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From l..@apache.org
Subject 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
Date Thu, 09 Apr 2009 15:23:12 GMT
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);
+      }
+
+}

Propchange: commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java
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Propchange: commons/proper/math/trunk/src/test/org/apache/commons/math/ode/NordsieckTransformerTest.java
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    svn:keywords = Author Date Id Revision



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