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    * Licensed to the Apache Software Foundation (ASF) under one or more
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    * 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.interpolation;
  
  import org.apache.commons.math4.legacy.analysis.UnivariateFunction;
  import org.apache.commons.math4.legacy.analysis.function.Expm1;
  import org.apache.commons.math4.legacy.analysis.function.Sin;
  import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  import org.junit.Assert;
  import org.junit.Test;
  
  
  /**
   * Test case for Divided Difference interpolator.
   * <p>
   * The error of polynomial interpolation is
   *     f(z) - p(z) = f^(n)(zeta) * (z-x[0])(z-x[1])...(z-x[n-1]) / n!
   * where f^(n) is the n-th derivative of the approximated function and
   * zeta is some point in the interval determined by x[] and z.
   * <p>
   * Since zeta is unknown, f^(n)(zeta) cannot be calculated. But we can bound
   * it and use the absolute value upper bound for estimates. For reference,
   * see <b>Introduction to Numerical Analysis</b>, ISBN 038795452X, chapter 2.
   *
   */
  public final class DividedDifferenceInterpolatorTest {
  
      /**
       * Test of interpolator for the sine function.
       * <p>
       * |sin^(n)(zeta)| <= 1.0, zeta in [0, 2*PI]
       */
      @Test
      public void testSinFunction() {
          UnivariateFunction f = new Sin();
          UnivariateInterpolator interpolator = new DividedDifferenceInterpolator();
          double[] x;
          double[] y;
          double z;
          double expected;
          double result;
          double tolerance;
  
          // 6 interpolating points on interval [0, 2*PI]
          int n = 6;
          double min = 0.0;
          double max = 2 * JdkMath.PI;
          x = new double[n];
          y = new double[n];
          for (int i = 0; i < n; i++) {
              x[i] = min + i * (max - min) / n;
              y[i] = f.value(x[i]);
          }
          double derivativebound = 1.0;
          UnivariateFunction p = interpolator.interpolate(x, y);
  
          z = JdkMath.PI / 4; expected = f.value(z); result = p.value(z);
          tolerance = JdkMath.abs(derivativebound * partialerror(x, z));
          Assert.assertEquals(expected, result, tolerance);
  
          z = JdkMath.PI * 1.5; expected = f.value(z); result = p.value(z);
          tolerance = JdkMath.abs(derivativebound * partialerror(x, z));
          Assert.assertEquals(expected, result, tolerance);
      }
  
      /**
       * Test of interpolator for the exponential function.
       * <p>
       * |expm1^(n)(zeta)| <= e, zeta in [-1, 1]
       */
      @Test
      public void testExpm1Function() {
          UnivariateFunction f = new Expm1();
          UnivariateInterpolator interpolator = new DividedDifferenceInterpolator();
          double[] x;
          double[] y;
          double z;
          double expected;
          double result;
          double tolerance;
  
          // 5 interpolating points on interval [-1, 1]
          int n = 5;
          double min = -1.0;
         double max = 1.0;
         x = new double[n];
         y = new double[n];
         for (int i = 0; i < n; i++) {
             x[i] = min + i * (max - min) / n;
             y[i] = f.value(x[i]);
         }
         double derivativebound = JdkMath.E;
         UnivariateFunction p = interpolator.interpolate(x, y);
 
         z = 0.0; expected = f.value(z); result = p.value(z);
         tolerance = JdkMath.abs(derivativebound * partialerror(x, z));
         Assert.assertEquals(expected, result, tolerance);
 
         z = 0.5; expected = f.value(z); result = p.value(z);
         tolerance = JdkMath.abs(derivativebound * partialerror(x, z));
         Assert.assertEquals(expected, result, tolerance);
 
         z = -0.5; expected = f.value(z); result = p.value(z);
         tolerance = JdkMath.abs(derivativebound * partialerror(x, z));
         Assert.assertEquals(expected, result, tolerance);
     }
 
     /**
      * Test of parameters for the interpolator.
      */
     @Test
     public void testParameters() {
         UnivariateInterpolator interpolator = new DividedDifferenceInterpolator();
 
         try {
             // bad abscissas array
             double x[] = { 1.0, 2.0, 2.0, 4.0 };
             double y[] = { 0.0, 4.0, 4.0, 2.5 };
             UnivariateFunction p = interpolator.interpolate(x, y);
             p.value(0.0);
             Assert.fail("Expecting NonMonotonicSequenceException - bad abscissas array");
         } catch (NonMonotonicSequenceException ex) {
             // expected
         }
     }
 
     /**
      * Returns the partial error term (z-x[0])(z-x[1])...(z-x[n-1])/n!
      */
     protected double partialerror(double x[], double z) throws
         IllegalArgumentException {
 
         if (x.length < 1) {
             throw new IllegalArgumentException
                 ("Interpolation array cannot be empty.");
         }
         double out = 1;
         for (int i = 0; i < x.length; i++) {
             out *= (z - x[i]) / (i + 1);
         }
         return out;
     }
 }