001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.math4.analysis.interpolation;
018
019import org.apache.commons.math4.analysis.TrivariateFunction;
020import org.apache.commons.math4.exception.DimensionMismatchException;
021import org.apache.commons.math4.exception.NoDataException;
022import org.apache.commons.math4.exception.NonMonotonicSequenceException;
023import org.apache.commons.math4.exception.OutOfRangeException;
024import org.apache.commons.math4.util.MathArrays;
025
026/**
027 * Function that implements the
028 * <a href="http://en.wikipedia.org/wiki/Tricubic_interpolation">
029 * tricubic spline interpolation</a>, as proposed in
030 * <blockquote>
031 *  Tricubic interpolation in three dimensions<br>
032 *  F. Lekien and J. Marsden<br>
033 *  <em>Int. J. Numer. Meth. Eng</em> 2005; <b>63</b>:455-471<br>
034 * </blockquote>
035 *
036 * @since 3.4.
037 */
038public class TricubicInterpolatingFunction
039    implements TrivariateFunction {
040    /**
041     * Matrix to compute the spline coefficients from the function values
042     * and function derivatives values
043     */
044    private static final double[][] AINV = {
045        { 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 },
046        { 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 },
047        { -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 },
048        { 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 },
049        { 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 },
050        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
051        { 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 },
052        { 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 },
053        { -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 },
054        { 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 },
055        { 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 },
056        { -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 },
057        { 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 },
058        { 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 },
059        { -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 },
060        { 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 },
061        { 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 },
062        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
063        { 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 },
064        { 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 },
065        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
066        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 },
067        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,3,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0 },
068        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,0,0,0,0,1,1,0,0,0,0,0,0 },
069        { 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 },
070        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,0,0,0,0 },
071        { 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 },
072        { 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 },
073        { 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 },
074        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0 },
075        { 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 },
076        { 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 },
077        {-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 },
078        { 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 },
079        { 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 },
080        { -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 },
081        { 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 },
082        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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 },
083        { 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 },
084        { 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 },
085        { 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 },
086        { 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 },
087        { -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 },
088        { 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 },
089        { -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 },
090        { 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 },
091        { 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 },
092        { -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 },
093        { 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 },
094        { 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 },
095        { -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 },
096        { 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 },
097        { 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 },
098        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-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 },
099        { 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 },
100        { 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 },
101        { -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 },
102        { 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 },
103        { 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 },
104        { -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 },
105        { 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 },
106        { 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 },
107        { -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 },
108        { 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 }
109    };
110
111    /** Samples x-coordinates */
112    private final double[] xval;
113    /** Samples y-coordinates */
114    private final double[] yval;
115    /** Samples z-coordinates */
116    private final double[] zval;
117    /** Set of cubic splines patching the whole data grid */
118    private final TricubicFunction[][][] splines;
119
120    /**
121     * @param x Sample values of the x-coordinate, in increasing order.
122     * @param y Sample values of the y-coordinate, in increasing order.
123     * @param z Sample values of the y-coordinate, in increasing order.
124     * @param f Values of the function on every grid point.
125     * @param dFdX Values of the partial derivative of function with respect to x on every grid point.
126     * @param dFdY Values of the partial derivative of function with respect to y on every grid point.
127     * @param dFdZ Values of the partial derivative of function with respect to z on every grid point.
128     * @param d2FdXdY Values of the cross partial derivative of function on every grid point.
129     * @param d2FdXdZ Values of the cross partial derivative of function on every grid point.
130     * @param d2FdYdZ Values of the cross partial derivative of function on every grid point.
131     * @param d3FdXdYdZ Values of the cross partial derivative of function on every grid point.
132     * @throws NoDataException if any of the arrays has zero length.
133     * @throws DimensionMismatchException if the various arrays do not contain the expected number of elements.
134     * @throws NonMonotonicSequenceException if {@code x}, {@code y} or {@code z} are not strictly increasing.
