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.math3.analysis.interpolation;
018
019import org.apache.commons.math3.analysis.TrivariateFunction;
020import org.apache.commons.math3.exception.DimensionMismatchException;
021import org.apache.commons.math3.exception.NoDataException;
022import org.apache.commons.math3.exception.OutOfRangeException;
023import org.apache.commons.math3.exception.NonMonotonicSequenceException;
024import org.apache.commons.math3.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 * <quote>
031 *  Tricubic interpolation in three dimensions<br/>
032 *  F. Lekien and J. Marsden<br/>
033 *  <em>Int. J. Numer. Meth. Engng</em> 2005; <b>63</b>:455-471
034 * </quote>
035 *
036 * @since 2.2
037 */
038public class TricubicSplineInterpolatingFunction
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 pacthing the whole data grid */
118    private final TricubicSplineFunction[][][] 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 TricubicSplineInterpolatingFunction(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 TricubicSplineFunction[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            for (int j = 0; j < lastJ; j++) {
223                if (f[i][j].length != zLen) {
224                    throw new DimensionMismatchException(f[i][j].length, zLen);
225                }
226                if (dFdX[i][j].length != zLen) {
227                    throw new DimensionMismatchException(dFdX[i][j].length, zLen);
228                }
229                if (dFdY[i][j].length != zLen) {
230                    throw new DimensionMismatchException(dFdY[i][j].length, zLen);
231                }
232                if (dFdZ[i][j].length != zLen) {
233                    throw new DimensionMismatchException(dFdZ[i][j].length, zLen);
234                }
235                if (d2FdXdY[i][j].length != zLen) {
236                    throw new DimensionMismatchException(d2FdXdY[i][j].length, zLen);
237                }
238                if (d2FdXdZ[i][j].length != zLen) {
239                    throw new DimensionMismatchException(d2FdXdZ[i][j].length, zLen);
240                }
241                if (d2FdYdZ[i][j].length != zLen) {
242                    throw new DimensionMismatchException(d2FdYdZ[i][j].length, zLen);
243                }
244                if (d3FdXdYdZ[i][j].length != zLen) {
245                    throw new DimensionMismatchException(d3FdXdYdZ[i][j].length, zLen);
246                }
247
248                final int jp1 = j + 1;
249                for (int k = 0; k < lastK; k++) {
250                    final int kp1 = k + 1;
251
252                    final double[] beta = new double[] {
253                        f[i][j][k], f[ip1][j][k],
254                        f[i][jp1][k], f[ip1][jp1][k],
255                        f[i][j][kp1], f[ip1][j][kp1],
256                        f[i][jp1][kp1], f[ip1][jp1][kp1],
257
258                        dFdX[i][j][k], dFdX[ip1][j][k],
259                        dFdX[i][jp1][k], dFdX[ip1][jp1][k],
260                        dFdX[i][j][kp1], dFdX[ip1][j][kp1],
261                        dFdX[i][jp1][kp1], dFdX[ip1][jp1][kp1],
