NewcombOperators.java
/* Copyright 2002-2016 CS Systèmes d'Information
* Licensed to CS Systèmes d'Information (CS) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
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*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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* See the License for the specific language governing permissions and
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*/
package org.orekit.propagation.semianalytical.dsst.utilities;
import java.util.ArrayList;
import java.util.List;
import java.util.Map;
import java.util.SortedMap;
import java.util.TreeMap;
import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;
import org.apache.commons.math3.util.FastMath;
/** Implementation of the Modified Newcomb Operators.
* <p>
* From equations 2.7.3 - (12)(13) of the Danielson paper, those operators are defined as:
* <pre>
* 4(ρ + σ)Y<sub>ρ,σ</sub><sup>n,s</sup> =
* 2(2s - n)Y<sub>ρ-1,σ</sub><sup>n,s+1</sup> + (s - n)Y<sub>ρ-2,σ</sub><sup>n,s+2</sup>
* - 2(2s + n)Y<sub>ρ,σ-1</sub><sup>n,s-1</sup> - (s+n)Y<sub>ρ,σ-2</sub><sup>n,s-2</sup>
* + 2(2ρ + 2σ + 2 + 3n)Y<sub>ρ-1,σ-1</sub><sup>n,s</sup>
* </pre>
* Initialization is given by : <pre>Y<sub>0,0</sub><sup>n,s</sup> = 1</pre>
* </p>
*
* Internally, the Modified Newcomb Operators are stored as an array of {@link PolynomialFunction} :
*
* <pre>
* Y<sub>ρ,σ</sub><sup>n,s</sup> = P<sub>k₀</sub> + P<sub>k₁</sub>n + ... + P<sub>k<sub>j</sub></sub>n<sup>j</sup>
* </pre>
*
* where the P<sub>k<sub>j</sub></sub> are given by
*
* <pre>
* P<sub>k<sub>j</sub></sub> = ∑<sub>j=0;ρ</sub> a<sub>j</sub>s<sup>j</sup>
* </pre>
*
* @author Romain Di Costanzo
* @author Pascal Parraud
*/
public class NewcombOperators {
/** Storage map. */
private static final Map<NewKey, Double> MAP = new TreeMap<NewKey, Double>();
/** Private constructor as class is a utility.
*/
private NewcombOperators() {
}
/** Get the Newcomb operator evaluated at n, s, ρ, σ.
* <p>
* This method is guaranteed to be thread-safe
* </p>
* @param rho ρ index
* @param sigma σ index
* @param n n index
* @param s s index
* @return Y<sub>ρ,σ</sub><sup>n,s</sup>
*/
public static double getValue(final int rho, final int sigma, final int n, final int s) {
final NewKey key = new NewKey(n, s, rho, sigma);
synchronized (MAP) {
if (MAP.containsKey(key)) {
return MAP.get(key);
}
}
// Get the Newcomb polynomials for the given rho and sigma
final List<PolynomialFunction> polynomials = PolynomialsGenerator.getPolynomials(rho, sigma);
// Compute the value from the list of polynomials for the given n and s
double nPower = 1.;
double value = 0.0;
for (final PolynomialFunction polynomial : polynomials) {
value += polynomial.value(s) * nPower;
nPower = n * nPower;
}
synchronized (MAP) {
MAP.put(key, value);
}
return value;
}
/** Generator for Newcomb polynomials. */
private static class PolynomialsGenerator {
/** Polynomials storage. */
private static final SortedMap<Couple, List<PolynomialFunction>> POLYNOMIALS =
new TreeMap<Couple, List<PolynomialFunction>>();
/** Private constructor as class is a utility.
*/
private PolynomialsGenerator() {
}
/** Get the list of polynomials representing the Newcomb Operator for the (ρ,σ) couple.
* <p>
* This method is guaranteed to be thread-safe
* </p>
* @param rho ρ value
* @param sigma σ value
* @return Polynomials representing the Newcomb Operator for the (ρ,σ) couple.
*/
private static List<PolynomialFunction> getPolynomials(final int rho, final int sigma) {
final Couple couple = new Couple(rho, sigma);
synchronized (POLYNOMIALS) {
if (POLYNOMIALS.isEmpty()) {
// Initialize lists
final List<PolynomialFunction> l00 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> l01 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> l10 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> l11 = new ArrayList<PolynomialFunction>();
// Y(rho = 0, sigma = 0) = 1
l00.add(new PolynomialFunction(new double[] {
1.
}));
// Y(rho = 0, sigma = 1) = -s - n/2
l01.add(new PolynomialFunction(new double[] {
0, -1.
