TricubicSplineInterpolatingFunction.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.apache.commons.math3.analysis.interpolation;

  19. import org.apache.commons.math3.analysis.TrivariateFunction;
  20. import org.apache.commons.math3.exception.DimensionMismatchException;
  21. import org.apache.commons.math3.exception.NoDataException;
  22. import org.apache.commons.math3.exception.OutOfRangeException;
  23. import org.apache.commons.math3.exception.NonMonotonicSequenceException;
  24. import org.apache.commons.math3.util.MathArrays;

  26. /**
  27.  * Function that implements the
  28.  * <a href="http://en.wikipedia.org/wiki/Tricubic_interpolation">
  29.  * tricubic spline interpolation</a>, as proposed in
  30.  * <quote>
  31.  *  Tricubic interpolation in three dimensions<br/>
  32.  *  F. Lekien and J. Marsden<br/>
  33.  *  <em>Int. J. Numer. Meth. Engng</em> 2005; <b>63</b>:455-471
  34.  * </quote>
  35.  *
  36.  * @since 2.2
  37.  * @deprecated To be removed in 4.0 (see MATH-1166).
  38.  */
  39. @Deprecated
  40. public class TricubicSplineInterpolatingFunction
  41.     implements TrivariateFunction {
  42.     /**
  43.      * Matrix to compute the spline coefficients from the function values
  44.      * and function derivatives values
  45.      */
  46.     private static final double[][] AINV = {
  47.         { 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  48.         { 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  49.         { -3,3,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  50.         { 2,-2,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  51.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  52.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  53.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  54.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  55.         { -3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  56.         { 0,0,0,0,0,0,0,0,-3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  57.         { 9,-9,-9,9,0,0,0,0,6,3,-6,-3,0,0,0,0,6,-6,3,-3,0,0,0,0,0,0,0,0,0,0,0,0,4,2,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  58.         { -6,6,6,-6,0,0,0,0,-3,-3,3,3,0,0,0,0,-4,4,-2,2,0,0,0,0,0,0,0,0,0,0,0,0,-2,-2,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  59.         { 2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  60.         { 0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  61.         { -6,6,6,-6,0,0,0,0,-4,-2,4,2,0,0,0,0,-3,3,-3,3,0,0,0,0,0,0,0,0,0,0,0,0,-2,-1,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  62.         { 4,-4,-4,4,0,0,0,0,2,2,-2,-2,0,0,0,0,2,-2,2,-2,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  63.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  64.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  65.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  66.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  67.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  68.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 },