135     */
136    public TricubicInterpolatingFunction(double[] x,
137                                         double[] y,
138                                         double[] z,
139                                         double[][][] f,
140                                         double[][][] dFdX,
141                                         double[][][] dFdY,
142                                         double[][][] dFdZ,
143                                         double[][][] d2FdXdY,
144                                         double[][][] d2FdXdZ,
145                                         double[][][] d2FdYdZ,
146                                         double[][][] d3FdXdYdZ)
147        throws NoDataException,
148               DimensionMismatchException,
149               NonMonotonicSequenceException {
150        final int xLen = x.length;
151        final int yLen = y.length;
152        final int zLen = z.length;
153
154        if (xLen == 0 || yLen == 0 || z.length == 0 || f.length == 0 || f[0].length == 0) {
155            throw new NoDataException();
156        }
157        if (xLen != f.length) {
158            throw new DimensionMismatchException(xLen, f.length);
159        }
160        if (xLen != dFdX.length) {
161            throw new DimensionMismatchException(xLen, dFdX.length);
162        }
163        if (xLen != dFdY.length) {
164            throw new DimensionMismatchException(xLen, dFdY.length);
165        }
166        if (xLen != dFdZ.length) {
167            throw new DimensionMismatchException(xLen, dFdZ.length);
168        }
169        if (xLen != d2FdXdY.length) {
170            throw new DimensionMismatchException(xLen, d2FdXdY.length);
171        }
172        if (xLen != d2FdXdZ.length) {
173            throw new DimensionMismatchException(xLen, d2FdXdZ.length);
174        }
175        if (xLen != d2FdYdZ.length) {
176            throw new DimensionMismatchException(xLen, d2FdYdZ.length);
177        }
178        if (xLen != d3FdXdYdZ.length) {
179            throw new DimensionMismatchException(xLen, d3FdXdYdZ.length);
180        }
181
182        MathArrays.checkOrder(x);
183        MathArrays.checkOrder(y);
184        MathArrays.checkOrder(z);
185
186        xval = x.clone();
187        yval = y.clone();
188        zval = z.clone();
189
190        final int lastI = xLen - 1;
191        final int lastJ = yLen - 1;
192        final int lastK = zLen - 1;
193        splines = new TricubicFunction[lastI][lastJ][lastK];
194
195        for (int i = 0; i < lastI; i++) {
196            if (f[i].length != yLen) {
197                throw new DimensionMismatchException(f[i].length, yLen);
198            }
199            if (dFdX[i].length != yLen) {
200                throw new DimensionMismatchException(dFdX[i].length, yLen);
201            }
202            if (dFdY[i].length != yLen) {
203                throw new DimensionMismatchException(dFdY[i].length, yLen);
204            }
205            if (dFdZ[i].length != yLen) {
206                throw new DimensionMismatchException(dFdZ[i].length, yLen);
207            }
208            if (d2FdXdY[i].length != yLen) {
209                throw new DimensionMismatchException(d2FdXdY[i].length, yLen);
210            }
211            if (d2FdXdZ[i].length != yLen) {
212                throw new DimensionMismatchException(d2FdXdZ[i].length, yLen);
213            }
214            if (d2FdYdZ[i].length != yLen) {
215                throw new DimensionMismatchException(d2FdYdZ[i].length, yLen);
216            }
217            if (d3FdXdYdZ[i].length != yLen) {
218                throw new DimensionMismatchException(d3FdXdYdZ[i].length, yLen);
219            }
220
221            final int ip1 = i + 1;
222            final double xR = xval[ip1] - xval[i];
223            for (int j = 0; j < lastJ; j++) {
224                if (f[i][j].length != zLen) {
225                    throw new DimensionMismatchException(f[i][j].length, zLen);
226                }
227                if (dFdX[i][j].length != zLen) {
228                    throw new DimensionMismatchException(dFdX[i][j].length, zLen);
229                }
230                if (dFdY[i][j].length != zLen) {
231                    throw new DimensionMismatchException(dFdY[i][j].length, zLen);