}));
l01.add(new PolynomialFunction(new double[] {
-0.5
}));
// Y(rho = 1, sigma = 0) = s - n/2
l10.add(new PolynomialFunction(new double[] {
0, 1.
}));
l10.add(new PolynomialFunction(new double[] {
-0.5
}));
// Y(rho = 1, sigma = 1) = 3/2 - s² + 5n/4 + n²/4
l11.add(new PolynomialFunction(new double[] {
1.5, 0., -1.
}));
l11.add(new PolynomialFunction(new double[] {
1.25
}));
l11.add(new PolynomialFunction(new double[] {
0.25
}));
// Initialize polynomials
POLYNOMIALS.put(new Couple(0, 0), l00);
POLYNOMIALS.put(new Couple(0, 1), l01);
POLYNOMIALS.put(new Couple(1, 0), l10);
POLYNOMIALS.put(new Couple(1, 1), l11);
}
// If order hasn't been computed yet, update the Newcomb polynomials
if (!POLYNOMIALS.containsKey(couple)) {
PolynomialsGenerator.computeFor(rho, sigma);
}
return POLYNOMIALS.get(couple);
}
}
/** Compute the Modified Newcomb Operators up to a given (ρ, σ) couple.
* <p>
* The recursive computation uses equation 2.7.3-(12) of the Danielson paper.
* </p>
* @param rho ρ value to reach
* @param sigma σ value to reach
*/
private static void computeFor(final int rho, final int sigma) {
// Initialize result :
List<PolynomialFunction> result = new ArrayList<PolynomialFunction>();
// Get the coefficient from the recurrence relation
final Map<Integer, List<PolynomialFunction>> map = generateRecurrenceCoefficients(rho, sigma);
// Compute (s - n) * Y[rho - 2, sigma][n, s + 2]
if (rho >= 2) {
final List<PolynomialFunction> poly = map.get(0);
final List<PolynomialFunction> list = getPolynomials(rho - 2, sigma);
result = multiplyPolynomialList(poly, shiftList(list, 2));
}
// Compute 2(2rho + 2sigma + 2 + 3n) * Y[rho - 1, sigma - 1][n, s]
if (rho >= 1 && sigma >= 1) {
final List<PolynomialFunction> poly = map.get(1);
final List<PolynomialFunction> list = getPolynomials(rho - 1, sigma - 1);
result = sumPolynomialList(result, multiplyPolynomialList(poly, list));
}
// Compute 2(2s - n) * Y[rho - 1, sigma][n, s + 1]
if (rho >= 1) {
final List<PolynomialFunction> poly = map.get(2);
final List<PolynomialFunction> list = getPolynomials(rho - 1, sigma);
result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, 1)));
}
// Compute -(s + n) * Y[rho, sigma - 2][n, s - 2]
if (sigma >= 2) {
final List<PolynomialFunction> poly = map.get(3);
final List<PolynomialFunction> list = getPolynomials(rho, sigma - 2);
result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, -2)));
}
// Compute -2(2s + n) * Y[rho, sigma - 1][n, s - 1]
if (sigma >= 1) {
final List<PolynomialFunction> poly = map.get(4);
final List<PolynomialFunction> list = getPolynomials(rho, sigma - 1);
result = sumPolynomialList(result, multiplyPolynomialList(poly, shiftList(list, -1)));
}
// Save polynomials for current (rho, sigma) couple
final Couple couple = new Couple(rho, sigma);
POLYNOMIALS.put(couple, result);
}
/** Multiply two lists of polynomials defined as the internal representation of the Newcomb Operator.
* <p>
* Let's call R<sub>s</sub>(n) the result returned by the method :
* <pre>
* R<sub>s</sub>(n) = (P<sub>s₀</sub> + P<sub>s₁</sub>n + ... + P<sub>s<sub>j</sub></sub>n<sup>j</sup>) *(Q<sub>s₀</sub> + Q<sub>s₁</sub>n + ... + Q<sub>s<sub>k</sub></sub>n<sup>k</sup>)
* </pre>
*
* where the P<sub>s<sub>j</sub></sub> and Q<sub>s<sub>k</sub></sub> are polynomials in s,
* s being the index of the Y<sub>ρ,σ</sub><sup>n,s</sup> function
*
* @param poly1 first list of polynomials
* @param poly2 second list of polynomials
* @return R<sub>s</sub>(n) as a list of {@link PolynomialFunction}
*/
private static List<PolynomialFunction> multiplyPolynomialList(final List<PolynomialFunction> poly1,
final List<PolynomialFunction> poly2) {
// Initialize the result list of polynomial function
final List<PolynomialFunction> result = new ArrayList<PolynomialFunction>();
initializeListOfPolynomials(poly1.size() + poly2.size() - 1, result);
int i = 0;
// Iterate over first polynomial list
for (PolynomialFunction f1 : poly1) {
// Iterate over second polynomial list
for (int j = i; j < poly2.size() + i; j++) {
final PolynomialFunction p2 = poly2.get(j - i);
// Get previous polynomial for current 'n' order
final PolynomialFunction previousP2 = result.get(j);
// Replace the current order by summing the product of both of the polynomials
result.set(j, previousP2.add(f1.multiply(p2)));
}
// shift polynomial order in 'n'
i++;
}
return result;
}
/** Sum two lists of {@link PolynomialFunction}.
