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Posted to commits@commons.apache.org by lu...@apache.org on 2009/04/09 17:23:12 UTC
svn commit: r763692 - in /commons/proper/math/trunk/src:
java/org/apache/commons/math/ode/NordsieckTransformer.java
test/org/apache/commons/math/ode/NordsieckTransformerTest.java
Author: luc
Date: Thu Apr 9 15:23:11 2009
New Revision: 763692
URL: http://svn.apache.org/viewvc?rev=763692&view=rev
Log:
added a transformer between Nordsieck form and multistep form
for state history in multistep ODE integrators
(this will help implementing adaptive stepsize multistep integrators)
Added:
commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java (with props)
commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java (with props)
Added: commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java
URL: http://svn.apache.org/viewvc/commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java?rev=763692&view=auto
==============================================================================
--- commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java (added)
+++ commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java Thu Apr 9 15:23:11 2009
@@ -0,0 +1,511 @@
+/*
+ * 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.math.ode;
+
+import java.io.Serializable;
+import java.math.BigInteger;
+import java.util.Arrays;
+
+import org.apache.commons.math.fraction.BigFraction;
+import org.apache.commons.math.linear.RealMatrix;
+import org.apache.commons.math.linear.RealMatrixImpl;
+
+/**
+ * This class transforms state history between multistep (with or without
+ * derivatives) and Nordsieck forms.
+ * <p>
+ * {@link MultistepIntegrator multistep integrators} use state history
+ * from several previous steps to compute the current state. They may also use
+ * the first derivative of current state. All states are separated by a fixed
+ * step size h from each other. Since these methods are based on polynomial
+ * interpolation, the information from the previous state may be represented
+ * in another equivalent way: using the state higher order derivatives at
+ * current step rather. This class transforms state history between these three
+ * equivalent forms.
+ * <p>
+ * <p>
+ * The supported forms for a dimension n history are:
+ * <ul>
+ * <li>multistep without derivatives:<br/>
+ * <pre>
+ * y<sub>k</sub>, y<sub>k-1</sub> ... y<sub>k-(n-2), y<sub>k-(n-1)</sub>
+ * </pre>
+ * </li>
+ * <li>multistep with first derivative at current step:<br/>
+ * <pre>
+ * y<sub>k</sub>, y'<sub>k</sub>, y<sub>k-1</sub> ... y<sub>k-(n-2)</sub>
+ * </pre>
+ * </li>
+ * <li>Nordsieck:
+ * <pre>
+ * y<sub>k</sub>, h y'<sub>k</sub>, h<sup>2</sup>/2 y''<sub>k</sub> ... h<sup>n-1</sup>/(n-1)! yn-1<sub>k</sub>
+ * </pre>
+ * </li>
+ * </ul>
+ * In these expressions, y<sub>k</sub> is the state at the current step. For each p,
+ * y<sub>k-p</sub> is the state at the p<sup>th</sup> previous step. y'<sub>k</sub>,
+ * y''<sub>k</sub> ... yn-1<sub>k</sub> are respectively the first, second, ...
+ * (n-1)<sup>th</sup> derivatives of the state at current step and h is the fixed
+ * step size.
+ * </p>
+ * <p>
+ * The transforms are exact for polynomials.
+ * </p>
+ * <p>
+ * In Nordsieck form, the state history can be converted from step size h to step
+ * size h' by rescaling each component by 1, h'/h, (h'/h)<sup>2</sup> ...
+ * (h'/h)<sup>n-1</sup>.
+ * </p>
+ * <p>
+ * Instances of this class are guaranteed to be immutable.
+ * </p>
+ * @see org.apache.commons.math.ode.MultistepIntegrator
+ * @see org.apache.commons.math.ode.nonstiff.AdamsBashforthIntegrator
+ * @see org.apache.commons.math.ode.nonstiff.AdamsMoultonIntegrator
+ * @version $Revision$ $Date$
+ * @since 2.0
+ */
+public class NordsieckTransformer implements Serializable {
+
+ /** Serializable version identifier. */
+ private static final long serialVersionUID = -2707468304560314664L;
+
+ /** Nordsieck to Multistep without derivatives matrix. */
+ private final RealMatrix matNtoMWD;
+
+ /** Multistep without derivatives to Nordsieck matrix. */
+ private final RealMatrix matMWDtoN;
+
+ /** Nordsieck to Multistep matrix. */
+ private final RealMatrix matNtoM;
+
+ /** Multistep to Nordsieck matrix. */
+ private final RealMatrix matMtoN;
+
+ /**
+ * Build a transformer for a specified order.