232                }
233                if (dFdZ[i][j].length != zLen) {
234                    throw new DimensionMismatchException(dFdZ[i][j].length, zLen);
235                }
236                if (d2FdXdY[i][j].length != zLen) {
237                    throw new DimensionMismatchException(d2FdXdY[i][j].length, zLen);
238                }
239                if (d2FdXdZ[i][j].length != zLen) {
240                    throw new DimensionMismatchException(d2FdXdZ[i][j].length, zLen);
241                }
242                if (d2FdYdZ[i][j].length != zLen) {
243                    throw new DimensionMismatchException(d2FdYdZ[i][j].length, zLen);
244                }
245                if (d3FdXdYdZ[i][j].length != zLen) {
246                    throw new DimensionMismatchException(d3FdXdYdZ[i][j].length, zLen);
247                }
248
249                final int jp1 = j + 1;
250                final double yR = yval[jp1] - yval[j];
251                final double xRyR = xR * yR;
252                for (int k = 0; k < lastK; k++) {
253                    final int kp1 = k + 1;
254                    final double zR = zval[kp1] - zval[k];
255                    final double xRzR = xR * zR;
256                    final double yRzR = yR * zR;
257                    final double xRyRzR = xR * yRzR;
258
259                    final double[] beta = new double[] {
260                        f[i][j][k], f[ip1][j][k],
261                        f[i][jp1][k], f[ip1][jp1][k],
262                        f[i][j][kp1], f[ip1][j][kp1],
263                        f[i][jp1][kp1], f[ip1][jp1][kp1],
264
265                        dFdX[i][j][k] * xR, dFdX[ip1][j][k] * xR,
266                        dFdX[i][jp1][k] * xR, dFdX[ip1][jp1][k] * xR,
267                        dFdX[i][j][kp1] * xR, dFdX[ip1][j][kp1] * xR,
268                        dFdX[i][jp1][kp1] * xR, dFdX[ip1][jp1][kp1] * xR,
269
270                        dFdY[i][j][k] * yR, dFdY[ip1][j][k] * yR,
271                        dFdY[i][jp1][k] * yR, dFdY[ip1][jp1][k] * yR,
272                        dFdY[i][j][kp1] * yR, dFdY[ip1][j][kp1] * yR,
273                        dFdY[i][jp1][kp1] * yR, dFdY[ip1][jp1][kp1] * yR,
274
275                        dFdZ[i][j][k] * zR, dFdZ[ip1][j][k] * zR,
276                        dFdZ[i][jp1][k] * zR, dFdZ[ip1][jp1][k] * zR,
277                        dFdZ[i][j][kp1] * zR, dFdZ[ip1][j][kp1] * zR,
278                        dFdZ[i][jp1][kp1] * zR, dFdZ[ip1][jp1][kp1] * zR,
279
280                        d2FdXdY[i][j][k] * xRyR, d2FdXdY[ip1][j][k] * xRyR,
281                        d2FdXdY[i][jp1][k] * xRyR, d2FdXdY[ip1][jp1][k] * xRyR,
282                        d2FdXdY[i][j][kp1] * xRyR, d2FdXdY[ip1][j][kp1] * xRyR,
283                        d2FdXdY[i][jp1][kp1] * xRyR, d2FdXdY[ip1][jp1][kp1] * xRyR,
284
285                        d2FdXdZ[i][j][k] * xRzR, d2FdXdZ[ip1][j][k] * xRzR,
286                        d2FdXdZ[i][jp1][k] * xRzR, d2FdXdZ[ip1][jp1][k] * xRzR,
287                        d2FdXdZ[i][j][kp1] * xRzR, d2FdXdZ[ip1][j][kp1] * xRzR,
288                        d2FdXdZ[i][jp1][kp1] * xRzR, d2FdXdZ[ip1][jp1][kp1] * xRzR,
289
290                        d2FdYdZ[i][j][k] * yRzR, d2FdYdZ[ip1][j][k] * yRzR,
291                        d2FdYdZ[i][jp1][k] * yRzR, d2FdYdZ[ip1][jp1][k] * yRzR,
292                        d2FdYdZ[i][j][kp1] * yRzR, d2FdYdZ[ip1][j][kp1] * yRzR,
293                        d2FdYdZ[i][jp1][kp1] * yRzR, d2FdYdZ[ip1][jp1][kp1] * yRzR,
294
295                        d3FdXdYdZ[i][j][k] * xRyRzR, d3FdXdYdZ[ip1][j][k] * xRyRzR,
296                        d3FdXdYdZ[i][jp1][k] * xRyRzR, d3FdXdYdZ[ip1][jp1][k] * xRyRzR,
297                        d3FdXdYdZ[i][j][kp1] * xRyRzR, d3FdXdYdZ[ip1][j][kp1] * xRyRzR,
298                        d3FdXdYdZ[i][jp1][kp1] * xRyRzR, d3FdXdYdZ[ip1][jp1][kp1] * xRyRzR,
299                    };
300
301                    splines[i][j][k] = new TricubicFunction(computeCoefficients(beta));
302                }
303            }
304        }
305    }
306
307    /**
308     * {@inheritDoc}
309     *
310     * @throws OutOfRangeException if any of the variables is outside its interpolation range.