262
263                        dFdY[i][j][k], dFdY[ip1][j][k],
264                        dFdY[i][jp1][k], dFdY[ip1][jp1][k],
265                        dFdY[i][j][kp1], dFdY[ip1][j][kp1],
266                        dFdY[i][jp1][kp1], dFdY[ip1][jp1][kp1],
267
268                        dFdZ[i][j][k], dFdZ[ip1][j][k],
269                        dFdZ[i][jp1][k], dFdZ[ip1][jp1][k],
270                        dFdZ[i][j][kp1], dFdZ[ip1][j][kp1],
271                        dFdZ[i][jp1][kp1], dFdZ[ip1][jp1][kp1],
272
273                        d2FdXdY[i][j][k], d2FdXdY[ip1][j][k],
274                        d2FdXdY[i][jp1][k], d2FdXdY[ip1][jp1][k],
275                        d2FdXdY[i][j][kp1], d2FdXdY[ip1][j][kp1],
276                        d2FdXdY[i][jp1][kp1], d2FdXdY[ip1][jp1][kp1],
277
278                        d2FdXdZ[i][j][k], d2FdXdZ[ip1][j][k],
279                        d2FdXdZ[i][jp1][k], d2FdXdZ[ip1][jp1][k],
280                        d2FdXdZ[i][j][kp1], d2FdXdZ[ip1][j][kp1],
281                        d2FdXdZ[i][jp1][kp1], d2FdXdZ[ip1][jp1][kp1],
282
283                        d2FdYdZ[i][j][k], d2FdYdZ[ip1][j][k],
284                        d2FdYdZ[i][jp1][k], d2FdYdZ[ip1][jp1][k],
285                        d2FdYdZ[i][j][kp1], d2FdYdZ[ip1][j][kp1],
286                        d2FdYdZ[i][jp1][kp1], d2FdYdZ[ip1][jp1][kp1],
287
288                        d3FdXdYdZ[i][j][k], d3FdXdYdZ[ip1][j][k],
289                        d3FdXdYdZ[i][jp1][k], d3FdXdYdZ[ip1][jp1][k],
290                        d3FdXdYdZ[i][j][kp1], d3FdXdYdZ[ip1][j][kp1],
291                        d3FdXdYdZ[i][jp1][kp1], d3FdXdYdZ[ip1][jp1][kp1],
292                    };
293
294                    splines[i][j][k] = new TricubicSplineFunction(computeSplineCoefficients(beta));
295                }
296            }
297        }
298    }
299
300    /**
301     * {@inheritDoc}
302     *
303     * @throws OutOfRangeException if any of the variables is outside its interpolation range.
304     */
305    public double value(double x, double y, double z)
306        throws OutOfRangeException {
307        final int i = searchIndex(x, xval);
308        if (i == -1) {
309            throw new OutOfRangeException(x, xval[0], xval[xval.length - 1]);
310        }
311        final int j = searchIndex(y, yval);
312        if (j == -1) {
313            throw new OutOfRangeException(y, yval[0], yval[yval.length - 1]);
314        }
315        final int k = searchIndex(z, zval);
316        if (k == -1) {
317            throw new OutOfRangeException(z, zval[0], zval[zval.length - 1]);
318        }
319
320        final double xN = (x - xval[i]) / (xval[i + 1] - xval[i]);
321        final double yN = (y - yval[j]) / (yval[j + 1] - yval[j]);
322        final double zN = (z - zval[k]) / (zval[k + 1] - zval[k]);
323
324        return splines[i][j][k].value(xN, yN, zN);
325    }
326
327    /**
328     * @param c Coordinate.
329     * @param val Coordinate samples.
330     * @return the index in {@code val} corresponding to the interval containing {@code c}, or {@code -1}
331     *   if {@code c} is out of the range defined by the end values of {@code val}.
