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 * @version $Id: TricubicSplineInterpolatingFunction.java 1385314 2012-09-16 16:35:49Z tn $
038 */
039public class TricubicSplineInterpolatingFunction
040    implements TrivariateFunction {
041    /**
042     * Matrix to compute the spline coefficients from the function values
043     * and function derivatives values
044     */
045    private static final double[][] AINV = {
046        { 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 },
047        { 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 },
048        { -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 },
049        { 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 },
050        { 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 },
051        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
052        { 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 },
053        { 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 },
054        { -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 },
055        { 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 },
056        { 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 },
057        { -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 },
058        { 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 },
059        { 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 },
060        { -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 },
061        { 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 },
062        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
063        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
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,-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 },
065        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-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 },
066        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 },
067        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0 },
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,-3,3,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0 },
069        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-2,0,0,0,0,0,0,1,1,0,0,0,0,0,0 },
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,-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 },
071        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,-1,0,0,0,0,0 },
072        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,-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 },
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,-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 },
074        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,-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 },
075        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0 },
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,-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 },
077        { 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 },
078        {-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 },
079        { 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 },
080        { 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 },
081        { -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 },
082        { 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 },
083        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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 },
084        { 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 },
085        { 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 },
086        { 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 },
087        { 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 },
088        { -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 },
089        { 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 },
090        { -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 },
091        { 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 },
092        { 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 },
093        { -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 },
094        { 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 },
095        { 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 },
096        { -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 },
097        { 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 },
098        { 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 },
099        { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,-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 },
100        { 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 },
101        { 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 },
102        { -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 },
103        { 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 },
104        { 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 },
105        { -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 },
106        { 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 },
107        { 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 },
108        { -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 },
109        { 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 }
110    };
111
112    /** Samples x-coordinates */
113    private final double[] xval;
114    /** Samples y-coordinates */
115    private final double[] yval;
116    /** Samples z-coordinates */
117    private final double[] zval;
118    /** Set of cubic splines pacthing the whole data grid */
119    private final TricubicSplineFunction[][][] splines;
120
121    /**
122     * @param x Sample values of the x-coordinate, in increasing order.
123     * @param y Sample values of the y-coordinate, in increasing order.
124     * @param z Sample values of the y-coordinate, in increasing order.
125     * @param f Values of the function on every grid point.
126     * @param dFdX Values of the partial derivative of function with respect to x on every grid point.
127     * @param dFdY Values of the partial derivative of function with respect to y on every grid point.
128     * @param dFdZ Values of the partial derivative of function with respect to z on every grid point.
129     * @param d2FdXdY Values of the cross partial derivative of function on every grid point.
130     * @param d2FdXdZ Values of the cross partial derivative of function on every grid point.
131     * @param d2FdYdZ Values of the cross partial derivative of function on every grid point.
132     * @param d3FdXdYdZ Values of the cross partial derivative of function on every grid point.