311     */
312    @Override
313    public double value(double x, double y, double z)
314        throws OutOfRangeException {
315        final int i = searchIndex(x, xval);
316        if (i == -1) {
317            throw new OutOfRangeException(x, xval[0], xval[xval.length - 1]);
318        }
319        final int j = searchIndex(y, yval);
320        if (j == -1) {
321            throw new OutOfRangeException(y, yval[0], yval[yval.length - 1]);
322        }
323        final int k = searchIndex(z, zval);
324        if (k == -1) {
325            throw new OutOfRangeException(z, zval[0], zval[zval.length - 1]);
326        }
327
328        final double xN = (x - xval[i]) / (xval[i + 1] - xval[i]);
329        final double yN = (y - yval[j]) / (yval[j + 1] - yval[j]);
330        final double zN = (z - zval[k]) / (zval[k + 1] - zval[k]);
331
332        return splines[i][j][k].value(xN, yN, zN);
333    }
334
335    /**
336     * Indicates whether a point is within the interpolation range.
337     *
338     * @param x First coordinate.
339     * @param y Second coordinate.
340     * @param z Third coordinate.
341     * @return {@code true} if (x, y, z) is a valid point.
342     */
343    public boolean isValidPoint(double x, double y, double z) {
344        if (x < xval[0] ||
345            x > xval[xval.length - 1] ||
346            y < yval[0] ||
347            y > yval[yval.length - 1] ||
348            z < zval[0] ||
349            z > zval[zval.length - 1]) {
350            return false;
351        } else {
352            return true;
353        }
354    }
355
356    /**
357     * @param c Coordinate.
358     * @param val Coordinate samples.
359     * @return the index in {@code val} corresponding to the interval containing {@code c}, or {@code -1}
360     *   if {@code c} is out of the range defined by the end values of {@code val}.