+ * @param n dimension of the history
+ */
+ public NordsieckTransformer(final int n) {
+
+ // from Nordsieck to multistep without derivatives
+ final BigInteger[][] bigNtoMWD = buildNordsieckToMultistepWithoutDerivatives(n);
+ double[][] dataNtoMWD = new double[n][n];
+ for (int i = 0; i < n; ++i) {
+ double[] dRow = dataNtoMWD[i];
+ BigInteger[] bRow = bigNtoMWD[i];
+ for (int j = 0; j < n; ++j) {
+ dRow[j] = bRow[j].doubleValue();
+ }
+ }
+ matNtoMWD = new RealMatrixImpl(dataNtoMWD, false);
+
+ // from multistep without derivatives to Nordsieck
+ final BigFraction[][] bigToN = buildMultistepWithoutDerivativesToNordsieck(n);
+ double[][] dataMWDtoN = new double[n][n];
+ for (int i = 0; i < n; ++i) {
+ double[] dRow = dataMWDtoN[i];
+ BigFraction[] bRow = bigToN[i];
+ for (int j = 0; j < n; ++j) {
+ dRow[j] = bRow[j].doubleValue();
+ }
+ }
+ matMWDtoN = new RealMatrixImpl(dataMWDtoN, false);
+
+ // from Nordsieck to multistep
+ final BigInteger[][] bigNtoM = buildNordsieckToMultistep(n);
+ double[][] dataNtoM = new double[n][n];
+ for (int i = 0; i < n; ++i) {
+ double[] dRow = dataNtoM[i];
+ BigInteger[] bRow = bigNtoM[i];
+ for (int j = 0; j < n; ++j) {
+ dRow[j] = bRow[j].doubleValue();
+ }
+ }
+ matNtoM = new RealMatrixImpl(dataNtoM, false);
+
+ // from multistep to Nordsieck
+ convertMWDtNtoMtN(bigToN);
+ double[][] dataMtoN = new double[n][n];
+ for (int i = 0; i < n; ++i) {
+ double[] dRow = dataMtoN[i];
+ BigFraction[] bRow = bigToN[i];
+ for (int j = 0; j < n; ++j) {
+ dRow[j] = bRow[j].doubleValue();
+ }
+ }
+ matMtoN = new RealMatrixImpl(dataMtoN, false);
+
+ }
+
+ /**
+ * Build the transform from Nordsieck to multistep without derivatives.
+ * @param n dimension of the history
+ * @return transform from Nordsieck to multistep without derivatives
+ */
+ public static BigInteger[][] buildNordsieckToMultistepWithoutDerivatives(final int n) {
+
+ final BigInteger[][] array = new BigInteger[n][n];
+
+ // row 0: [1 0 0 0 ... 0 ]
+ array[0][0] = BigInteger.ONE;
+ Arrays.fill(array[0], 1, n, BigInteger.ZERO);
+
+ // the following expressions are direct applications of Taylor series
+ // rows 1 to n-1: aij = (-i)^j
+ // [ 1 -1 1 -1 1 ...]
+ // [ 1 -2 4 -8 16 ...]
+ // [ 1 -3 9 -27 81 ...]
+ // [ 1 -4 16 -64 256 ...]
+ for (int i = 1; i < n; ++i) {
+ final BigInteger[] row = array[i];
+ final BigInteger factor = BigInteger.valueOf(-i);
+ BigInteger aj = BigInteger.ONE;
+ for (int j = 0; j < n; ++j) {
+ row[j] = aj;
+ aj = aj.multiply(factor);
+ }
+ }
+
+ return array;
+
+ }
+
+ /**
+ * Build the transform from multistep without derivatives to Nordsieck.
+ * @param n dimension of the history
+ * @return transform from multistep without derivatives to Nordsieck
+ */
+ public static BigFraction[][] buildMultistepWithoutDerivativesToNordsieck(final int n) {
+
+ final BigInteger[][] iArray = new BigInteger[n][n];
+
+ // row 0: [1 0 0 0 ... 0 ]
+ iArray[0][0] = BigInteger.ONE;
+ Arrays.fill(iArray[0], 1, n, BigInteger.ZERO);
+
+ // We use recursive definitions of triangular integer series for each column.
