MATH-1650: Clamped spline interpolation (WIP).

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

@@ -0,0 +1,161 @@
+/*
+ * 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.NonMonotonicSequenceException;
+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 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 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 = 3d * (y[1] - y[0]) / h[0] - 3d * fpStart;
+        final double alphaN = 3d * fpEnd - 3d * (y[n] - y[n - 1]) / h[n - 1];
+
+        mu[0] = 0.5d;
+        final double ell0 = 2d * h[0];
+        z[0] = alpha0 / ell0;
+
+        for (int i = 1; i < n; i++) {
+            final double alpha = (3d / h[i]) * (y[i + 1] - y[i]) - (3d / h[i - 1]) * (y[i] - y[i - 1]);
+            final double ell = 2d * (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] * (2d - 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] + 2d * c[j]) / 3d;
+            d[j] = (c[j + 1] - c[j]) / (3d * 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);
+    }
+}

commons-math-legacy/src/test/java/org/apache/commons/math4/legacy/analysis/interpolation/ClampedSplineInterpolatorTest.java

@@ -0,0 +1,176 @@
+/*
+ * 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.TestUtils;
+import org.apache.commons.math4.legacy.analysis.UnivariateFunction;
+import org.apache.commons.math4.legacy.analysis.integration.SimpsonIntegrator;
+import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialFunction;
+import org.apache.commons.math4.legacy.analysis.polynomials.PolynomialSplineFunction;
+import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
+import org.apache.commons.math4.legacy.exception.NonMonotonicSequenceException;
+import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
+import org.junit.Assert;
+import org.junit.Test;
+
+/**
+ * Test the ClampedSplineInterpolator.
+ */
+public class ClampedSplineInterpolatorTest {
+    /** Error tolerance for spline interpolator value at knot points. */
+    private static final double KNOT_TOL = 1e-14;
+    /** Error tolerance for interpolating polynomial coefficients. */
+    private static final double COEF_TOL = 1e-14;
+
+    @Test
+    public void testInterpolateLinearDegenerateTwoSegment() {
+        final double[] x = {0, 0.5, 1};
+        final double[] y = {1, Math.exp(0.5), Math.exp(1)};
+        final double fpo = 1;
+        final double fpn = Math.exp(1);
+        final ClampedSplineInterpolator i = new ClampedSplineInterpolator();
+        final PolynomialSplineFunction f = i.interpolate(x, y, fpo, fpn);
+        verifyInterpolation(f, x, y);
+        verifyConsistency(f, x);
+
+        // Verify coefficients using analytical values
+        final PolynomialFunction[] polynomials = f.getPolynomials();
+
+        final double[] target0 = {1, 1, 0.4889506772539256, 0.21186881109317435};
+        final double[] target1 = {1.6487212707001282, 1.6478522855738063, 0.8067538938936871, 0.35156753198873575};
+        TestUtils.assertEquals(polynomials[0].getCoefficients(), target0, COEF_TOL);
+        TestUtils.assertEquals(polynomials[1].getCoefficients(), target1, COEF_TOL);
+    }
+
+    @Test
+    public void testInterpolateLinearDegenerateThreeSegment() {
+        final double[] x = {0, 1, 2, 3};
+        final double[] y = {1, Math.exp(1), Math.exp(2), Math.exp(3)};
+        final double fpo = 1;
+        final double fpn = Math.exp(3);
+        final ClampedSplineInterpolator i = new ClampedSplineInterpolator();
+        final PolynomialSplineFunction f = i.interpolate(x, y, fpo, fpn);
+        verifyInterpolation(f, x, y);
+        verifyConsistency(f, x);
+
+        // Verify coefficients using analytical values
+        final PolynomialFunction[] polynomials = f.getPolynomials();
+
+        final double[] target0 = {1, 0.9999999999999999, 0.4446824969658283, 0.27359933149321697};
+        final double[] target1 = {2.718281828459045, 2.710162988411307, 1.2654804914454791, 0.6951307906148195};
+        final double[] target2 = {7.38905609893065, 7.326516343146723, 3.3508728632899376, 2.019091617820356};
+        TestUtils.assertEquals(polynomials[0].getCoefficients(), target0, COEF_TOL);
+        TestUtils.assertEquals(polynomials[1].getCoefficients(), target1, COEF_TOL);
+        TestUtils.assertEquals(polynomials[2].getCoefficients(), target2, COEF_TOL);
+    }
+
+    @Test(expected=DimensionMismatchException.class)
+    public void testArrayLengthMismatch() {
+        // Data set arrays of different size.
+        new ClampedSplineInterpolator().interpolate(new double[] {1, 2, 3, 4},
+                                                    new double[] {2, 3, 5},
+                                                    2, 1);
+    }
+    @Test(expected=NonMonotonicSequenceException.class)
+    public void testUnsortedArray() {
+        // Knot values not sorted.
+        new ClampedSplineInterpolator().interpolate(new double[] {1, 3, 2 },
