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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.math4.legacy.analysis.integration.gauss;
  
  import org.apache.commons.math4.legacy.analysis.UnivariateFunction;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  import org.junit.Test;
  import org.junit.Assert;
  
  /**
   * Test of the {@link HermiteRuleFactory}.
   *
   */
  public class HermiteTest {
      private static final GaussIntegratorFactory factory = new GaussIntegratorFactory();
  
      @Test
      public void testNormalDistribution() {
          final double oneOverSqrtPi = 1 / JdkMath.sqrt(Math.PI);
  
          // By definition, Gauss-Hermite quadrature readily provides the
          // integral of the normal distribution density.
          final int numPoints = 1;
  
          // Change of variable:
          //   y = (x - mu) / (sqrt(2) *  sigma)
          // such that the integrand
          //   N(x, mu, sigma)
          // is transformed to
          //   f(y) * exp(-y^2)
          final UnivariateFunction f = new UnivariateFunction() {
                  @Override
                  public double value(double y) {
                      return oneOverSqrtPi; // Constant function.
                  }
              };
  
          final GaussIntegrator integrator = factory.hermite(numPoints);
          final double result = integrator.integrate(f);
          final double expected = 1;
          Assert.assertEquals(expected, result, Math.ulp(expected));
      }
  
      @Test
      public void testNormalMean() {
          final double sqrtTwo = JdkMath.sqrt(2);
          final double oneOverSqrtPi = 1 / JdkMath.sqrt(Math.PI);
  
          final double mu = 12345.6789;
          final double sigma = 987.654321;
          final int numPoints = 5;
  
          // Change of variable:
          //   y = (x - mu) / (sqrt(2) *  sigma)
          // such that the integrand
          //   x * N(x, mu, sigma)
          // is transformed to
          //   f(y) * exp(-y^2)
          final UnivariateFunction f = new UnivariateFunction() {
                  @Override
                  public double value(double y) {
                      return oneOverSqrtPi * (sqrtTwo * sigma * y + mu);
                  }
              };
  
          final GaussIntegrator integrator = factory.hermite(numPoints);
          final double result = integrator.integrate(f);
          final double expected = mu;
          Assert.assertEquals(expected, result, Math.ulp(expected));
      }
  
      @Test
      public void testNormalVariance() {
          final double twoOverSqrtPi = 2 / JdkMath.sqrt(Math.PI);
  
          final double sigma = 987.654321;
          final double sigma2 = sigma * sigma;
          final int numPoints = 5;
  
          // Change of variable:
          //   y = (x - mu) / (sqrt(2) *  sigma)
          // such that the integrand
          //   (x - mu)^2 * N(x, mu, sigma)
          // is transformed to
          //   f(y) * exp(-y^2)
         final UnivariateFunction f = new UnivariateFunction() {
                 @Override
                 public double value(double y) {
                     return twoOverSqrtPi * sigma2 * y * y;
                 }
             };
 
         final GaussIntegrator integrator = factory.hermite(numPoints);
         final double result = integrator.integrate(f);
         final double expected = sigma2;
         Assert.assertEquals(expected, result, 10 * Math.ulp(expected));
     }
 }