  69.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,3,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0 },
  70.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,0,0,0,0,1,1,0,0,0,0,0,0 },
  71.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  72.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,0,0,0,0 },
  73.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,-9,-9,9,0,0,0,0,0,0,0,0,0,0,0,0,6,3,-6,-3,0,0,0,0,6,-6,3,-3,0,0,0,0,4,2,2,1,0,0,0,0 },
  74.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6,6,6,-6,0,0,0,0,0,0,0,0,0,0,0,0,-3,-3,3,3,0,0,0,0,-4,4,-2,2,0,0,0,0,-2,-2,-1,-1,0,0,0,0 },
  75.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  76.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0 },
  77.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6,6,6,-6,0,0,0,0,0,0,0,0,0,0,0,0,-4,-2,4,2,0,0,0,0,-3,3,-3,3,0,0,0,0,-2,-1,-2,-1,0,0,0,0 },
  78.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,-4,-4,4,0,0,0,0,0,0,0,0,0,0,0,0,2,2,-2,-2,0,0,0,0,2,-2,2,-2,0,0,0,0,1,1,1,1,0,0,0,0 },
  79.         {-3,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  80.         { 0,0,0,0,0,0,0,0,-3,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  81.         { 9,-9,0,0,-9,9,0,0,6,3,0,0,-6,-3,0,0,0,0,0,0,0,0,0,0,6,-6,0,0,3,-3,0,0,0,0,0,0,0,0,0,0,4,2,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  82.         { -6,6,0,0,6,-6,0,0,-3,-3,0,0,3,3,0,0,0,0,0,0,0,0,0,0,-4,4,0,0,-2,2,0,0,0,0,0,0,0,0,0,0,-2,-2,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  83.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0 },
  84.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,-1,0,0,0 },
  85.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,-9,0,0,-9,9,0,0,0,0,0,0,0,0,0,0,6,3,0,0,-6,-3,0,0,0,0,0,0,0,0,0,0,6,-6,0,0,3,-3,0,0,4,2,0,0,2,1,0,0 },
  86.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6,6,0,0,6,-6,0,0,0,0,0,0,0,0,0,0,-3,-3,0,0,3,3,0,0,0,0,0,0,0,0,0,0,-4,4,0,0,-2,2,0,0,-2,-2,0,0,-1,-1,0,0 },
  87.         { 9,0,-9,0,-9,0,9,0,0,0,0,0,0,0,0,0,6,0,3,0,-6,0,-3,0,6,0,-6,0,3,0,-3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,2,0,2,0,1,0,0,0,0,0,0,0,0,0 },
  88.         { 0,0,0,0,0,0,0,0,9,0,-9,0,-9,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,3,0,-6,0,-3,0,6,0,-6,0,3,0,-3,0,0,0,0,0,0,0,0,0,4,0,2,0,2,0,1,0 },
  89.         { -27,27,27,-27,27,-27,-27,27,-18,-9,18,9,18,9,-18,-9,-18,18,-9,9,18,-18,9,-9,-18,18,18,-18,-9,9,9,-9,-12,-6,-6,-3,12,6,6,3,-12,-6,12,6,-6,-3,6,3,-12,12,-6,6,-6,6,-3,3,-8,-4,-4,-2,-4,-2,-2,-1 },
  90.         { 18,-18,-18,18,-18,18,18,-18,9,9,-9,-9,-9,-9,9,9,12,-12,6,-6,-12,12,-6,6,12,-12,-12,12,6,-6,-6,6,6,6,3,3,-6,-6,-3,-3,6,6,-6,-6,3,3,-3,-3,8,-8,4,-4,4,-4,2,-2,4,4,2,2,2,2,1,1 },
  91.         { -6,0,6,0,6,0,-6,0,0,0,0,0,0,0,0,0,-3,0,-3,0,3,0,3,0,-4,0,4,0,-2,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-2,0,-1,0,-1,0,0,0,0,0,0,0,0,0 },
