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added a transformer between Nordsieck form and multistep form 

for state history in multistep ODE integrators
(this will help implementing adaptive stepsize multistep integrators)

git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@763692 13f79535-47bb-0310-9956-ffa450edef68
This commit is contained in:
Luc Maisonobe 2009-04-09 15:23:11 +00:00
parent e94434768e
commit b6703d294c
2 changed files with 779 additions and 0 deletions
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/*
* 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);
}
}

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/*
* 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);
}
}