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newspark110/lib/math/rs_math.cpp
Chenwenxuan edac2715f0 init
2024-03-06 14:54:30 +08:00

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/****************************************************************************
**
** This file is part of the LibreCAD project, a 2D CAD program
**
** Copyright (C) 2010 R. van Twisk (librecad@rvt.dds.nl)
** Copyright (C) 2001-2003 RibbonSoft. All rights reserved.
**
**
** This file may be distributed and/or modified under the terms of the
** GNU General Public License version 2 as published by the Free Software
** Foundation and appearing in the file gpl-2.0.txt included in the
** packaging of this file.
**
** This program is distributed in the hope that it will be useful,
** but WITHOUT ANY WARRANTY; without even the implied warranty of
** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
** GNU General Public License for more details.
**
** You should have received a copy of the GNU General Public License
** along with this program; if not, write to the Free Software
** Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
**
** This copyright notice MUST APPEAR in all copies of the script!
**
**********************************************************************/
#include <boost/numeric/ublas/matrix.hpp>
#include <boost/numeric/ublas/io.hpp>
#include <boost/numeric/ublas/lu.hpp>
#include <boost/math/special_functions/ellint_2.hpp>
#include <cmath>
#include <muParser.h>
#include <QString>
#include <QDebug>
#include "rs_math.h"
#include "rs_vector.h"
#include "rs_debug.h"
#ifdef EMU_C99
#include "emu_c99.h"
#endif
namespace {
constexpr double m_piX2 = M_PI*2; //2*PI
}
/**
* Rounds the given double to the closest int.
*/
int RS_Math::round(double v) {
return (int) lrint(v);
}
double RS_Math::round(const double v, const double precision)
{
return precision * lround(v / precision);
}
/**
* Save pow function
*/
double RS_Math::pow(double x, double y) {
errno = 0;
double ret = ::pow(x, y);
if (errno==EDOM) {
RS_DEBUG->print(RS_Debug::D_ERROR,
"RS_Math::pow: EDOM in pow");
ret = 0.0;
}
else if (errno==ERANGE) {
RS_DEBUG->print(RS_Debug::D_WARNING,
"RS_Math::pow: ERANGE in pow");
ret = 0.0;
}
return ret;
}
/* pow of vector components */
RS_Vector RS_Math::pow(RS_Vector vp, double y) {
return RS_Vector(pow(vp.x,y),pow(vp.y,y));
}
/**
* Save equal function for real types
*/
bool RS_Math::equal(const double d1, const double d2)
{
return fabs(d1 - d2) < RS_TOLERANCE;
}
/**
* Converts radians to degrees.
*/
double RS_Math::rad2deg(double a) {
return 180./M_PI*a;
}
/**
* Converts degrees to radians.
*/
double RS_Math::deg2rad(double a) {
return M_PI/180.0*a;
}
/**
* Converts radians to gradians.
*/
double RS_Math::rad2gra(double a) {
return 200./M_PI*a;
}
double RS_Math::gra2rad(double a) {
return M_PI/200.*a;
}
/**
* Finds greatest common divider using Euclid's algorithm.
*/
unsigned RS_Math::findGCD(unsigned a, unsigned b) {
while (b) {
unsigned rem = a % b;
a = b;
b = rem;
}
return a;
}
/**
* Tests if angle a is between a1 and a2. a, a1 and a2 must be in the
* range between 0 and 2*PI.
* All angles in rad.
*
* @param reversed true for clockwise testing. false for ccw testing.
* @return true if the angle a is between a1 and a2.
*/
bool RS_Math::isAngleBetween(double a,
double a1, double a2,
bool reversed) {
if (reversed) std::swap(a1,a2);
if(getAngleDifferenceU(a2, a1 ) < RS_TOLERANCE_ANGLE) return true;
const double tol=0.5*RS_TOLERANCE_ANGLE;
const double diff0=correctAngle(a2 -a1) + tol;
return diff0 >= correctAngle(a - a1) || diff0 >= correctAngle(a2 - a);
}
/**
* Corrects the given angle to the range of 0 to +PI*2.0.
*/
double RS_Math::correctAngle(double a) {
return fmod(M_PI + remainder(a - M_PI, m_piX2), m_piX2);
}
/**
* Corrects the given angle to the range of -PI to +PI.
*/
double RS_Math::correctAngle2(double a) {
return remainder(a, m_piX2);
}
/**
* Returns the given angle as an Unsigned Angle in the range of 0 to +PI.
