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.legacy.analysis.interpolation; 018 019import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunction; 020import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialSplineFunction; 021import org.apache.commons.math4.legacy.core.MathArrays; 022import org.apache.commons.math4.legacy.exception.DimensionMismatchException; 023import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException; 024import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException; 025import org.apache.commons.math4.legacy.exception.util.LocalizedFormats; 026 027/** 028 * Computes a clamped cubic spline interpolation for the data set. 029 * <p> 030 * The {@link #interpolate(double[], double[], double, double)} method returns a {@link PolynomialSplineFunction} 031 * consisting of n cubic polynomials, defined over the subintervals determined by the x values, 032 * {@code x[0] < x[i] ... < x[n]}. The x values are referred to as "knot points."</p> 033 * <p> 034 * The value of the PolynomialSplineFunction at a point x that is greater than or equal to the smallest 035 * knot point and strictly less than the largest knot point is computed by finding the subinterval to which 036 * x belongs and computing the value of the corresponding polynomial at <code>x - x[i]</code> where 037 * <code>i</code> is the index of the subinterval. See {@link PolynomialSplineFunction} for more details. 038 * </p> 039 * <p> 040 * The interpolating polynomials satisfy: <ol> 041 * <li>The value of the PolynomialSplineFunction at each of the input x values equals the 042 * corresponding y value.</li> 043 * <li>Adjacent polynomials are equal through two derivatives at the knot points (i.e., adjacent polynomials 044 * "match up" at the knot points, as do their first and second derivatives).</li> 045 * <li>The <i>clamped boundary condition</i>, i.e., the PolynomialSplineFunction takes "a specific direction" at both 046 * its start point and its end point by providing the desired first derivative values (slopes) as function parameters to 047 * {@link #interpolate(double[], double[], double, double)}.</li> 048 * </ol> 049 * <p> 050 * The clamped cubic spline interpolation algorithm implemented is as described in R.L. Burden, J.D. Faires, 051 * <u>Numerical Analysis</u>, 9th Ed., 2010, Cengage Learning, ISBN 0-538-73351-9, pp 153-156. 052 * </p> 053 * 054 */ 055public class ClampedSplineInterpolator extends SplineInterpolator { 056 /** 057 * Computes an interpolating function for the data set. 058 * @param x the arguments for the interpolation points 059 * @param y the values for the interpolation points 060 * @param fpo first derivative at the starting point of the returned spline function (starting slope), satisfying 061 * clamped boundary condition S′(x0) = f′(x0) 062 * @param fpn first derivative at the ending point of the returned spline function (ending slope), satisfying 063 * clamped boundary condition S′(xn) = f′(xn) 064 * @return a function which interpolates the data set 065 * @throws DimensionMismatchException if {@code x} and {@code y} 066 * have different sizes. 067 * @throws NumberIsTooSmallException if the size of {@code x < 3}. 068 * @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException 069 * if {@code x} is not sorted in strict increasing order. 070 */ 071 public PolynomialSplineFunction interpolate(final double[] x, final double[] y, 072 final double fpo, final double fpn) 073 throws DimensionMismatchException, NumberIsTooSmallException, NonMonotonicSequenceException { 074 if (x.length != y.length) { 075 throw new DimensionMismatchException(x.length, y.length); 076 } 077 078 if (x.length < 3) { 079 throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS, 080 x.length, 3, true); 081 } 082 083 // Number of intervals. The number of data points is n + 1. 084 final int n = x.length - 1; 085 086 MathArrays.checkOrder(x); 087 088 // Differences between knot points 089 final double h[] = new double[n]; 090 for (int i = 0; i < n; i++) { 091 h[i] = x[i + 1] - x[i]; 092 } 093 094 final double mu[] = new double[n]; 095 final double z[] = new double[n + 1]; 096 final double alpha[] = new double[n + 1]; 097 final double l[] = new double[n + 1]; 098 099 alpha[0] = 3d * (y[1] - y[0]) / h[0] - 3d * fpo; 100 alpha[n] = 3d * fpn - 3d * (y[n] - y[n - 1]) / h[n - 1]; 101 102 mu[0] = 0.5d; 103 l[0] = 2d * h[0]; 104 z[0] = alpha[0] / l[0]; 105 106 for (int i = 1; i < n; i++) { 107 108 alpha[i] = (3d / h[i]) * (y[i + 1] - y[i]) - (3d / h[i - 1]) * (y[i] - y[i - 1]); 109 l[i] = 2d * (x[i + 1] - x[i - 1]) - h[i - 1] * mu[i - 1]; 110 mu[i] = h[i] / l[i]; 111 z[i] = (alpha[i] - h[i - 1] * z[i - 1]) / l[i]; 112 } 113 // cubic spline coefficients -- b is linear, c quadratic, d is cubic (original y's are constants) 114 final double b[] = new double[n]; 115 final double c[] = new double[n + 1]; 116 final double d[] = new double[n]; 117 l[n] = h[n - 1] * (2d - mu[n - 1]); 118 z[n] = (alpha[n] - h[n - 1] * z[n - 1]) / l[n]; 119 c[n] = z[n]; 120 121 for (int j = n - 1; j >= 0; j--) { 122 c[j] = z[j] - mu[j] * c[j + 1]; 123 b[j] = ((y[j + 1] - y[j]) / h[j]) - h[j] * (c[j + 1] + 2d * c[j]) / 3d; 124 d[j] = (c[j + 1] - c[j]) / (3d * h[j]); 125 } 126 127 final PolynomialFunction polynomials[] = new PolynomialFunction[n]; 128 final double coefficients[] = new double[4]; 129 for (int i = 0; i < n; i++) { 130 coefficients[0] = y[i]; 131 coefficients[1] = b[i]; 132 coefficients[2] = c[i]; 133 coefficients[3] = d[i]; 134 polynomials[i] = new PolynomialFunction(coefficients); 135 } 136 return new PolynomialSplineFunction(x, polynomials); 137 } 138}