+ // For example column 0 of matrices of increasing dimensions are:
+ // 1/0! for dimension 1
+ // 1/1!, 1/1! for dimension 2
+ // 2/2!, 3/2!, 1/2! for dimension 3
+ // 6/3!, 11/3!, 6/3!, 1/3! for dimension 4
+ // 24/4!, 50/4!, 35/4!, 10/4!, 1/4! for dimension 5
+ // The numerators are the Stirling numbers of the first kind, (A008275 in
+ // Sloane's encyclopedia http://www.research.att.com/~njas/sequences/A008275)
+ // with a multiplicative factor of +/-1 (which we will write +/-binomial(n-1, 0)).
+ // In the same way, column 1 is A049444 with a multiplicative factor of
+ // +/-binomial(n-1, 1) and column 2 is A123319 with a multiplicative factor of
+ // +/-binomial(n-1, 2). The next columns are defined by similar definitions but
+ // are not identified in Sloane's encyclopedia.
+ // Another interesting observation is that for each dimension k, the last column
+ // (except the initial 0) is a copy of the first column of the dimension k-1 matrix,
+ // possibly with an opposite sign (i.e. these columns are also linked to Stirling
+ // numbers of the first kind).
+ for (int i = 1; i < n; ++i) {
+
+ final BigInteger bigI = BigInteger.valueOf(i);
+
+ // row i
+ BigInteger[] rowK = iArray[i];
+ BigInteger[] rowKm1 = iArray[i - 1];
+ for (int j = 0; j < i; ++j) {
+ rowK[j] = BigInteger.ONE;
+ }
+ rowK[i] = rowKm1[0];
+
+ // rows i-1 to 1
+ for (int k = i - 1; k > 0; --k) {
+
+ // select rows
+ rowK = rowKm1;
+ rowKm1 = iArray[k - 1];
+
+ // apply recursive defining formula
+ for (int j = 0; j < i; ++j) {
+ rowK[j] = rowK[j].multiply(bigI).add(rowKm1[j]);
+ }
+
+ // initialize new last column
+ rowK[i] = rowKm1[0];
+
+ }
+ rowKm1[0] = rowKm1[0].multiply(bigI);
+
+ }
+
+ // apply column specific factors
+ final BigInteger factorial = iArray[0][0];
+ final BigFraction[][] fArray = new BigFraction[n][n];
+ for (int i = 0; i < n; ++i) {
+ final BigFraction[] fRow = fArray[i];
+ final BigInteger[] iRow = iArray[i];
+ BigInteger binomial = BigInteger.ONE;
+ for (int j = 0; j < n; ++j) {
+ fRow[j] = new BigFraction(binomial.multiply(iRow[j]), factorial);
+ binomial = binomial.negate().multiply(BigInteger.valueOf(n - j - 1)).divide(BigInteger.valueOf(j + 1));
+ }
+ }
+
+ return fArray;
+
+ }
+
+ /**
+ * Build the transform from Nordsieck to multistep.
+ * @param n dimension of the history
+ * @return transform from Nordsieck to multistep
+ */
+ public static BigInteger[][] buildNordsieckToMultistep(final int n) {
+
+ final BigInteger[][] array = new BigInteger[n][n];
+
+ // row 0: [1 0 0 0 ... 0 ]
+ array[0][0] = BigInteger.ONE;
+ Arrays.fill(array[0], 1, n, BigInteger.ZERO);
+
+ if (n > 1) {
+
+ // row 1: [0 1 0 0 ... 0 ]
+ array[1][0] = BigInteger.ZERO;
+ array[1][1] = BigInteger.ONE;
+ Arrays.fill(array[1], 2, n, BigInteger.ZERO);
+
+ // the following expressions are direct applications of Taylor series
+ // rows 2 to n-1: aij = (1-i)^j
+ // [ 1 -1 1 -1 1 ...]
+ // [ 1 -2 4 -8 16 ...]
+ // [ 1 -3 9 -27 81 ...]
+ // [ 1 -4 16 -64 256 ...]