*/
double RS_Math::correctAngleU(double a) {
return fabs(remainder(a, m_piX2));
}
/**
* @return The angle that needs to be added to a1 to reach a2.
* Always positive and less than 2*pi.
*/
double RS_Math::getAngleDifference(double a1, double a2, bool reversed) {
if(reversed) std::swap(a1, a2);
return correctAngle(a2 - a1);
}
double RS_Math::getAngleDifferenceU(double a1, double a2)
{
return correctAngleU(a1 - a2);
}
/**
* Makes a text constructed with the given angle readable. Used
* for dimension texts and for mirroring texts.
*
* @param readable true: make angle readable, false: unreadable
* @param corrected Will point to true if the given angle was
* corrected, false otherwise.
*
* @return The given angle or the given angle+PI, depending which on
* is readable from the bottom or right.
*/
double RS_Math::makeAngleReadable(double angle, bool readable,
bool* corrected) {
double ret=correctAngle(angle);
bool cor = isAngleReadable(ret) ^ readable;
// quadrant 1 & 4
if (cor) {
// ret = angle;
// }
// quadrant 2 & 3
// else {
ret = correctAngle(angle+M_PI);
}
if (corrected) {
*corrected = cor;
}
return ret;
}
/**
* @return true: if the given angle is in a range that is readable
* for texts created with that angle.
*/
bool RS_Math::isAngleReadable(double angle) {
const double tolerance=0.001;
if (angle>M_PI_2)
return fabs(remainder(angle, m_piX2)) < (M_PI_2 - tolerance);
else
return fabs(remainder(angle, m_piX2)) < (M_PI_2 + tolerance);
}
/**
* @param tol Tolerance in rad.
* @retval true The two angles point in the same direction.
*/
bool RS_Math::isSameDirection(double dir1, double dir2, double tol) {
return getAngleDifferenceU(dir1, dir2) < tol;
}
/**
* Evaluates a mathematical expression and returns the result.
* If an error occurred, the given default value 'def' will be returned.
*/
double RS_Math::eval(const QString& expr, double def) {
bool ok;
double res = RS_Math::eval(expr, &ok);
if (!ok) {
//std::cerr << "RS_Math::evaluate: Parse error at col "
//<< ret << ": " << fp.ErrorMsg() << "\n";
return def;
}
return res;
}
/**
* Evaluates a mathematical expression and returns the result.
* If an error occurred, ok will be set to false (if ok isn't NULL).
*/
double RS_Math::eval(const QString& expr, bool* ok) {
bool okTmp(false);
if(!ok) ok=&okTmp;
if (expr.isEmpty()) {
*ok = false;
return 0.0;
}
double ret(0.);
try{
mu::Parser p;
p.DefineConst(_T("pi"),M_PI);
#ifdef _UNICODE
p.SetExpr(expr.toStdWString());
#else
p.SetExpr(expr.toStdString());
#endif
ret=p.Eval();
*ok=true;
}
catch (mu::Parser::exception_type &e)
{
mu::console() << e.GetMsg() << std::endl;
*ok=false;
}
return ret;
}
/**
* Converts a double into a string which is as short as possible
*
* @param value The double value
* @param prec Precision e.g. a precision of 1 would mean that a
* value of 2.12030 will be converted to "2.1". 2.000 is always just "2").
*/
QString RS_Math::doubleToString(double value, double prec) {
if (prec< RS_TOLERANCE ) {
RS_DEBUG->print(RS_Debug::D_ERROR,
"RS_Math::doubleToString: invalid precision");
return QString().setNum(value, prec);
}
double const num = RS_Math::round(value / prec)*prec;
QString exaStr = RS_Math::doubleToString(1./prec, 10);
int const dotPos = exaStr.indexOf('.');
if (dotPos==-1) {
//big numbers for the precision
return QString().setNum(RS_Math::round(num));
} else {
//number of digits after the point
int digits = dotPos - 1;
return RS_Math::doubleToString(num, digits);
}
}
/**
* Converts a double into a string which is as short as possible.
*
* @param value The double value
* @param prec Precision
*/
QString RS_Math::doubleToString(double value, int prec) {
QString valStr;
valStr.setNum(value, 'f', prec);
if(valStr.contains('.')) {
// Remove tailing point and zeros:
// valStr.replace(QRegExp("0*$"), "");
// valStr.replace(QRegExp(R"(\.$)"), "");
// while (valStr.at(valStr.length()-1)=='0') {
// valStr.truncate(valStr.length()-1);
// }
if(valStr.at(valStr.length()-1)=='.') {
valStr.truncate(valStr.length()-1);
}
}
return valStr;
}
/**
* Performs some testing for the math class.