133     * @throws NoDataException if any of the arrays has zero length.
134     * @throws DimensionMismatchException if the various arrays do not contain the expected number of elements.
135     * @throws NonMonotonicSequenceException if {@code x}, {@code y} or {@code z} are not strictly increasing.
136     */
137    public TricubicSplineInterpolatingFunction(double[] x,
138                                               double[] y,
139                                               double[] z,
140                                               double[][][] f,
141                                               double[][][] dFdX,
142                                               double[][][] dFdY,
143                                               double[][][] dFdZ,
144                                               double[][][] d2FdXdY,
145                                               double[][][] d2FdXdZ,
146                                               double[][][] d2FdYdZ,
147                                               double[][][] d3FdXdYdZ)
148        throws NoDataException,
149               DimensionMismatchException,
150               NonMonotonicSequenceException {
151        final int xLen = x.length;
152        final int yLen = y.length;
153        final int zLen = z.length;
154
155        if (xLen == 0 || yLen == 0 || z.length == 0 || f.length == 0 || f[0].length == 0) {
156            throw new NoDataException();
157        }
158        if (xLen != f.length) {
159            throw new DimensionMismatchException(xLen, f.length);
160        }
161        if (xLen != dFdX.length) {
162            throw new DimensionMismatchException(xLen, dFdX.length);
163        }
164        if (xLen != dFdY.length) {
165            throw new DimensionMismatchException(xLen, dFdY.length);
166        }
167        if (xLen != dFdZ.length) {
168            throw new DimensionMismatchException(xLen, dFdZ.length);
169        }
170        if (xLen != d2FdXdY.length) {
171            throw new DimensionMismatchException(xLen, d2FdXdY.length);
172        }
173        if (xLen != d2FdXdZ.length) {
174            throw new DimensionMismatchException(xLen, d2FdXdZ.length);
175        }
176        if (xLen != d2FdYdZ.length) {
177            throw new DimensionMismatchException(xLen, d2FdYdZ.length);
178        }
179        if (xLen != d3FdXdYdZ.length) {
180            throw new DimensionMismatchException(xLen, d3FdXdYdZ.length);
181        }
182
183        MathArrays.checkOrder(x);
184        MathArrays.checkOrder(y);
185        MathArrays.checkOrder(z);
186
187        xval = x.clone();
188        yval = y.clone();
189        zval = z.clone();
190
191        final int lastI = xLen - 1;
192        final int lastJ = yLen - 1;
193        final int lastK = zLen - 1;
194        splines = new TricubicSplineFunction[lastI][lastJ][lastK];
195
196        for (int i = 0; i < lastI; i++) {
197            if (f[i].length != yLen) {
198                throw new DimensionMismatchException(f[i].length, yLen);
199            }
200            if (dFdX[i].length != yLen) {
201                throw new DimensionMismatchException(dFdX[i].length, yLen);
202            }
203            if (dFdY[i].length != yLen) {
204                throw new DimensionMismatchException(dFdY[i].length, yLen);
205            }
206            if (dFdZ[i].length != yLen) {
207                throw new DimensionMismatchException(dFdZ[i].length, yLen);
208            }
209            if (d2FdXdY[i].length != yLen) {
210                throw new DimensionMismatchException(d2FdXdY[i].length, yLen);
211            }
212            if (d2FdXdZ[i].length != yLen) {
213                throw new DimensionMismatchException(d2FdXdZ[i].length, yLen);
214            }
215            if (d2FdYdZ[i].length != yLen) {
216                throw new DimensionMismatchException(d2FdYdZ[i].length, yLen);
217            }
218            if (d3FdXdYdZ[i].length != yLen) {
219                throw new DimensionMismatchException(d3FdXdYdZ[i].length, yLen);
220            }
221
222            final int ip1 = i + 1;
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                for (int k = 0; k < lastK; k++) {
251                    final int kp1 = k + 1;
252
253                    final double[] beta = new double[] {
254                        f[i][j][k], f[ip1][j][k],
255                        f[i][jp1][k], f[ip1][jp1][k],
256                        f[i][j][kp1], f[ip1][j][kp1],
257                        f[i][jp1][kp1], f[ip1][jp1][kp1],
258
259                        dFdX[i][j][k], dFdX[ip1][j][k],
260                        dFdX[i][jp1][k], dFdX[ip1][jp1][k],
261                        dFdX[i][j][kp1], dFdX[ip1][j][kp1],
262                        dFdX[i][jp1][kp1], dFdX[ip1][jp1][kp1],
263
264                        dFdY[i][j][k], dFdY[ip1][j][k],
265                        dFdY[i][jp1][k], dFdY[ip1][jp1][k],
266                        dFdY[i][j][kp1], dFdY[ip1][j][kp1],
267                        dFdY[i][jp1][kp1], dFdY[ip1][jp1][kp1],
268
269                        dFdZ[i][j][k], dFdZ[ip1][j][k],
270                        dFdZ[i][jp1][k], dFdZ[ip1][jp1][k],
271                        dFdZ[i][j][kp1], dFdZ[ip1][j][kp1],
272                        dFdZ[i][jp1][kp1], dFdZ[ip1][jp1][kp1],
273
274                        d2FdXdY[i][j][k], d2FdXdY[ip1][j][k],
275                        d2FdXdY[i][jp1][k], d2FdXdY[ip1][jp1][k],
276                        d2FdXdY[i][j][kp1], d2FdXdY[ip1][j][kp1],
277                        d2FdXdY[i][jp1][kp1], d2FdXdY[ip1][jp1][kp1],
278
279                        d2FdXdZ[i][j][k], d2FdXdZ[ip1][j][k],
280                        d2FdXdZ[i][jp1][k], d2FdXdZ[ip1][jp1][k],
281                        d2FdXdZ[i][j][kp1], d2FdXdZ[ip1][j][kp1],
282                        d2FdXdZ[i][jp1][kp1], d2FdXdZ[ip1][jp1][kp1],
283
284                        d2FdYdZ[i][j][k], d2FdYdZ[ip1][j][k],
285                        d2FdYdZ[i][jp1][k], d2FdYdZ[ip1][jp1][k],
286                        d2FdYdZ[i][j][kp1], d2FdYdZ[ip1][j][kp1],
287                        d2FdYdZ[i][jp1][kp1], d2FdYdZ[ip1][jp1][kp1],
288
289                        d3FdXdYdZ[i][j][k], d3FdXdYdZ[ip1][j][k],
290                        d3FdXdYdZ[i][jp1][k], d3FdXdYdZ[ip1][jp1][k],
291                        d3FdXdYdZ[i][j][kp1], d3FdXdYdZ[ip1][j][kp1],
292                        d3FdXdYdZ[i][jp1][kp1], d3FdXdYdZ[ip1][jp1][kp1],
293                    };
294
295                    splines[i][j][k] = new TricubicSplineFunction(computeSplineCoefficients(beta));
296                }
297            }
298        }
299    }
300
301    /**
302     * {@inheritDoc}
303     *
304     * @throws OutOfRangeException if any of the variables is outside its interpolation range.
