MATH-1650.

Author: Gilles Sadowski

commons-math-legacy/src/main/java/org/apache/commons/math4/legacy/analysis/interpolation/ClampedSplineInterpolator.java

@@ -0,0 +1,160 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+package org.apache.commons.math4.legacy.analysis.interpolation;
+
+import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunction;
+import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialSplineFunction;
+import org.apache.commons.math4.legacy.core.MathArrays;
+import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
+import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
+import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
+
+/**
+ * Clamped cubic spline interpolator.
+ * The interpolating function consists in cubic polynomial functions defined over the
+ * subintervals determined by the "knot points".
+ *
+ * The interpolating polynomials satisfy:
+ * <ol>
+ *  <li>The value of the interpolating function at each of the input {@code x} values
+ *   equals the corresponding input {@code y} value.</li>
+ *  <li>Adjacent polynomials are equal through two derivatives at the knot points
+ *   (i.e., adjacent polynomials "match up" at the knot points, as do their first and
+ *   second derivatives).</li>
+ *  <li>The clamped boundary condition, i.e. the interpolating function takes "a
+ *   specific direction" at both its start point and its end point by providing the
+ *   desired first derivative values (slopes) as function parameters to
+ *   {@link #interpolate(double[], double[], double, double)}.</li>
+ * </ol>
+ *
+ * The algorithm is implemented as described in
+ * <blockquote>
+ *  R.L. Burden, J.D. Faires,
+ *  <em>Numerical Analysis</em>, 9th Ed., 2010, Cengage Learning, ISBN 0-538-73351-9, pp 153-156.
+ * </blockquote>
+ *
+ */
+public class ClampedSplineInterpolator implements UnivariateInterpolator {
+    /**
+     *
+     * The first derivatives evaluated at the first and last knot points are
+     * approximated from a natural/unclamped spline that passes through the same
+     * set of points.
+     *
+     * @param x Arguments for the interpolation points.
+     * @param y Values for the interpolation points.
+     * @return the interpolating function.
+     * @throws DimensionMismatchException if {@code x} and {@code y} have different sizes.
+     * @throws NumberIsTooSmallException if the size of {@code x < 3}.
+     * @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException if {@code x} is not sorted in strict increasing order.
+     */
+    @Override
+    public PolynomialSplineFunction interpolate(final double[] x,
+                                                final double[] y) {
+        final SplineInterpolator spliner = new SplineInterpolator();
+        final PolynomialSplineFunction spline = spliner.interpolate(x, y);
+        final PolynomialSplineFunction derivativeSpline = spline.polynomialSplineDerivative();
+        final double fpStart = derivativeSpline.value(x[0]);
+        final double fpEnd = derivativeSpline.value(x[x.length - 1]);
+
+        return this.interpolate(x, y, fpStart, fpEnd);
+    }
+
+    /**
+     * Computes an interpolating function for the data set with defined
+     * boundary conditions.
+     *
+     * @param x Arguments for the interpolation points.
+     * @param y Values for the interpolation points.
+     * @param fpStart First derivative at the starting point of the returned
+     * spline function (starting slope).
+     * @param fpEnd First derivative at the ending point of the returned
+     * spline function (ending slope).
+     * @return the interpolating function.
+     * @throws DimensionMismatchException if {@code x} and {@code y} have different sizes.
+     * @throws NumberIsTooSmallException if the size of {@code x < 3}.
+     * @throws org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException if {@code x} is not sorted in strict increasing order.
+     */
+    public PolynomialSplineFunction interpolate(final double[] x,
+                                                final double[] y,
+                                                final double fpStart,
+                                                final double fpEnd) {
+        if (x.length != y.length) {
+            throw new DimensionMismatchException(x.length, y.length);
+        }
+
+        if (x.length < 3) {
+            throw new NumberIsTooSmallException(LocalizedFormats.NUMBER_OF_POINTS,
+                                                x.length, 3, true);
+        }
+
+        // Number of intervals.  The number of data points is n + 1.
+        final int n = x.length - 1;
+
+        MathArrays.checkOrder(x);
+
+        // Differences between knot points.
+        final double[] h = new double[n];
+        for (int i = 0; i < n; i++) {
+            h[i] = x[i + 1] - x[i];
+        }
+
+        final double[] mu = new double[n];
+        final double[] z = new double[n + 1];
+
+        final double alpha0 = 3 * ((y[1] - y[0]) / h[0] - fpStart);
+        final double alphaN = 3 * (fpEnd - (y[n] - y[n - 1]) / h[n - 1]);
+
+        mu[0] = 0.5;
+        final double ell0 = 2 * h[0];
+        z[0] = alpha0 / ell0;
+
+        for (int i = 1; i < n; i++) {
+            final double alpha = 3 * ((y[i + 1] - y[i]) / h[i] - (y[i] - y[i - 1]) / h[i - 1]);
+            final double ell = 2 * (x[i + 1] - x[i - 1]) - h[i - 1] * mu[i - 1];
+            mu[i] = h[i] / ell;
+            z[i] = (alpha - h[i - 1] * z[i - 1]) / ell;
+        }
+
+        // Cubic spline coefficients.
+        final double[] b = new double[n]; // Linear.
+        final double[] c = new double[n + 1]; // Quadratic.
+        final double[] d = new double[n]; // Cubic.
+
+        final double ellN = h[n - 1] * (2 - mu[n - 1]);
+        z[n] = (alphaN - h[n - 1] * z[n - 1]) / ellN;
+        c[n] = z[n];
+
+        for (int j = n - 1; j >= 0; j--) {
+            c[j] = z[j] - mu[j] * c[j + 1];
+            b[j] = ((y[j + 1] - y[j]) / h[j]) - h[j] * (c[j + 1] + 2 * c[j]) / 3;
+            d[j] = (c[j + 1] - c[j]) / (3 * h[j]);
+        }
+
+        final PolynomialFunction[] polynomials = new PolynomialFunction[n];
+        final double[] coefficients = new double[4];
+        for (int i = 0; i < n; i++) {
+            coefficients[0] = y[i];
+            coefficients[1] = b[i];
+            coefficients[2] = c[i];
+            coefficients[3] = d[i];
+            polynomials[i] = new PolynomialFunction(coefficients);
+        }
+
+        return new PolynomialSplineFunction(x, polynomials);
+    }
+}

src/changes/changes.xml

@@ -96,6 +96,9 @@ Caveat:
  to support the whole codebase (it was one of the main reasons for
  creating more focused components).
 ">
+      <action dev="erans" type="add" issue="MATH-1650" due-to="Michael Scholz">
+        New class: "ClampedSplineInterpolator".
+      </action>
       <action dev="aherbert" type="update" issue="MATH-1671">
         Update o.a.c.m.legacy.stat.descriptive package. Functionality has been
         partially transferred to Commons Statistics descriptive.