/*
* Copyright (c) 2012-2014, Peter Abeles. All Rights Reserved.
*
* This file is part of DDogleg (http://ddogleg.org).
*
* Licensed 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.ddogleg.solver.impl;
import org.ddogleg.solver.Polynomial;
import org.ddogleg.solver.PolynomialOps;
import org.junit.Test;
import java.util.ArrayList;
import java.util.List;
import java.util.Random;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertTrue;
/**
* @author Peter Abeles
*/
public class TestSturmSequence {
Random rand = new Random(234);
@Test
public void countRealRoots() {
Polynomial poly = Polynomial.wrap(-1,0,3,1);
assertEquals(1, countRealRoots(poly, -3, -2));
assertEquals(0, countRealRoots(poly, 2, 3));
assertEquals(3, countRealRoots(poly, -3, 3));
assertEquals(3, countRealRoots(poly, Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY));
// see if it handles small polynomials correctly
assertEquals(0, countRealRoots(Polynomial.wrap(2), Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY));
assertEquals(1, countRealRoots(Polynomial.wrap(2, 3), Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY));
assertEquals(0, countRealRoots(Polynomial.wrap(2, 3), 0, Double.POSITIVE_INFINITY));
assertEquals(0, countRealRoots(Polynomial.wrap(2, 3), Double.NEGATIVE_INFINITY, -10));
assertEquals(1, countRealRoots(Polynomial.wrap(2, 3), -2, -0.5));
assertEquals(0, countRealRoots(Polynomial.wrap(2, 3, 4), Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY));
assertEquals(2, countRealRoots(Polynomial.wrap(2, -1, -4), Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY));
// more difficult case where special code is required for infinite bounds
poly = Polynomial.wrap(-1.322309e+02 , 3.713984e+02 , -5.007874e+02 , 3.744386e+02 ,-1.714667e+02 ,
4.865014e+01 ,-1.059870e+01 , 1.642273e+00 ,-2.304341e-01,2.112391e-03,-2.273737e-13);
assertEquals(2, countRealRoots(poly, Double.NEGATIVE_INFINITY, Double.POSITIVE_INFINITY));
}
public int countRealRoots( Polynomial p , double low , double high ) {
SturmSequence alg = new SturmSequence(p.size);
alg.initialize(p);
return alg.countRealRoots(low,high);
}
/**
* Check sequence against a hand selected sequence
*/
@Test
public void checkSequence() {
Polynomial poly = Polynomial.wrap(-1,0,3,1,5,-3);
SturmSequence alg = new SturmSequence(10);
alg.initialize(poly);
alg.computeFunctions(-3);
List expected = computeSturm(poly,-3);
assertEquals(expected.size(),alg.sequenceLength);
for( int i = 0; i < expected.size(); i++ ) {
assertEquals(expected.get(i),alg.f[i],1e-8);
}
}
@Test
public void checkCount() {
SturmSequence alg = new SturmSequence(10);
alg.f = new double[]{0,1,1,-1,0,-1,-1,0,1,1};
alg.sequenceLength = alg.f.length;
assertEquals(2,alg.countSignChanges());
alg.f = new double[]{1,1,1,-1,0,-1,-1,0,1,1,0,-1,-1,1};
alg.sequenceLength = alg.f.length;
assertEquals(4,alg.countSignChanges());
}
/**
* Compare the computed sequence against randomly generated polynomials of different length
*/
@Test
public void checkSequence_Random() {
for( int i = 3; i < 20; i++ ) {
Polynomial p = new Polynomial(i);
for( int trial = 0; trial < 20; trial++ ) {
for( int j = 0; j < p.size; j++ ) {
p.c[j] = (rand.nextDouble()-0.5)*2;
}
double value = (rand.nextDouble()-0.5)*4;
compareSequences( p, value);
compareSequences( p, Double.POSITIVE_INFINITY);
compareSequences( p, Double.NEGATIVE_INFINITY);
}
}
}
/**
* Several consistency checks on random polynomials to the number of roots within a random interval
*/
@Test
public void rootCountConsistency_Random() {
for( int i = 3; i < 20; i++ ) {
Polynomial p = new Polynomial(i);
for( int trial = 0; trial < 20; trial++ ) {
for( int j = 0; j < p.size; j++ ) {
p.c[j] = (rand.nextDouble()-0.5)*2;
}
SturmSequence alg = new SturmSequence(p.size());
double low = (rand.nextDouble() - 0.5)*200;
double high = low + rand.nextDouble()*200;
double middle = (low+high)/2.0;
alg.initialize(p);
int every = alg.countRealRoots(Double.NEGATIVE_INFINITY,Double.POSITIVE_INFINITY);
