // to this method.
          final BigDecimal[] previousPoints = getRuleInternal(numberOfPoints - 1).getFirst();
  
          // Compute next rule.
          final BigDecimal[] points = new BigDecimal[numberOfPoints];
          final BigDecimal[] weights = new BigDecimal[numberOfPoints];
  
          // Find i-th root of P[n+1] by bracketing.
          final int iMax = numberOfPoints / 2;
          for (int i = 0; i < iMax; i++) {
              // Lower-bound of the interval.
              BigDecimal a = (i == 0) ? minusOne : previousPoints[i - 1];
              // Upper-bound of the interval.
              BigDecimal b = (iMax == 1) ? BigDecimal.ONE : previousPoints[i];
              // P[j-1](a)
              BigDecimal pma = BigDecimal.ONE;
              // P[j](a)
              BigDecimal pa = a;
              // P[j-1](b)
              BigDecimal pmb = BigDecimal.ONE;
              // P[j](b)
              BigDecimal pb = b;
              for (int j = 1; j < numberOfPoints; j++) {
                  final BigDecimal b_two_j_p_1 = new BigDecimal(2 * j + 1, mContext);
                  final BigDecimal b_j = new BigDecimal(j, mContext);
                  final BigDecimal b_j_p_1 = new BigDecimal(j + 1, mContext);
  
                  // Compute P[j+1](a)
                 // ppa = ((2 * j + 1) * a * pa - j * pma) / (j + 1);
 
                 BigDecimal tmp1 = a.multiply(b_two_j_p_1, mContext);
                 tmp1 = pa.multiply(tmp1, mContext);
                 BigDecimal tmp2 = pma.multiply(b_j, mContext);
                 // P[j+1](a)
                 BigDecimal ppa = tmp1.subtract(tmp2, mContext);
                 ppa = ppa.divide(b_j_p_1, mContext);
 
                 // Compute P[j+1](b)
                 // ppb = ((2 * j + 1) * b * pb - j * pmb) / (j + 1);
 
                 tmp1 = b.multiply(b_two_j_p_1, mContext);
                 tmp1 = pb.multiply(tmp1, mContext);
                 tmp2 = pmb.multiply(b_j, mContext);
                 // P[j+1](b)
                 BigDecimal ppb = tmp1.subtract(tmp2, mContext);
                 ppb = ppb.divide(b_j_p_1, mContext);
 
                 pma = pa;
                 pa = ppa;
                 pmb = pb;
                 pb = ppb;
             }
             // Now pa = P[n+1](a), and pma = P[n](a). Same holds for b.
             // Middle of the interval.
             BigDecimal c = a.add(b, mContext).multiply(oneHalf, mContext);
             // P[j-1](c)
             BigDecimal pmc = BigDecimal.ONE;
             // P[j](c)
             BigDecimal pc = c;
             boolean done = false;
             while (!done) {
                 BigDecimal tmp1 = b.subtract(a, mContext);
                 BigDecimal tmp2 = c.ulp().multiply(BigDecimal.TEN, mContext);
                 done = tmp1.compareTo(tmp2) <= 0;
                 pmc = BigDecimal.ONE;
                 pc = c;
                 for (int j = 1; j < numberOfPoints; j++) {
                     final BigDecimal b_two_j_p_1 = new BigDecimal(2 * j + 1, mContext);
                     final BigDecimal b_j = new BigDecimal(j, mContext);
                     final BigDecimal b_j_p_1 = new BigDecimal(j + 1, mContext);
 
                     // Compute P[j+1](c)
                     tmp1 = c.multiply(b_two_j_p_1, mContext);
                     tmp1 = pc.multiply(tmp1, mContext);
                     tmp2 = pmc.multiply(b_j, mContext);
                     // P[j+1](c)
                     BigDecimal ppc = tmp1.subtract(tmp2, mContext);
                     ppc = ppc.divide(b_j_p_1, mContext);
 
                     pmc = pc;
                     pc = ppc;
                 }
                 // Now pc = P[n+1](c) and pmc = P[n](c).
                 if (!done) {
                     if (pa.signum() * pc.signum() <= 0) {
                         b = c;
                         pmb = pmc;
                         pb = pc;
                     } else {
                         a = c;
                         pma = pmc;
                         pa = pc;
                     }
                     c = a.add(b, mContext).multiply(oneHalf, mContext);
                 }
             }
             final BigDecimal nP = new BigDecimal(numberOfPoints, mContext);
             BigDecimal tmp1 = pmc.subtract(c.multiply(pc, mContext), mContext);
             tmp1 = tmp1.multiply(nP);
             tmp1 = tmp1.pow(2, mContext);
             BigDecimal tmp2 = c.pow(2, mContext);
             tmp2 = BigDecimal.ONE.subtract(tmp2, mContext);
             tmp2 = tmp2.multiply(two, mContext);
             tmp2 = tmp2.divide(tmp1, mContext);
 
             points[i] = c;
             weights[i] = tmp2;
 
             final int idx = numberOfPoints - i - 1;
             points[idx] = c.negate(mContext);
             weights[idx] = tmp2;
         }
         // If "numberOfPoints" is odd, 0 is a root.
         // Note: as written, the test for oddness will work for negative
         // integers too (although it is not necessary here), preventing
         // a FindBugs warning.
         if ((numberOfPoints & 1) != 0) {
             BigDecimal pmc = BigDecimal.ONE;
             for (int j = 1; j < numberOfPoints; j += 2) {
                 final BigDecimal b_j = new BigDecimal(j, mContext);
                 final BigDecimal b_j_p_1 = new BigDecimal(j + 1, mContext);
 
                 // pmc = -j * pmc / (j + 1);
                 pmc = pmc.multiply(b_j, mContext);
                 pmc = pmc.divide(b_j_p_1, mContext);
                 pmc = pmc.negate(mContext);
             }
 
             // 2 / pow(numberOfPoints * pmc, 2);
             final BigDecimal nP = new BigDecimal(numberOfPoints, mContext);
             BigDecimal tmp1 = pmc.multiply(nP, mContext);
             tmp1 = tmp1.pow(2, mContext);
             BigDecimal tmp2 = two.divide(tmp1, mContext);
 
             points[iMax] = BigDecimal.ZERO;
             weights[iMax] = tmp2;
         }
 
         return new Pair<>(points, weights);
     }
 }