+ for (int i = 2; i < n; ++i) {
+ final BigInteger[] row = array[i];
+ final BigInteger factor = BigInteger.valueOf(1 - i);
+ BigInteger aj = BigInteger.ONE;
+ for (int j = 0; j < n; ++j) {
+ row[j] = aj;
+ aj = aj.multiply(factor);
+ }
+ }
+
+ }
+
+ return array;
+
+ }
+
+ /**
+ * Build the transform from multistep to Nordsieck.
+ * @param n dimension of the history
+ * @return transform from multistep to Nordsieck
+ */
+ public static BigFraction[][] buildMultistepToNordsieck(final int n) {
+ final BigFraction[][] array = buildMultistepWithoutDerivativesToNordsieck(n);
+ convertMWDtNtoMtN(array);
+ return array;
+ }
+
+ /**
+ * Convert a transform from multistep without derivatives to Nordsieck to
+ * multistep to Nordsieck.
+ * @param work array, contains tansform from multistep without derivatives
+ * to Nordsieck on input, will be overwritten with tansform from multistep
+ * to Nordsieck on output
+ */
+ private static void convertMWDtNtoMtN(BigFraction[][] array) {
+
+ final int n = array.length;
+ if (n == 1) {
+ return;
+ }
+
+ // the second row of the matrix without derivatives represents the linear equation:
+ // hy' = a0 yk + a1 yk-1 + ... + a(n-1) yk-(n-1)
+ // we solve it with respect to the oldest state yk-(n-1) and get
+ // yk-(n-1) = -a0/a(n-1) yk + 1/a(n-1) hy' - a1/a(n-1) yk-1 - ...
+ final BigFraction[] secondRow = array[1];
+ final BigFraction[] solved = new BigFraction[n];
+ final BigFraction f = secondRow[n - 1].reciprocal().negate();
+ solved[0] = secondRow[0].multiply(f);
+ solved[1] = f.negate();
+ for (int j = 2; j < n; ++j) {
+ solved[j] = secondRow[j - 1].multiply(f);
+ }
+
+ // update the matrix so it expects hy' in second element
+ // rather than yk-(n-1) in last elements when post-multiplied
+ for (int i = 0; i < n; ++i) {
+ final BigFraction[] rowI = array[i];
+ final BigFraction last = rowI[n - 1];
+ for (int j = n - 1; j > 1; --j) {
+ rowI[j] = rowI[j - 1].add(last.multiply(solved[j]));
+ }
+ rowI[1] = last.multiply(solved[1]);
+ rowI[0] = rowI[0].add(last.multiply(solved[0]));
+ }
+
+ }
+
+ /**
+ * Transform a scalar state history from multistep form to Nordsieck form.
+ * <p>
+ * The input state history must be in multistep form with element 0 for
+ * current state, element 1 for current state scaled first derivative, element
+ * 2 for previous state ... element n-1 for (n-2)<sup>th</sup> previous state.
+ * The output state history will be in Nordsieck form with element 0 for
+ * current state, element 1 for current state scaled first derivative, element
+ * 2 for current state scaled second derivative ... element n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * </p>
+ * @param multistepHistory scalar state history in multistep form
+ * @return scalar state history in Nordsieck form
+ */
+ public double[] multistepToNordsieck(final double[] multistepHistory) {
+ return matMtoN.operate(multistepHistory);
+ }
+
+ /**
+ * Transform a vectorial state history from multistep form to Nordsieck form.
+ * <p>
+ * The input state history must be in multistep form with row 0 for
+ * current state, row 1 for current state scaled first derivative, row
+ * 2 for previous state ... row n-1 for (n-2)<sup>th</sup> previous state.
+ * The output state history will be in Nordsieck form with row 0 for
+ * current state, row 1 for current state scaled first derivative, row
+ * 2 for current state scaled second derivative ... row n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * </p>
+ * @param multistepHistory vectorial state history in multistep form
+ * @return vectorial state history in Nordsieck form
+ */
+ public RealMatrix multistepToNordsieck(final RealMatrix multistepHistory) {
+ return matMtoN.multiply(multistepHistory);
+ }
+
+ /**
+ * Transform a scalar state history from Nordsieck form to multistep form.
+ * <p>
+ * The input state history must be in Nordsieck form with element 0 for
+ * current state, element 1 for current state scaled first derivative, element
+ * 2 for current state scaled second derivative ... element n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * The output state history will be in multistep form with element 0 for
+ * current state, element 1 for current state scaled first derivative, element
+ * 2 for previous state ... element n-1 for (n-2)<sup>th</sup> previous state.