* @param poly1 first list
* @param poly2 second list
* @return the summed list
*/
private static List<PolynomialFunction> sumPolynomialList(final List<PolynomialFunction> poly1,
final List<PolynomialFunction> poly2) {
// identify the lowest degree polynomial
final int lowLength = FastMath.min(poly1.size(), poly2.size());
final int highLength = FastMath.max(poly1.size(), poly2.size());
// Initialize the result list of polynomial function
final List<PolynomialFunction> result = new ArrayList<PolynomialFunction>();
initializeListOfPolynomials(highLength, result);
for (int i = 0; i < lowLength; i++) {
// Add polynomials by increasing order of 'n'
result.set(i, poly1.get(i).add(poly2.get(i)));
}
// Complete the list if lists are of different size:
for (int i = lowLength; i < highLength; i++) {
if (poly1.size() < poly2.size()) {
result.set(i, poly2.get(i));
} else {
result.set(i, poly1.get(i));
}
}
return result;
}
/** Initialize an empty list of polynomials.
* @param i order
* @param result list into which polynomials should be added
*/
private static void initializeListOfPolynomials(final int i,
final List<PolynomialFunction> result) {
for (int k = 0; k < i; k++) {
result.add(new PolynomialFunction(new double[] {0.}));
}
}
/** Shift a list of {@link PolynomialFunction}.
*
* @param polynomialList list of {@link PolynomialFunction}
* @param shift shift value
* @return new list of shifted {@link PolynomialFunction}
*/
private static List<PolynomialFunction> shiftList(final List<PolynomialFunction> polynomialList,
final int shift) {
final List<PolynomialFunction> shiftedList = new ArrayList<PolynomialFunction>();
for (PolynomialFunction function : polynomialList) {
shiftedList.add(new PolynomialFunction(shift(function.getCoefficients(), shift)));
}
return shiftedList;
}
/**
* Compute the coefficients of the polynomial \(P_s(x)\)
* whose values at point {@code x} will be the same as the those from the
* original polynomial \(P(x)\) when computed at {@code x + shift}.
* <p>
* More precisely, let \(\Delta = \) {@code shift} and let
* \(P_s(x) = P(x + \Delta)\). The returned array
* consists of the coefficients of \(P_s\). So if \(a_0, ..., a_{n-1}\)
* are the coefficients of \(P\), then the returned array
* \(b_0, ..., b_{n-1}\) satisfies the identity
* \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\).
* </p>
* <p>
* This method is a modified version of the method with the same name
* in Apache Commons Math {@code PolynomialsUtils} class, simply changing
* computation of binomial coefficients so degrees higher than 66 can be used.
* </p>
*
* @param coefficients Coefficients of the original polynomial.
* @param shift Shift value.
* @return the coefficients \(b_i\) of the shifted
* polynomial.
*/
public static double[] shift(final double[] coefficients,
final double shift) {
final int dp1 = coefficients.length;
final double[] newCoefficients = new double[dp1];
// Pascal triangle.
final double[][] coeff = new double[dp1][dp1];
coeff[0][0] = 1;
for (int i = 1; i < dp1; i++) {
coeff[i][0] = 1;
for (int j = 1; j < i; j++) {
coeff[i][j] = coeff[i - 1][j - 1] + coeff[i - 1][j];
}
coeff[i][i] = 1;
}
// First polynomial coefficient.
double shiftI = 1;
for (int i = 0; i < dp1; i++) {
newCoefficients[0] += coefficients[i] * shiftI;
shiftI *= shift;
}
// Superior order.
final int d = dp1 - 1;
for (int i = 0; i < d; i++) {
double shiftJmI = 1;
for (int j = i; j < d; j++) {
newCoefficients[i + 1] += coeff[j + 1][j - i] * coefficients[j + 1] * shiftJmI;
shiftJmI *= shift;
}
}
return newCoefficients;
}
/** Generate recurrence coefficients.