  92.         { 0,0,0,0,0,0,0,0,-6,0,6,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,-3,0,3,0,3,0,-4,0,4,0,-2,0,2,0,0,0,0,0,0,0,0,0,-2,0,-2,0,-1,0,-1,0 },
  93.         { 18,-18,-18,18,-18,18,18,-18,12,6,-12,-6,-12,-6,12,6,9,-9,9,-9,-9,9,-9,9,12,-12,-12,12,6,-6,-6,6,6,3,6,3,-6,-3,-6,-3,8,4,-8,-4,4,2,-4,-2,6,-6,6,-6,3,-3,3,-3,4,2,4,2,2,1,2,1 },
  94.         { -12,12,12,-12,12,-12,-12,12,-6,-6,6,6,6,6,-6,-6,-6,6,-6,6,6,-6,6,-6,-8,8,8,-8,-4,4,4,-4,-3,-3,-3,-3,3,3,3,3,-4,-4,4,4,-2,-2,2,2,-4,4,-4,4,-2,2,-2,2,-2,-2,-2,-2,-1,-1,-1,-1 },
  95.         { 2,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  96.         { 0,0,0,0,0,0,0,0,2,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  97.         { -6,6,0,0,6,-6,0,0,-4,-2,0,0,4,2,0,0,0,0,0,0,0,0,0,0,-3,3,0,0,-3,3,0,0,0,0,0,0,0,0,0,0,-2,-1,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  98.         { 4,-4,0,0,-4,4,0,0,2,2,0,0,-2,-2,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,2,-2,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
  99.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0 },
  100.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0 },
  101.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6,6,0,0,6,-6,0,0,0,0,0,0,0,0,0,0,-4,-2,0,0,4,2,0,0,0,0,0,0,0,0,0,0,-3,3,0,0,-3,3,0,0,-2,-1,0,0,-2,-1,0,0 },
  102.         { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,-4,0,0,-4,4,0,0,0,0,0,0,0,0,0,0,2,2,0,0,-2,-2,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,2,-2,0,0,1,1,0,0,1,1,0,0 },
  103.         { -6,0,6,0,6,0,-6,0,0,0,0,0,0,0,0,0,-4,0,-2,0,4,0,2,0,-3,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,-2,0,-1,0,0,0,0,0,0,0,0,0 },
  104.         { 0,0,0,0,0,0,0,0,-6,0,6,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,-2,0,4,0,2,0,-3,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,0,-2,0,-1,0,-2,0,-1,0 },
  105.         { 18,-18,-18,18,-18,18,18,-18,12,6,-12,-6,-12,-6,12,6,12,-12,6,-6,-12,12,-6,6,9,-9,-9,9,9,-9,-9,9,8,4,4,2,-8,-4,-4,-2,6,3,-6,-3,6,3,-6,-3,6,-6,3,-3,6,-6,3,-3,4,2,2,1,4,2,2,1 },
  106.         { -12,12,12,-12,12,-12,-12,12,-6,-6,6,6,6,6,-6,-6,-8,8,-4,4,8,-8,4,-4,-6,6,6,-6,-6,6,6,-6,-4,-4,-2,-2,4,4,2,2,-3,-3,3,3,-3,-3,3,3,-4,4,-2,2,-4,4,-2,2,-2,-2,-1,-1,-2,-2,-1,-1 },
  107.         { 4,0,-4,0,-4,0,4,0,0,0,0,0,0,0,0,0,2,0,2,0,-2,0,-2,0,2,0,-2,0,2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0 },
  108.         { 0,0,0,0,0,0,0,0,4,0,-4,0,-4,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,-2,0,-2,0,2,0,-2,0,2,0,-2,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0 },
  109.         { -12,12,12,-12,12,-12,-12,12,-8,-4,8,4,8,4,-8,-4,-6,6,-6,6,6,-6,6,-6,-6,6,6,-6,-6,6,6,-6,-4,-2,-4,-2,4,2,4,2,-4,-2,4,2,-4,-2,4,2,-3,3,-3,3,-3,3,-3,3,-2,-1,-2,-1,-2,-1,-2,-1 },
  110.         { 8,-8,-8,8,-8,8,8,-8,4,4,-4,-4,-4,-4,4,4,4,-4,4,-4,-4,4,-4,4,4,-4,-4,4,4,-4,-4,4,2,2,2,2,-2,-2,-2,-2,2,2,-2,-2,2,2,-2,-2,2,-2,2,-2,2,-2,2,-2,1,1,1,1,1,1,1,1 }
  111.     };