+ * </p>
+ * @param nordsieckHistory scalar state history in Nordsieck form
+ * @return scalar state history in multistep form
+ */
+ public double[] nordsieckToMultistep(final double[] nordsieckHistory) {
+ return matNtoM.operate(nordsieckHistory);
+ }
+
+ /**
+ * Transform a vectorial state history from Nordsieck form to multistep form.
+ * <p>
+ * The input state history must be in Nordsieck form with row 0 for
+ * current state, row 1 for current state scaled first derivative, row
+ * 2 for current state scaled second derivative ... row n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * The output state history will be in multistep form with row 0 for
+ * current state, row 1 for current state scaled first derivative, row
+ * 2 for previous state ... row n-1 for (n-2)<sup>th</sup> previous state.
+ * </p>
+ * @param nordsieckHistory vectorial state history in Nordsieck form
+ * @return vectorial state history in multistep form
+ */
+ public RealMatrix nordsieckToMultistep(final RealMatrix nordsieckHistory) {
+ return matNtoM.multiply(nordsieckHistory);
+ }
+
+ /**
+ * Transform a scalar state history from multistep without derivatives form
+ * to Nordsieck form.
+ * <p>
+ * The input state history must be in multistep without derivatives form with
+ * element 0 for current state, element 1 for previous state ... element n-1
+ * for (n-1)<sup>th</sup> previous state.
+ * The output state history will be in Nordsieck form with element 0 for
+ * current state, element 1 for current state scaled first derivative, element
+ * 2 for current state scaled second derivative ... element n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * </p>
+ * @param mwdHistory scalar state history in multistep without derivatives form
+ * @return scalar state history in Nordsieck form
+ */
+ public double[] multistepWithoutDerivativesToNordsieck(final double[] mwdHistory) {
+ return matMWDtoN.operate(mwdHistory);
+ }
+
+ /**
+ * Transform a vectorial state history from multistep without derivatives form
+ * to Nordsieck form.
+ * <p>
+ * The input state history must be in multistep without derivatives form with
+ * row 0 for current state, row 1 for previous state ... row n-1
+ * for (n-1)<sup>th</sup> previous state.
+ * The output state history will be in Nordsieck form with row 0 for
+ * current state, row 1 for current state scaled first derivative, row
+ * 2 for current state scaled second derivative ... row n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * </p>
+ * @param mwdHistory vectorial state history in multistep without derivatives form
+ * @return vectorial state history in Nordsieck form
+ */
+ public RealMatrix multistepWithoutDerivativesToNordsieck(final RealMatrix mwdHistory) {
+ return matMWDtoN.multiply(mwdHistory);
+ }
+
+ /**
+ * Transform a scalar state history from Nordsieck form to multistep without
+ * derivatives form.
+ * <p>
+ * The input state history must be in Nordsieck form with element 0 for
+ * current state, element 1 for current state scaled first derivative, element
+ * 2 for current state scaled second derivative ... element n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * The output state history will be in multistep without derivatives form with
+ * element 0 for current state, element 1 for previous state ... element n-1
+ * for (n-1)<sup>th</sup> previous state.
+ * </p>
+ * @param nordsieckHistory scalar state history in Nordsieck form
+ * @return scalar state history in multistep without derivatives form
+ */
+ public double[] nordsieckToMultistepWithoutDerivatives(final double[] nordsieckHistory) {
+ return matNtoMWD.operate(nordsieckHistory);
+ }
+
+ /**
+ * Transform a vectorial state history from Nordsieck form to multistep without
+ * derivatives form.
+ * <p>
+ * The input state history must be in Nordsieck form with row 0 for
+ * current state, row 1 for current state scaled first derivative, row
+ * 2 for current state scaled second derivative ... row n-1 for current state
+ * scaled (n-1)<sup>th</sup> derivative.
+ * The output state history will be in multistep without derivatives form with
+ * row 0 for current state, row 1 for previous state ... row n-1
+ * for (n-1)<sup>th</sup> previous state.