361     */
362    private int searchIndex(double c, double[] val) {
363        if (c < val[0]) {
364            return -1;
365        }
366
367        final int max = val.length;
368        for (int i = 1; i < max; i++) {
369            if (c <= val[i]) {
370                return i - 1;
371            }
372        }
373
374        return -1;
375    }
376
377    /**
378     * Compute the spline coefficients from the list of function values and
379     * function partial derivatives values at the four corners of a grid
380     * element. They must be specified in the following order:
381     * <ul>
382     *  <li>f(0,0,0)</li>
383     *  <li>f(1,0,0)</li>
384     *  <li>f(0,1,0)</li>
385     *  <li>f(1,1,0)</li>
386     *  <li>f(0,0,1)</li>
387     *  <li>f(1,0,1)</li>
388     *  <li>f(0,1,1)</li>
389     *  <li>f(1,1,1)</li>
390     *
391     *  <li>f<sub>x</sub>(0,0,0)</li>
392     *  <li>... <em>(same order as above)</em></li>
393     *  <li>f<sub>x</sub>(1,1,1)</li>
394     *
395     *  <li>f<sub>y</sub>(0,0,0)</li>
396     *  <li>... <em>(same order as above)</em></li>
397     *  <li>f<sub>y</sub>(1,1,1)</li>
398     *
399     *  <li>f<sub>z</sub>(0,0,0)</li>
400     *  <li>... <em>(same order as above)</em></li>
401     *  <li>f<sub>z</sub>(1,1,1)</li>
402     *
403     *  <li>f<sub>xy</sub>(0,0,0)</li>
404     *  <li>... <em>(same order as above)</em></li>
405     *  <li>f<sub>xy</sub>(1,1,1)</li>
406     *
407     *  <li>f<sub>xz</sub>(0,0,0)</li>
408     *  <li>... <em>(same order as above)</em></li>
409     *  <li>f<sub>xz</sub>(1,1,1)</li>
410     *
411     *  <li>f<sub>yz</sub>(0,0,0)</li>
412     *  <li>... <em>(same order as above)</em></li>
413     *  <li>f<sub>yz</sub>(1,1,1)</li>
414     *
415     *  <li>f<sub>xyz</sub>(0,0,0)</li>
416     *  <li>... <em>(same order as above)</em></li>
417     *  <li>f<sub>xyz</sub>(1,1,1)</li>
418     * </ul>
419     * where the subscripts indicate the partial derivative with respect to
420     * the corresponding variable(s).
421     *
422     * @param beta List of function values and function partial derivatives values.
423     * @return the spline coefficients.
424     */
425    private double[] computeCoefficients(double[] beta) {
426        final int sz = 64;
427        final double[] a = new double[sz];
428
429        for (int i = 0; i < sz; i++) {
430            double result = 0;
431            final double[] row = AINV[i];
432            for (int j = 0; j < sz; j++) {
433                result += row[j] * beta[j];
434            }
435            a[i] = result;
436        }
437
438        return a;
439    }
440}
441
442/**
443 * 3D-spline function.
444 *
445 */
446class TricubicFunction
447    implements TrivariateFunction {
448    /** Number of points. */
449    private static final short N = 4;
450    /** Coefficients */
451    private final double[][][] a = new double[N][N][N];
452
453    /**
454     * @param aV List of spline coefficients.
455     */
456    TricubicFunction(double[] aV) {
457        for (int i = 0; i < N; i++) {
458            for (int j = 0; j < N; j++) {
459                for (int k = 0; k < N; k++) {
460                    a[i][j][k] = aV[i + N * (j + N * k)];
461                }
462            }
463        }
464    }
465
466    /**
467     * @param x x-coordinate of the interpolation point.
468     * @param y y-coordinate of the interpolation point.
469     * @param z z-coordinate of the interpolation point.
470     * @return the interpolated value.
471     * @throws OutOfRangeException if {@code x}, {@code y} or
472     * {@code z} are not in the interval {@code [0, 1]}.
473     */
474    @Override
475    public double value(double x, double y, double z) throws OutOfRangeException {
476        if (x < 0 || x > 1) {
477            throw new OutOfRangeException(x, 0, 1);
478        }
479        if (y < 0 || y > 1) {
480            throw new OutOfRangeException(y, 0, 1);
481        }
482        if (z < 0 || z > 1) {
483            throw new OutOfRangeException(z, 0, 1);
484        }
485
486        final double x2 = x * x;
487        final double x3 = x2 * x;
488        final double[] pX = { 1, x, x2, x3 };
489
490        final double y2 = y * y;
491        final double y3 = y2 * y;
492        final double[] pY = { 1, y, y2, y3 };
493
494        final double z2 = z * z;
495        final double z3 = z2 * z;
496        final double[] pZ = { 1, z, z2, z3 };
497
498        double result = 0;
499        for (int i = 0; i < N; i++) {
500            for (int j = 0; j < N; j++) {
501                for (int k = 0; k < N; k++) {
502                    result += a[i][j][k] * pX[i] * pY[j] * pZ[k];
503                }
504            }
505        }
506
507        return result;
508    }
509}