  113.     /** Samples x-coordinates */
  114.     private final double[] xval;
  115.     /** Samples y-coordinates */
  116.     private final double[] yval;
  117.     /** Samples z-coordinates */
  118.     private final double[] zval;
  119.     /** Set of cubic splines pacthing the whole data grid */
  120.     private final TricubicSplineFunction[][][] splines;

  122.     /**
  123.      * @param x Sample values of the x-coordinate, in increasing order.
  124.      * @param y Sample values of the y-coordinate, in increasing order.
  125.      * @param z Sample values of the y-coordinate, in increasing order.
  126.      * @param f Values of the function on every grid point.
  127.      * @param dFdX Values of the partial derivative of function with respect to x on every grid point.
  128.      * @param dFdY Values of the partial derivative of function with respect to y on every grid point.
  129.      * @param dFdZ Values of the partial derivative of function with respect to z on every grid point.
  130.      * @param d2FdXdY Values of the cross partial derivative of function on every grid point.
  131.      * @param d2FdXdZ Values of the cross partial derivative of function on every grid point.
  132.      * @param d2FdYdZ Values of the cross partial derivative of function on every grid point.
  133.      * @param d3FdXdYdZ Values of the cross partial derivative of function on every grid point.
  134.      * @throws NoDataException if any of the arrays has zero length.
  135.      * @throws DimensionMismatchException if the various arrays do not contain the expected number of elements.
  136.      * @throws NonMonotonicSequenceException if {@code x}, {@code y} or {@code z} are not strictly increasing.
  137.      */
  138.     public TricubicSplineInterpolatingFunction(double[] x,
  139.                                                double[] y,
  140.                                                double[] z,
  141.                                                double[][][] f,
  142.                                                double[][][] dFdX,
  143.                                                double[][][] dFdY,
  144.                                                double[][][] dFdZ,
  145.                                                double[][][] d2FdXdY,
  146.                                                double[][][] d2FdXdZ,
  147.                                                double[][][] d2FdYdZ,
  148.                                                double[][][] d3FdXdYdZ)
  149.         throws NoDataException,
  150.                DimensionMismatchException,
  151.                NonMonotonicSequenceException {
  152.         final int xLen = x.length;
  153.         final int yLen = y.length;
  154.         final int zLen = z.length;

  156.         if (xLen == 0 || yLen == 0 || z.length == 0 || f.length == 0 || f[0].length == 0) {
  157.             throw new NoDataException();
  158.         }
  159.         if (xLen != f.length) {
  160.             throw new DimensionMismatchException(xLen, f.length);
  161.         }
  162.         if (xLen != dFdX.length) {
  163.             throw new DimensionMismatchException(xLen, dFdX.length);
  164.         }
  165.         if (xLen != dFdY.length) {
  166.             throw new DimensionMismatchException(xLen, dFdY.length);
  167.         }
  168.         if (xLen != dFdZ.length) {
  169.             throw new DimensionMismatchException(xLen, dFdZ.length);
  170.         }
  171.         if (xLen != d2FdXdY.length) {
  172.             throw new DimensionMismatchException(xLen, d2FdXdY.length);
  173.         }
  174.         if (xLen != d2FdXdZ.length) {
  175.             throw new DimensionMismatchException(xLen, d2FdXdZ.length);
  176.         }
  177.         if (xLen != d2FdYdZ.length) {
  178.             throw new DimensionMismatchException(xLen, d2FdYdZ.length);
  179.         }
  180.         if (xLen != d3FdXdYdZ.length) {
  181.             throw new DimensionMismatchException(xLen, d3FdXdYdZ.length);
  182.         }

  184.         MathArrays.checkOrder(x);
  185.         MathArrays.checkOrder(y);
  186.         MathArrays.checkOrder(z);

  188.         xval = x.clone();
  189.         yval = y.clone();
  190.         zval = z.clone();

  192.         final int lastI = xLen - 1;
  193.         final int lastJ = yLen - 1;
  194.         final int lastK = zLen - 1;
  195.         splines = new TricubicSplineFunction[lastI][lastJ][lastK];

  197.         for (int i = 0; i < lastI; i++) {
  198.             if (f[i].length != yLen) {
  199.                 throw new DimensionMismatchException(f[i].length, yLen);
  200.             }
  201.             if (dFdX[i].length != yLen) {
  202.                 throw new DimensionMismatchException(dFdX[i].length, yLen);
  203.             }
  204.             if (dFdY[i].length != yLen) {
  205.                 throw new DimensionMismatchException(dFdY[i].length, yLen);
  206.             }
  207.             if (dFdZ[i].length != yLen) {
  208.                 throw new DimensionMismatchException(dFdZ[i].length, yLen);
  209.             }
  210.             if (d2FdXdY[i].length != yLen) {
  211.                 throw new DimensionMismatchException(d2FdXdY[i].length, yLen);
  212.             }
  213.             if (d2FdXdZ[i].length != yLen) {
  214.                 throw new DimensionMismatchException(d2FdXdZ[i].length, yLen);
  215.             }
  216.             if (d2FdYdZ[i].length != yLen) {
  217.                 throw new DimensionMismatchException(d2FdYdZ[i].length, yLen);
  218.             }
  219.             if (d3FdXdYdZ[i].length != yLen) {
  220.                 throw new DimensionMismatchException(d3FdXdYdZ[i].length, yLen);
  221.             }