332     */
333    private int searchIndex(double c, double[] val) {
334        if (c < val[0]) {
335            return -1;
336        }
337
338        final int max = val.length;
339        for (int i = 1; i < max; i++) {
340            if (c <= val[i]) {
341                return i - 1;
342            }
343        }
344
345        return -1;
346    }
347
348    /**
349     * Compute the spline coefficients from the list of function values and
350     * function partial derivatives values at the four corners of a grid
351     * element. They must be specified in the following order:
352     * <ul>
353     *  <li>f(0,0,0)</li>
354     *  <li>f(1,0,0)</li>
355     *  <li>f(0,1,0)</li>
356     *  <li>f(1,1,0)</li>
357     *  <li>f(0,0,1)</li>
358     *  <li>f(1,0,1)</li>
359     *  <li>f(0,1,1)</li>
360     *  <li>f(1,1,1)</li>
361     *
362     *  <li>f<sub>x</sub>(0,0,0)</li>
363     *  <li>... <em>(same order as above)</em></li>
364     *  <li>f<sub>x</sub>(1,1,1)</li>
365     *
366     *  <li>f<sub>y</sub>(0,0,0)</li>
367     *  <li>... <em>(same order as above)</em></li>
368     *  <li>f<sub>y</sub>(1,1,1)</li>
369     *
370     *  <li>f<sub>z</sub>(0,0,0)</li>
371     *  <li>... <em>(same order as above)</em></li>
372     *  <li>f<sub>z</sub>(1,1,1)</li>
373     *
374     *  <li>f<sub>xy</sub>(0,0,0)</li>
375     *  <li>... <em>(same order as above)</em></li>
376     *  <li>f<sub>xy</sub>(1,1,1)</li>
377     *
378     *  <li>f<sub>xz</sub>(0,0,0)</li>
379     *  <li>... <em>(same order as above)</em></li>
380     *  <li>f<sub>xz</sub>(1,1,1)</li>
381     *
382     *  <li>f<sub>yz</sub>(0,0,0)</li>
383     *  <li>... <em>(same order as above)</em></li>
384     *  <li>f<sub>yz</sub>(1,1,1)</li>
385     *
386     *  <li>f<sub>xyz</sub>(0,0,0)</li>
387     *  <li>... <em>(same order as above)</em></li>
388     *  <li>f<sub>xyz</sub>(1,1,1)</li>
389     * </ul>
390     * where the subscripts indicate the partial derivative with respect to
391     * the corresponding variable(s).
392     *
393     * @param beta List of function values and function partial derivatives values.
394     * @return the spline coefficients.
395     */
396    private double[] computeSplineCoefficients(double[] beta) {
397        final int sz = 64;
398        final double[] a = new double[sz];
399
400        for (int i = 0; i < sz; i++) {
401            double result = 0;
402            final double[] row = AINV[i];
403            for (int j = 0; j < sz; j++) {
404                result += row[j] * beta[j];
405            }
406            a[i] = result;
407        }
408
409        return a;
410    }
411}
412
413/**
414 * 3D-spline function.
415 *
416 */
417class TricubicSplineFunction
418    implements TrivariateFunction {
419    /** Number of points. */
420    private static final short N = 4;
421    /** Coefficients */
422    private final double[][][] a = new double[N][N][N];
423
424    /**
425     * @param aV List of spline coefficients.
426     */
427    public TricubicSplineFunction(double[] aV) {
428        for (int i = 0; i < N; i++) {
429            for (int j = 0; j < N; j++) {
430                for (int k = 0; k < N; k++) {
431                    a[i][j][k] = aV[i + N * (j + N * k)];
432                }
433            }
434        }
435    }
436
437    /**
438     * @param x x-coordinate of the interpolation point.
439     * @param y y-coordinate of the interpolation point.
440     * @param z z-coordinate of the interpolation point.
441     * @return the interpolated value.
442     * @throws OutOfRangeException if {@code x}, {@code y} or
443     * {@code z} are not in the interval {@code [0, 1]}.
444     */
445    public double value(double x, double y, double z)
446        throws OutOfRangeException {
447        if (x < 0 || x > 1) {
448            throw new OutOfRangeException(x, 0, 1);
449        }
450        if (y < 0 || y > 1) {
451            throw new OutOfRangeException(y, 0, 1);
452        }
453        if (z < 0 || z > 1) {
454            throw new OutOfRangeException(z, 0, 1);
455        }
456
457        final double x2 = x * x;
458        final double x3 = x2 * x;
459        final double[] pX = { 1, x, x2, x3 };
460
461        final double y2 = y * y;
462        final double y3 = y2 * y;
463        final double[] pY = { 1, y, y2, y3 };
464
465        final double z2 = z * z;
466        final double z3 = z2 * z;
467        final double[] pZ = { 1, z, z2, z3 };
468
469        double result = 0;
470        for (int i = 0; i < N; i++) {
471            for (int j = 0; j < N; j++) {
472                for (int k = 0; k < N; k++) {
473                    result += a[i][j][k] * pX[i] * pY[j] * pZ[k];
474                }
475            }
476        }
477
478        return result;
479    }
480}