int all = alg.countRealRoots(low,high);
int lowN = alg.countRealRoots(low,middle);
int highN = alg.countRealRoots(middle,high);
assertTrue(all >= 0);
assertTrue(lowN >= 0);
assertTrue(highN >= 0);
assertEquals(all, lowN + highN);
assertTrue(all <= every);
}
}
}
/**
* Examine a case which was found to cause problems
*/
@Test
public void checkSpecificPoly01() {
Polynomial poly = Polynomial.wrap(-41.118263303597175,-120.95384505825373,-417.8477600492497,-634.5308297409192,
-347.7885168491812,6.771313016808563,79.70258790927392,31.68212813610444,5.0248961592587875,
0.2879701466217739,0.0);
// Compare computed sequence to the standard
compareSequences( poly, -500);
compareSequences( poly, Double.NEGATIVE_INFINITY);
SturmSequence alg = new SturmSequence(poly.size);
alg.initialize(poly);
int N = alg.countRealRoots(Double.NEGATIVE_INFINITY,Double.POSITIVE_INFINITY);
int M = alg.countRealRoots(-500,500);
assertTrue(M <= N);
}
/**
* Examine a case which was found to cause problems. It would get stuck in a loop indefinitely
*/
@Test
public void checkSpecificPoly02() {
Polynomial poly = Polynomial.wrap(-0.3497655753671151,2.9621784756210587,-12.9131723419964320,
8.0038345403612960,31.0841473414231300,-90.1765840283034800,62.2630323196965500,
-17.4634213565573200,1.4562328842432635,0.1227493996590678,-0.0133645583045475);
FindRealRootsSturm sturm = new FindRealRootsSturm(11,-1,1e-10,20,20);
try {
sturm.process(poly);
} catch( RuntimeException ignore ) {}
}
// This test is commented out because the equation is so degenerate it would require a significant change to the Sturm
// sequence algorithm to work.
//
// @Test
// public void checkSpecificPoly02() {
// Polynomial poly = Polynomial.wrap(0.06496129844668003,-0.20388125146277708,-0.5346822141623102,3.8451325925247914,
// -8.125384551749551,7.281661653591961,-0.1827681555908356,-4.918274060516843,
// 3.6136415842421954,-0.8418091530846867,5.662137425588298E-15);
//
// // Compare computed sequence to the standard
// compareSequences( poly, 0.012207);
//
// SturmSequence alg = new SturmSequence(poly.size);
// alg.initialize(poly);
//
// double low = 0.01220703125;
// double high = 1.5743255615234375;
// double middle = (low+high)/2.0;
//
// int every = alg.countRealRoots(Double.NEGATIVE_INFINITY,Double.POSITIVE_INFINITY);
// int all = alg.countRealRoots(low,high);
// int lowN = alg.countRealRoots(low,middle);
// int highN = alg.countRealRoots(middle,high);
//
// RootFinderCompanion alt = new RootFinderCompanion();
// alt.process(poly);
//
// List l = alt.getRoots();
// for( Complex64F c : l ) {
// System.out.println(c);
// if( c.isReal() ) {
// System.out.println(" value = "+poly.evaluate(c.real));
// }
// }
//
// assertTrue(all >= 0);
// assertTrue(lowN>=0);
// assertTrue(highN>=0);
// assertEquals(all,lowN+highN);
// assertTrue(all <= every );
// }
private void compareSequences( Polynomial p, double value) {
List expected = computeSturm(p,value);
SturmSequence alg = new SturmSequence(p.size);
alg.initialize(p);
alg.computeFunctions(value);
assertEquals(expected.size(),alg.sequenceLength);
for( int j = 0; j < expected.size(); j++ ) {
if( Double.isInfinite(expected.get(j)) ) {
assertTrue(expected.get(j) == alg.f[j]);
} else {
assertEquals(expected.get(j),alg.f[j],Math.abs(alg.f[j])*1e-6);
}
}
}
/**
* Compute the sturm sequence using a straight forward method
*/
private List computeSturm( Polynomial poly , double x ) {
Polynomial d = new Polynomial(poly.size);
PolynomialOps.derivative(poly, d);
List found = new ArrayList();
found.add( poly.evaluate(x));
found.add( d.evaluate(x));
Polynomial q = new Polynomial(poly.size);
Polynomial r = new Polynomial(poly.size);
Polynomial p1 = new Polynomial(poly.size);
Polynomial p2 = new Polynomial(poly.size);
p1.setTo(poly);
p2.setTo(d);
do {
PolynomialOps.divide(p1,p2,q,r);
for( int i = 0; i < r.size; i++ ) {
r.c[i] = -r.c[i];
}
found.add(r.evaluate(x));
p1.setTo(p2);
p2.setTo(r);
} while( r.computeDegree() > 0 );
return found;
}
}
Ads help maintain this website.