  223.             final int ip1 = i + 1;
  224.             for (int j = 0; j < lastJ; j++) {
  225.                 if (f[i][j].length != zLen) {
  226.                     throw new DimensionMismatchException(f[i][j].length, zLen);
  227.                 }
  228.                 if (dFdX[i][j].length != zLen) {
  229.                     throw new DimensionMismatchException(dFdX[i][j].length, zLen);
  230.                 }
  231.                 if (dFdY[i][j].length != zLen) {
  232.                     throw new DimensionMismatchException(dFdY[i][j].length, zLen);
  233.                 }
  234.                 if (dFdZ[i][j].length != zLen) {
  235.                     throw new DimensionMismatchException(dFdZ[i][j].length, zLen);
  236.                 }
  237.                 if (d2FdXdY[i][j].length != zLen) {
  238.                     throw new DimensionMismatchException(d2FdXdY[i][j].length, zLen);
  239.                 }
  240.                 if (d2FdXdZ[i][j].length != zLen) {
  241.                     throw new DimensionMismatchException(d2FdXdZ[i][j].length, zLen);
  242.                 }
  243.                 if (d2FdYdZ[i][j].length != zLen) {
  244.                     throw new DimensionMismatchException(d2FdYdZ[i][j].length, zLen);
  245.                 }
  246.                 if (d3FdXdYdZ[i][j].length != zLen) {
  247.                     throw new DimensionMismatchException(d3FdXdYdZ[i][j].length, zLen);
  248.                 }

  250.                 final int jp1 = j + 1;
  251.                 for (int k = 0; k < lastK; k++) {
  252.                     final int kp1 = k + 1;

  254.                     final double[] beta = new double[] {
  255.                         f[i][j][k], f[ip1][j][k],
  256.                         f[i][jp1][k], f[ip1][jp1][k],
  257.                         f[i][j][kp1], f[ip1][j][kp1],
  258.                         f[i][jp1][kp1], f[ip1][jp1][kp1],

  260.                         dFdX[i][j][k], dFdX[ip1][j][k],
  261.                         dFdX[i][jp1][k], dFdX[ip1][jp1][k],
  262.                         dFdX[i][j][kp1], dFdX[ip1][j][kp1],
  263.                         dFdX[i][jp1][kp1], dFdX[ip1][jp1][kp1],

  265.                         dFdY[i][j][k], dFdY[ip1][j][k],
  266.                         dFdY[i][jp1][k], dFdY[ip1][jp1][k],
  267.                         dFdY[i][j][kp1], dFdY[ip1][j][kp1],
  268.                         dFdY[i][jp1][kp1], dFdY[ip1][jp1][kp1],

  270.                         dFdZ[i][j][k], dFdZ[ip1][j][k],
  271.                         dFdZ[i][jp1][k], dFdZ[ip1][jp1][k],
  272.                         dFdZ[i][j][kp1], dFdZ[ip1][j][kp1],
  273.                         dFdZ[i][jp1][kp1], dFdZ[ip1][jp1][kp1],

  275.                         d2FdXdY[i][j][k], d2FdXdY[ip1][j][k],
  276.                         d2FdXdY[i][jp1][k], d2FdXdY[ip1][jp1][k],
  277.                         d2FdXdY[i][j][kp1], d2FdXdY[ip1][j][kp1],
  278.                         d2FdXdY[i][jp1][kp1], d2FdXdY[ip1][jp1][kp1],

  280.                         d2FdXdZ[i][j][k], d2FdXdZ[ip1][j][k],
  281.                         d2FdXdZ[i][jp1][k], d2FdXdZ[ip1][jp1][k],
  282.                         d2FdXdZ[i][j][kp1], d2FdXdZ[ip1][j][kp1],
  283.                         d2FdXdZ[i][jp1][kp1], d2FdXdZ[ip1][jp1][kp1],