+ * </p>
+ * @param nordsieckHistory vectorial state history in Nordsieck form
+ * @return vectorial state history in multistep without derivatives form
+ */
+ public RealMatrix nordsieckToMultistepWithoutDerivatives(final RealMatrix nordsieckHistory) {
+ return matNtoMWD.multiply(nordsieckHistory);
+ }
+
+}
Propchange: commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java
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Propchange: commons/proper/math/trunk/src/java/org/apache/commons/math/ode/NordsieckTransformer.java
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svn:keywords = Author Date Id Revision
Added: commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java
URL: http://svn.apache.org/viewvc/commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java?rev=763692&view=auto
==============================================================================
--- commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java (added)
+++ commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java Thu Apr 9 15:23:11 2009
@@ -0,0 +1,268 @@
+/*
+ * 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.math.ode;
+
+import java.math.BigInteger;
+import java.util.Random;
+
+import junit.framework.Test;
+import junit.framework.TestCase;
+import junit.framework.TestSuite;
+
+import org.apache.commons.math.analysis.polynomials.PolynomialFunction;
+import org.apache.commons.math.fraction.BigFraction;
+import org.apache.commons.math.linear.RealMatrix;
+import org.apache.commons.math.linear.RealMatrixImpl;
+
+public class NordsieckTransformerTest
+extends TestCase {
+
+ public NordsieckTransformerTest(String name) {
+ super(name);
+ }
+
+ public void testDimension2() {
+ NordsieckTransformer transformer = new NordsieckTransformer(2);
+ double[] nordsieckHistory = new double[] { 1.0, 2.0 };
+ double[] mwdHistory = new double[] { 1.0, -1.0 };
+ double[] multistepHistory = new double[] { 1.0, 2.0 };
+ checkVector(nordsieckHistory, transformer.multistepWithoutDerivativesToNordsieck(mwdHistory));
+ checkVector(mwdHistory, transformer.nordsieckToMultistepWithoutDerivatives(nordsieckHistory));
+ checkVector(nordsieckHistory, transformer.multistepToNordsieck(multistepHistory));
+ checkVector(multistepHistory, transformer.nordsieckToMultistep(nordsieckHistory));
+ }
+
+ public void testDimension3() {
+ NordsieckTransformer transformer = new NordsieckTransformer(3);
+ double[] nordsieckHistory = new double[] { 1.0, 4.0, 18.0 };
+ double[] mwdHistory = new double[] { 1.0, 15.0, 65.0 };
+ double[] multistepHistory = new double[] { 1.0, 4.0, 15.0 };
+ checkVector(nordsieckHistory, transformer.multistepWithoutDerivativesToNordsieck(mwdHistory));
+ checkVector(mwdHistory, transformer.nordsieckToMultistepWithoutDerivatives(nordsieckHistory));
+ checkVector(nordsieckHistory, transformer.multistepToNordsieck(multistepHistory));
+ checkVector(multistepHistory, transformer.nordsieckToMultistep(nordsieckHistory));
+ }
+
+ public void testDimension7() {
+ NordsieckTransformer transformer = new NordsieckTransformer(7);
+ RealMatrix nordsieckHistory =
+ new RealMatrixImpl(new double[][] {
+ { 1, 2, 3 },
+ { -2, 1, 0 },
+ { 1, 1, 1 },
+ { 0, -1, 1 },
+ { 1, -1, 2 },
+ { 2, 0, 1 },
+ { 1, 1, 2 }
+ }, false);
+ RealMatrix mwdHistory =
+ new RealMatrixImpl(new double[][] {
+ { 1, 2, 3 },
+ { 4, 3, 6 },
+ { 25, 60, 127 },
+ { 340, 683, 1362 },
+ { 2329, 3918, 7635 },
+ { 10036, 15147, 29278 },
+ { 32449, 45608, 87951 }
+ }, false);
+ RealMatrix multistepHistory =
+ new RealMatrixImpl(new double[][] {