*
* @param rho ρ value
* @param sigma σ value
* @return recurrence coefficients
*/
private static Map<Integer, List<PolynomialFunction>> generateRecurrenceCoefficients(final int rho, final int sigma) {
final double den = 1. / (4. * (rho + sigma));
final double denx2 = 2. * den;
final double denx4 = 4. * den;
// Initialization :
final Map<Integer, List<PolynomialFunction>> list = new TreeMap<Integer, List<PolynomialFunction>>();
final List<PolynomialFunction> poly0 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> poly1 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> poly2 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> poly3 = new ArrayList<PolynomialFunction>();
final List<PolynomialFunction> poly4 = new ArrayList<PolynomialFunction>();
// (s - n)
poly0.add(new PolynomialFunction(new double[] {0., den}));
poly0.add(new PolynomialFunction(new double[] {-den}));
// 2(2 * rho + 2 * sigma + 2 + 3*n)
poly1.add(new PolynomialFunction(new double[] {1. + denx4}));
poly1.add(new PolynomialFunction(new double[] {denx2 + denx4}));
// 2(2s - n)
poly2.add(new PolynomialFunction(new double[] {0., denx4}));
poly2.add(new PolynomialFunction(new double[] {-denx2}));
// - (s + n)
poly3.add(new PolynomialFunction(new double[] {0., -den}));
poly3.add(new PolynomialFunction(new double[] {-den}));
// - 2(2s + n)
poly4.add(new PolynomialFunction(new double[] {0., -denx4}));
poly4.add(new PolynomialFunction(new double[] {-denx2}));
// Fill the map :
list.put(0, poly0);
list.put(1, poly1);
list.put(2, poly2);
list.put(3, poly3);
list.put(4, poly4);
return list;
}
}
/** Private class to define a couple of value. */
private static class Couple implements Comparable<Couple> {
/** first couple value. */
private final int rho;
/** second couple value. */
private final int sigma;
/** Constructor.
* @param rho first couple value
* @param sigma second couple value
*/
private Couple(final int rho, final int sigma) {
this.rho = rho;
this.sigma = sigma;
}
/** {@inheritDoc} */
public int compareTo(final Couple c) {
int result = 1;
if (rho == c.rho) {
if (sigma < c.sigma) {
result = -1;
} else if (sigma == c.sigma) {
result = 0;
}
} else if (rho < c.rho) {
result = -1;
}
return result;
}
/** {@inheritDoc} */
public boolean equals(final Object couple) {
if (couple == this) {
// first fast check
return true;
}
if ((couple != null) && (couple instanceof Couple)) {
return (rho == ((Couple) couple).rho) && (sigma == ((Couple) couple).sigma);
}
return false;
}
/** {@inheritDoc} */
public int hashCode() {
return 0x7ab17c0c ^ (rho << 8) ^ sigma;
}
}
/** Newcomb operator's key. */
private static class NewKey implements Comparable<NewKey> {
/** n value. */
private final int n;
/** s value. */
private final int s;
/** ρ value. */
private final int rho;
/** σ value. */
private final int sigma;
/** Simpleconstructor.
* @param n n
* @param s s
* @param rho ρ
* @param sigma σ
*/
NewKey(final int n, final int s, final int rho, final int sigma) {
this.n = n;
this.s = s;
this.rho = rho;
this.sigma = sigma;
}
/** {@inheritDoc} */
public int compareTo(final NewKey key) {
int result = 1;
if (n == key.n) {
if (s == key.s) {
if (rho == key.rho) {
if (sigma < key.sigma) {
result = -1;
} else if (sigma == key.sigma) {
result = 0;
} else {
result = 1;
}
} else if (rho < key.rho) {
result = -1;
} else {
result = 1;
}
} else if (s < key.s) {
result = -1;
} else {
result = 1;
}
} else if (n < key.n) {
result = -1;
}
return result;
}
/** {@inheritDoc} */
public boolean equals(final Object key) {
if (key == this) {
// first fast check
return true;
}
if ((key != null) && (key instanceof NewKey)) {
return (n == ((NewKey) key).n) &&
(s == ((NewKey) key).s) &&
(rho == ((NewKey) key).rho) &&
(sigma == ((NewKey) key).sigma);
}
return false;
}
/** {@inheritDoc} */
public int hashCode() {
return 0x25baa451 ^ (n << 24) ^ (s << 16) ^ (rho << 8) ^ sigma;
}
}
}