  285.                         d2FdYdZ[i][j][k], d2FdYdZ[ip1][j][k],
  286.                         d2FdYdZ[i][jp1][k], d2FdYdZ[ip1][jp1][k],
  287.                         d2FdYdZ[i][j][kp1], d2FdYdZ[ip1][j][kp1],
  288.                         d2FdYdZ[i][jp1][kp1], d2FdYdZ[ip1][jp1][kp1],

  290.                         d3FdXdYdZ[i][j][k], d3FdXdYdZ[ip1][j][k],
  291.                         d3FdXdYdZ[i][jp1][k], d3FdXdYdZ[ip1][jp1][k],
  292.                         d3FdXdYdZ[i][j][kp1], d3FdXdYdZ[ip1][j][kp1],
  293.                         d3FdXdYdZ[i][jp1][kp1], d3FdXdYdZ[ip1][jp1][kp1],
  294.                     };

  296.                     splines[i][j][k] = new TricubicSplineFunction(computeSplineCoefficients(beta));
  297.                 }
  298.             }
  299.         }
  300.     }

  302.     /**
  303.      * {@inheritDoc}
  304.      *
  305.      * @throws OutOfRangeException if any of the variables is outside its interpolation range.
  306.      */
  307.     public double value(double x, double y, double z)
  308.         throws OutOfRangeException {
  309.         final int i = searchIndex(x, xval);
  310.         if (i == -1) {
  311.             throw new OutOfRangeException(x, xval[0], xval[xval.length - 1]);
  312.         }
  313.         final int j = searchIndex(y, yval);
  314.         if (j == -1) {
  315.             throw new OutOfRangeException(y, yval[0], yval[yval.length - 1]);
  316.         }
  317.         final int k = searchIndex(z, zval);
  318.         if (k == -1) {
  319.             throw new OutOfRangeException(z, zval[0], zval[zval.length - 1]);
  320.         }

  322.         final double xN = (x - xval[i]) / (xval[i + 1] - xval[i]);
  323.         final double yN = (y - yval[j]) / (yval[j + 1] - yval[j]);
  324.         final double zN = (z - zval[k]) / (zval[k + 1] - zval[k]);

  326.         return splines[i][j][k].value(xN, yN, zN);
  327.     }

  329.     /**
  330.      * @param c Coordinate.
  331.      * @param val Coordinate samples.
  332.      * @return the index in {@code val} corresponding to the interval containing {@code c}, or {@code -1}
  333.      *   if {@code c} is out of the range defined by the end values of {@code val}.
  334.      */
  335.     private int searchIndex(double c, double[] val) {
  336.         if (c < val[0]) {
  337.             return -1;
  338.         }

  340.         final int max = val.length;
  341.         for (int i = 1; i < max; i++) {
  342.             if (c <= val[i]) {
  343.                 return i - 1;
  344.             }
  345.         }

  347.         return -1;
  348.     }

  350.     /**
  351.      * Compute the spline coefficients from the list of function values and
  352.      * function partial derivatives values at the four corners of a grid
  353.      * element. They must be specified in the following order:
  354.      * <ul>
  355.      *  <li>f(0,0,0)</li>
  356.      *  <li>f(1,0,0)</li>
  357.      *  <li>f(0,1,0)</li>
  358.      *  <li>f(1,1,0)</li>
  359.      *  <li>f(0,0,1)</li>
  360.      *  <li>f(1,0,1)</li>
  361.      *  <li>f(0,1,1)</li>
  362.      *  <li>f(1,1,1)</li>
  363.      *
  364.      *  <li>f<sub>x</sub>(0,0,0)</li>
  365.      *  <li>... <em>(same order as above)</em></li>
  366.      *  <li>f<sub>x</sub>(1,1,1)</li>
  367.      *
  368.      *  <li>f<sub>y</sub>(0,0,0)</li>
  369.      *  <li>... <em>(same order as above)</em></li>
  370.      *  <li>f<sub>y</sub>(1,1,1)</li>
  371.      *
  372.      *  <li>f<sub>z</sub>(0,0,0)</li>
  373.      *  <li>... <em>(same order as above)</em></li>
  374.      *  <li>f<sub>z</sub>(1,1,1)</li>
  375.      *
  376.      *  <li>f<sub>xy</sub>(0,0,0)</li>
  377.      *  <li>... <em>(same order as above)</em></li>
  378.      *  <li>f<sub>xy</sub>(1,1,1)</li>
  379.      *
  380.      *  <li>f<sub>xz</sub>(0,0,0)</li>
  381.      *  <li>... <em>(same order as above)</em></li>
  382.      *  <li>f<sub>xz</sub>(1,1,1)</li>
  383.      *
  384.      *  <li>f<sub>yz</sub>(0,0,0)</li>
  385.      *  <li>... <em>(same order as above)</em></li>
  386.      *  <li>f<sub>yz</sub>(1,1,1)</li>
  387.      *
  388.      *  <li>f<sub>xyz</sub>(0,0,0)</li>
  389.      *  <li>... <em>(same order as above)</em></li>
  390.      *  <li>f<sub>xyz</sub>(1,1,1)</li>
  391.      * </ul>
  392.      * where the subscripts indicate the partial derivative with respect to
  393.      * the corresponding variable(s).
  394.      *
  395.      * @param beta List of function values and function partial derivatives values.
  396.      * @return the spline coefficients.
  397.      */
  398.     private double[] computeSplineCoefficients(double[] beta) {
  399.         final int sz = 64;
  400.         final double[] a = new double[sz];