+ { 1, 2, 3 },
+ { -2, 1, 0 },
+ { 4, 3, 6 },
+ { 25, 60, 127 },
+ { 340, 683, 1362 },
+ { 2329, 3918, 7635 },
+ { 10036, 15147, 29278 }
+ }, false);
+
+ RealMatrix m = transformer.multistepWithoutDerivativesToNordsieck(mwdHistory);
+ assertEquals(0.0, m.subtract(nordsieckHistory).getNorm(), 1.0e-11);
+ m = transformer.nordsieckToMultistepWithoutDerivatives(nordsieckHistory);
+ assertEquals(0.0, m.subtract(mwdHistory).getNorm(), 1.0e-11);
+ m = transformer.multistepToNordsieck(multistepHistory);
+ assertEquals(0.0, m.subtract(nordsieckHistory).getNorm(), 1.0e-11);
+ m = transformer.nordsieckToMultistep(nordsieckHistory);
+ assertEquals(0.0, m.subtract(multistepHistory).getNorm(), 1.0e-11);
+
+ }
+
+ public void testInverseWithoutDerivatives() {
+ for (int n = 1; n < 20; ++n) {
+ BigInteger[][] nTom =
+ NordsieckTransformer.buildNordsieckToMultistepWithoutDerivatives(n);
+ BigFraction[][] mTon =
+ NordsieckTransformer.buildMultistepWithoutDerivativesToNordsieck(n);
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ BigFraction s = BigFraction.ZERO;
+ for (int k = 0; k < n; ++k) {
+ s = s.add(mTon[i][k].multiply(nTom[k][j]));
+ }
+ assertEquals((i == j) ? BigFraction.ONE : BigFraction.ZERO, s);
+ }
+ }
+ }
+ }
+
+ public void testInverse() {
+ for (int n = 1; n < 20; ++n) {
+ BigInteger[][] nTom =
+ NordsieckTransformer.buildNordsieckToMultistep(n);
+ BigFraction[][] mTon =
+ NordsieckTransformer.buildMultistepToNordsieck(n);
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ BigFraction s = BigFraction.ZERO;
+ for (int k = 0; k < n; ++k) {
+ s = s.add(mTon[i][k].multiply(nTom[k][j]));
+ }
+ assertEquals((i == j) ? BigFraction.ONE : BigFraction.ZERO, s);
+ }
+ }
+ }
+ }
+
+ public void testMatrices1() {
+ checkMatrix(1, new int[][] { { 1 } },
+ NordsieckTransformer.buildMultistepWithoutDerivativesToNordsieck(1));
+ checkMatrix(new int[][] { { 1 } },
+ NordsieckTransformer.buildNordsieckToMultistepWithoutDerivatives(1));
+ checkMatrix(1, new int[][] { { 1 } },
+ NordsieckTransformer.buildMultistepToNordsieck(1));
+ checkMatrix(new int[][] { { 1 } },
+ NordsieckTransformer.buildNordsieckToMultistep(1));
+ }
+
+ public void testMatrices2() {
+ checkMatrix(1, new int[][] { { 1, 0 }, { 1, -1 } },
+ NordsieckTransformer.buildMultistepWithoutDerivativesToNordsieck(2));
+ checkMatrix(new int[][] { { 1, 0 }, { 1, -1 } },
+ NordsieckTransformer.buildNordsieckToMultistepWithoutDerivatives(2));
+ checkMatrix(1, new int[][] { { 1, 0 }, { 0, 1 } },
+ NordsieckTransformer.buildMultistepToNordsieck(2));
+ checkMatrix(new int[][] { { 1, 0 }, { 0, 1 } },
+ NordsieckTransformer.buildNordsieckToMultistep(2));
+ }
+
+ public void testMatrices3() {
+ checkMatrix(2, new int[][] { { 2, 0, 0 }, { 3, -4, 1 }, { 1, -2, 1 } },
+ NordsieckTransformer.buildMultistepWithoutDerivativesToNordsieck(3));
+ checkMatrix(new int[][] { { 1, 0, 0 }, { 1, -1, 1 }, { 1, -2, 4 } },
+ NordsieckTransformer.buildNordsieckToMultistepWithoutDerivatives(3));
+ checkMatrix(1, new int[][] { { 1, 0, 0 }, { 0, 1, 0 }, { -1, 1, 1} },
+ NordsieckTransformer.buildMultistepToNordsieck(3));
+ checkMatrix(new int[][] { { 1, 0, 0 }, { 0, 1, 0 }, { 1, -1, 1 } },
+ NordsieckTransformer.buildNordsieckToMultistep(3));
+ }
+
+ public void testMatrices4() {
+ checkMatrix(6, new int[][] { { 6, 0, 0, 0 }, { 11, -18, 9, -2 }, { 6, -15, 12, -3 }, { 1, -3, 3, -1 } },
+ NordsieckTransformer.buildMultistepWithoutDerivativesToNordsieck(4));
+ checkMatrix(new int[][] { { 1, 0, 0, 0 }, { 1, -1, 1, -1 }, { 1, -2, 4, -8 }, { 1, -3, 9, -27 } },