+                                                    new double[] {2, 3, 5},
+                                                    2, 1);
+    }
+    @Test(expected=NumberIsTooSmallException.class)
+    public void testInsufficientData() {
+        // Not enough data to interpolate.
+        new ClampedSplineInterpolator().interpolate(new double[] {1, 2 },
+                                                    new double[] {2, 3},
+                                                    2, 1);
+    }
+
+    /**
+     * Verifies that a clamped spline <i>without</i> specified boundary conditions behaves similar to a natural
+     * ("unclamped") spline.
+     *
+     * <p>
+     * Using the exponential function <code>e<sup>x</sup></code> over the interval <code>[0, 3]</code>, the test
+     * evaluates:
+     * <ol>
+     * <li>The integral of a clamped spline with specified boundary conditions (endpoint slopes/derivatives).</li>
+     * <li>The integral of a clamped spline without specified boundary conditions.</li>
+     * <li>The integral of a natural spline (without boundary conditions by default).</li>
+     * </ol>
+     *
+     * These integrals are compared against the direct integral of the function <code>e<sup>x</sup></code> over the same
+     * interval.</p>
+     *
+     * <p>
+     * This test is based on "Example 4" in R.L. Burden, J.D. Faires,
+     * <u>Numerical Analysis</u>, 9th Ed., 2010, Cengage Learning, ISBN 0-538-73351-9, pp 156-157.
+     * </p>
+     */
+    @Test
+    public void testIntegral() {
+        final double[] x = {0, 1, 2, 3};
+        final double[] y = {1, Math.exp(1), Math.exp(2), Math.exp(3)};
+        final double fpo = 1;
+        final double fpn = Math.exp(3);
+
+        final ClampedSplineInterpolator clampedSplineInterpolator = new ClampedSplineInterpolator();
+        final PolynomialSplineFunction clampedSpline = clampedSplineInterpolator.interpolate(x, y, fpo, fpn);
+        final PolynomialSplineFunction clampedSplineAsNaturalSpline = clampedSplineInterpolator.interpolate(x, y);
+
+        final SplineInterpolator naturalSplineInterpolator = new SplineInterpolator();
+        final PolynomialSplineFunction naturalSpline = naturalSplineInterpolator.interpolate(x, y);
+
+        final SimpsonIntegrator integrator = new SimpsonIntegrator();
+
+        final double clampedSplineIntegral = integrator.integrate(1000, clampedSpline, 0, 3);
+        final double clampedSplineAsNaturalSplineIntegral = integrator.integrate(1000, clampedSplineAsNaturalSpline, 0, 3);
+        final double naturalSplineIntegral = integrator.integrate(1000, naturalSpline, 0, 3);
+        final double exponentialFunctionIntegral = integrator.integrate(1000, (arg) -> Math.exp(arg), 0, 3);
+
+        Assert.assertEquals(Math.abs(clampedSplineAsNaturalSplineIntegral - naturalSplineIntegral), 0, 0);
+        Assert.assertEquals(Math.abs(exponentialFunctionIntegral - clampedSplineIntegral), 0.02589, 0.1);
+        Assert.assertEquals(Math.abs(exponentialFunctionIntegral - naturalSplineIntegral), 0.46675, 0.1);
+    }
+
+    /**
+     * Verifies that f(x[i]) = y[i] for i = 0, ..., n-1 (where n is common length).
+     */
+    private void verifyInterpolation(PolynomialSplineFunction f,
+                                     double[] x, double[] y) {
+        for (int i = 0; i < x.length; i++) {
+            Assert.assertEquals(f.value(x[i]), y[i], KNOT_TOL);
+        }
+    }
+
+    /**
+     * Verifies that interpolating polynomials satisfy consistency requirement: adjacent polynomials must agree through
+     * two derivatives at knot points.
+     */
+    private void verifyConsistency(PolynomialSplineFunction f,
+                                   double[] x) {
+        PolynomialFunction polynomials[] = f.getPolynomials();
+        for (int i = 1; i < x.length - 2; i++) {
+            // evaluate polynomials and derivatives at x[i + 1]
+            Assert.assertEquals(polynomials[i].value(x[i + 1] - x[i]), polynomials[i + 1].value(0), 0.1);
+            Assert.assertEquals(polynomials[i].polynomialDerivative().value(x[i + 1] - x[i]),
+                                polynomials[i + 1].polynomialDerivative().value(0), 0.5);
+            Assert.assertEquals(polynomials[i].polynomialDerivative().polynomialDerivative().value(x[i + 1] - x[i]),
+                                polynomials[i + 1].polynomialDerivative().polynomialDerivative().value(0), 0.5);
+        }
+    }
+}

pom.xml

@@ -1054,6 +1054,9 @@ This is avoided by creating an empty directory when svn is not available.
     <contributor>
       <name>Samy Badjoudj</name>
     </contributor>
+    <contributor>
+      <name>Michael Scholz</name>
+    </contributor>
   </contributors>
 
 </project>

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 feature: "ClampedSplineInterpolator".
+      </action>
       <action dev="erans" type="update" issue="MATH-1669" due-to="Wolff Bock von Wuelfingen">
         Javadoc: Fix broken link.
       </action>