  402.         for (int i = 0; i < sz; i++) {
  403.             double result = 0;
  404.             final double[] row = AINV[i];
  405.             for (int j = 0; j < sz; j++) {
  406.                 result += row[j] * beta[j];
  407.             }
  408.             a[i] = result;
  409.         }

  411.         return a;
  412.     }
  413. }

  415. /**
  416.  * 3D-spline function.
  417.  *
  418.  */
  419. class TricubicSplineFunction
  420.     implements TrivariateFunction {
  421.     /** Number of points. */
  422.     private static final short N = 4;
  423.     /** Coefficients */
  424.     private final double[][][] a = new double[N][N][N];

  426.     /**
  427.      * @param aV List of spline coefficients.
  428.      */
  429.     public TricubicSplineFunction(double[] aV) {
  430.         for (int i = 0; i < N; i++) {
  431.             for (int j = 0; j < N; j++) {
  432.                 for (int k = 0; k < N; k++) {
  433.                     a[i][j][k] = aV[i + N * (j + N * k)];
  434.                 }
  435.             }
  436.         }
  437.     }

  439.     /**
  440.      * @param x x-coordinate of the interpolation point.
  441.      * @param y y-coordinate of the interpolation point.
  442.      * @param z z-coordinate of the interpolation point.
  443.      * @return the interpolated value.
  444.      * @throws OutOfRangeException if {@code x}, {@code y} or
  445.      * {@code z} are not in the interval {@code [0, 1]}.
  446.      */
  447.     public double value(double x, double y, double z)
  448.         throws OutOfRangeException {
  449.         if (x < 0 || x > 1) {
  450.             throw new OutOfRangeException(x, 0, 1);
  451.         }
  452.         if (y < 0 || y > 1) {
  453.             throw new OutOfRangeException(y, 0, 1);
  454.         }
  455.         if (z < 0 || z > 1) {
  456.             throw new OutOfRangeException(z, 0, 1);
  457.         }

  459.         final double x2 = x * x;
  460.         final double x3 = x2 * x;
  461.         final double[] pX = { 1, x, x2, x3 };

  463.         final double y2 = y * y;
  464.         final double y3 = y2 * y;
  465.         final double[] pY = { 1, y, y2, y3 };

  467.         final double z2 = z * z;
  468.         final double z3 = z2 * z;
  469.         final double[] pZ = { 1, z, z2, z3 };

  471.         double result = 0;
  472.         for (int i = 0; i < N; i++) {
  473.             for (int j = 0; j < N; j++) {
  474.                 for (int k = 0; k < N; k++) {
  475.                     result += a[i][j][k] * pX[i] * pY[j] * pZ[k];
  476.                 }
  477.             }
  478.         }

  480.         return result;
  481.     }
  482. }
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