+ NordsieckTransformer.buildNordsieckToMultistepWithoutDerivatives(4));
+ checkMatrix(4, new int[][] { { 4, 0, 0, 0 }, { 0, 4, 0, 0 }, { -7, 6, 8, -1 }, { -3, 2, 4, -1 } },
+ NordsieckTransformer.buildMultistepToNordsieck(4));
+ checkMatrix(new int[][] { { 1, 0, 0, 0 }, { 0, 1, 0, 0 }, { 1, -1, 1, -1 }, { 1, -2, 4, -8 } },
+ NordsieckTransformer.buildNordsieckToMultistep(4));
+ }
+
+ public void testPolynomial() {
+ Random r = new Random(1847222905841997856l);
+ for (int n = 2; n < 9; ++n) {
+
+ // build a polynomial and its derivatives
+ double[] coeffs = new double[n + 1];
+ for (int i = 0; i < n; ++i) {
+ coeffs[i] = 2 * r.nextDouble() - 1.0;
+ }
+ PolynomialFunction[] polynomials = new PolynomialFunction[n];
+ polynomials[0] = new PolynomialFunction(coeffs);
+ for (int k = 1; k < polynomials.length; ++k) {
+ polynomials[k] = (PolynomialFunction) polynomials[k - 1].derivative();
+ }
+ double h = 0.01;
+
+ // build a state history in multistep form
+ double[] multistepHistory = new double[n];
+ multistepHistory[0] = polynomials[0].value(1.0);
+ multistepHistory[1] = h * polynomials[1].value(1.0);
+ for (int i = 2; i < multistepHistory.length; ++i) {
+ multistepHistory[i] = polynomials[0].value(1.0 - (i - 1) * h);
+ }
+
+ // build the same state history in multistep without derivatives form
+ double[] mwdHistory = new double[n];
+ for (int i = 0; i < multistepHistory.length; ++i) {
+ mwdHistory[i] = polynomials[0].value(1.0 - i * h);
+ }
+
+ // build the same state history in Nordsieck form
+ double[] nordsieckHistory = new double[n];
+ double scale = 1.0;
+ for (int i = 0; i < nordsieckHistory.length; ++i) {
+ nordsieckHistory[i] = scale * polynomials[i].value(1.0);
+ scale *= h / (i + 1);
+ }
+
+ // check the transform is exact for these polynomials states
+ NordsieckTransformer transformer = new NordsieckTransformer(n);
+ checkVector(nordsieckHistory, transformer.multistepWithoutDerivativesToNordsieck(mwdHistory));
+ checkVector(mwdHistory, transformer.nordsieckToMultistepWithoutDerivatives(nordsieckHistory));
+ checkVector(nordsieckHistory, transformer.multistepToNordsieck(multistepHistory));
+ checkVector(multistepHistory, transformer.nordsieckToMultistep(nordsieckHistory));
+
+ }
+ }
+
+ private void checkVector(double[] reference, double[] candidate) {
+ assertEquals(reference.length, candidate.length);
+ for (int i = 0; i < reference.length; ++i) {
+ assertEquals(reference[i], candidate[i], 1.0e-14);
+ }
+ }
+
+ private void checkMatrix(int[][] reference, BigInteger[][] candidate) {
+ assertEquals(reference.length, candidate.length);
+ for (int i = 0; i < reference.length; ++i) {
+ int[] rRow = reference[i];
+ BigInteger[] cRow = candidate[i];
+ assertEquals(rRow.length, cRow.length);
+ for (int j = 0; j < rRow.length; ++j) {
+ assertEquals(rRow[j], cRow[j].intValue());
+ }
+ }
+ }
+
+ private void checkMatrix(int denominator, int[][] reference, BigFraction[][] candidate) {
+ assertEquals(reference.length, candidate.length);
+ for (int i = 0; i < reference.length; ++i) {
+ int[] rRow = reference[i];
+ BigFraction[] cRow = candidate[i];
+ assertEquals(rRow.length, cRow.length);
+ for (int j = 0; j < rRow.length; ++j) {
+ assertEquals(new BigFraction(rRow[j], denominator), cRow[j]);
+ }
+ }
+ }
+
+ public static Test suite() {
+ return new TestSuite(NordsieckTransformerTest.class);
+ }
+
+}
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Propchange: commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java
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