305     */
306    public double value(double x, double y, double z)
307        throws OutOfRangeException {
308        final int i = searchIndex(x, xval);
309        if (i == -1) {
310            throw new OutOfRangeException(x, xval[0], xval[xval.length - 1]);
311        }
312        final int j = searchIndex(y, yval);
313        if (j == -1) {
314            throw new OutOfRangeException(y, yval[0], yval[yval.length - 1]);
315        }
316        final int k = searchIndex(z, zval);
317        if (k == -1) {
318            throw new OutOfRangeException(z, zval[0], zval[zval.length - 1]);
319        }
320
321        final double xN = (x - xval[i]) / (xval[i + 1] - xval[i]);
322        final double yN = (y - yval[j]) / (yval[j + 1] - yval[j]);
323        final double zN = (z - zval[k]) / (zval[k + 1] - zval[k]);
324
325        return splines[i][j][k].value(xN, yN, zN);
326    }
327
328    /**
329     * @param c Coordinate.
330     * @param val Coordinate samples.
331     * @return the index in {@code val} corresponding to the interval containing {@code c}, or {@code -1}
332     *   if {@code c} is out of the range defined by the end values of {@code val}.
333     */
334    private int searchIndex(double c, double[] val) {
335        if (c < val[0]) {
336            return -1;
337        }
338
339        final int max = val.length;
340        for (int i = 1; i < max; i++) {
341            if (c <= val[i]) {
342                return i - 1;
343            }
344        }
345
346        return -1;
347    }
348
349    /**
350     * Compute the spline coefficients from the list of function values and
351     * function partial derivatives values at the four corners of a grid
352     * element. They must be specified in the following order:
353     * <ul>
354     *  <li>f(0,0,0)</li>
355     *  <li>f(1,0,0)</li>
356     *  <li>f(0,1,0)</li>
357     *  <li>f(1,1,0)</li>
358     *  <li>f(0,0,1)</li>
359     *  <li>f(1,0,1)</li>
360     *  <li>f(0,1,1)</li>
361     *  <li>f(1,1,1)</li>
362     *
363     *  <li>f<sub>x</sub>(0,0,0)</li>
364     *  <li>... <em>(same order as above)</em></li>
365     *  <li>f<sub>x</sub>(1,1,1)</li>
366     *
367     *  <li>f<sub>y</sub>(0,0,0)</li>
368     *  <li>... <em>(same order as above)</em></li>
369     *  <li>f<sub>y</sub>(1,1,1)</li>
370     *
371     *  <li>f<sub>z</sub>(0,0,0)</li>
372     *  <li>... <em>(same order as above)</em></li>
373     *  <li>f<sub>z</sub>(1,1,1)</li>
374     *
375     *  <li>f<sub>xy</sub>(0,0,0)</li>
376     *  <li>... <em>(same order as above)</em></li>
377     *  <li>f<sub>xy</sub>(1,1,1)</li>
378     *
379     *  <li>f<sub>xz</sub>(0,0,0)</li>
380     *  <li>... <em>(same order as above)</em></li>
381     *  <li>f<sub>xz</sub>(1,1,1)</li>
382     *
383     *  <li>f<sub>yz</sub>(0,0,0)</li>
384     *  <li>... <em>(same order as above)</em></li>
385     *  <li>f<sub>yz</sub>(1,1,1)</li>
386     *
387     *  <li>f<sub>xyz</sub>(0,0,0)</li>
388     *  <li>... <em>(same order as above)</em></li>
389     *  <li>f<sub>xyz</sub>(1,1,1)</li>
390     * </ul>
391     * where the subscripts indicate the partial derivative with respect to
392     * the corresponding variable(s).
393     *
394     * @param beta List of function values and function partial derivatives values.
395     * @return the spline coefficients.
396     */
397    private double[] computeSplineCoefficients(double[] beta) {
398        final int sz = 64;
399        final double[] a = new double[sz];
400
401        for (int i = 0; i < sz; i++) {
402            double result = 0;
403            final double[] row = AINV[i];
404            for (int j = 0; j < sz; j++) {
405                result += row[j] * beta[j];
406            }
407            a[i] = result;
408        }
409
410        return a;
411    }
412}
413
414/**
415 * 3D-spline function.
416 *
417 * @version $Id: TricubicSplineInterpolatingFunction.java 1385314 2012-09-16 16:35:49Z tn $
418 */
419class 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];
425
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    }
438
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        }
458
459        final double x2 = x * x;
460        final double x3 = x2 * x;
461        final double[] pX = { 1, x, x2, x3 };
462
463        final double y2 = y * y;
464        final double y3 = y2 * y;
465        final double[] pY = { 1, y, y2, y3 };
466
467        final double z2 = z * z;
468        final double z3 = z2 * z;
469        final double[] pZ = { 1, z, z2, z3 };
470
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        }
479
480        return result;
481    }
482}