*/
void RS_Math::test() {
{
std::cout<<"testing quadratic solver"<<std::endl;
//equations x^2 + v[0] x + v[1] = 0
std::vector<std::vector<double>> const eqns{
{-1., -1.},
{-101., -1.},
{-1., -100.},
{2., 1.},
{-2., 1.}
};
//expected roots
std::vector<std::vector<double>> roots{
{-0.6180339887498948, 1.6180339887498948},
{-0.0099000196991084878, 101.009900019699108},
{-9.5124921972503929, 10.5124921972503929},
{-1.},
{1.}
};
for(size_t i=0; i < eqns.size(); i++) {
std::cout<<"Test quadratic solver, test case: x^2 + ("
<<eqns[i].front()<<") x + ("
<<eqns[i].back()<<") = 0"<<std::endl;
auto sol = quadraticSolver(eqns[i]);
assert(sol.size()==roots[i].size());
if (sol.front() > sol.back())
std::swap(sol[0], sol[1]);
auto expected=roots[i];
if (expected.front() > expected.back())
std::swap(expected[0], expected[1]);
for (size_t j=0; j < sol.size(); j++) {
double x0 = sol[j];
double x1 = expected[j];
double const prec = (x0 - x1)/(fabs(x0 + x1) + RS_TOLERANCE2);
std::cout<<"root "<<j<<" : precision level = "<<prec<<std::endl;
std::cout<<std::setprecision(17)<<"found: "<<x0<<"\texpected: "<<x1<<std::endl;
assert(prec < RS_TOLERANCE);
}
std::cout<<std::endl;
}
return;
}
QString s;
double v;
std::cout << "RS_Math::test: doubleToString:\n";
v = 0.1;
s = RS_Math::doubleToString(v, 0.1);
assert(s=="0.1");
s = RS_Math::doubleToString(v, 0.01);
assert(s=="0.10");
v = 0.01;
s = RS_Math::doubleToString(v, 0.1);
assert(s=="0.0");
s = RS_Math::doubleToString(v, 0.01);
assert(s=="0.01");
s = RS_Math::doubleToString(v, 0.001);
assert(s=="0.010");
v = 0.001;
s = RS_Math::doubleToString(v, 0.1);
assert(s=="0.0");
s = RS_Math::doubleToString(v, 0.01);
assert(s=="0.00");
s = RS_Math::doubleToString(v, 0.001);
assert(s=="0.001");
std::cout << "RS_Math::test: complete"<<std::endl;
}
//Equation solvers
// quadratic, cubic, and quartic equation solver
// @ ce[] contains coefficient of the cubic equation:
// @ returns a vector contains real roots
//
// solvers assume arguments are valid, and there's no attempt to verify validity of the argument pointers
//
// @author Dongxu Li <dongxuli2011@gmail.com>
std::vector<double> RS_Math::quadraticSolver(const std::vector<double>& ce)
//quadratic solver for
// x^2 + ce[0] x + ce[1] =0
{
std::vector<double> ans(0,0.);
if (ce.size() != 2) return ans;
using LDouble = long double;
LDouble const b = -0.5L * ce[0];
LDouble const c = ce[1];
// x^2 -2 b x + c=0
// (x - b)^2 = b^2 - c
// b^2 >= fabs(c)
// x = b \pm b sqrt(1. - c/(b^2))
LDouble const b2= b * b;
LDouble const discriminant= b2 - c;
LDouble const fc = std::abs(c);
//TODO, fine tune to tolerance level
LDouble const TOL = 1e-24L;
if (discriminant < 0.L)
//negative discriminant, no real root
return ans;
//find the radical
LDouble r;
// given |p| >= |q|
// sqrt(p^2 \pm q^2) = p sqrt(1 \pm q^2/p^2)
if (b2 >= fc)
r = std::abs(b) * std::sqrt(1.L - c/b2);
else
// c is negative, because b2 - c is non-negative
r = std::sqrt(fc) * std::sqrt(1.L + b2/fc);
if (r >= TOL*std::abs(b)) {
//two roots
if (b >= 0.L)
//since both (b,r)>=0, avoid (b - r) loss of significance
ans.push_back(b + r);
else
//since b<0, r>=0, avoid (b + r) loss of significance
ans.push_back(b - r);
//Vieta's formulas for the second root
ans.push_back(c/ans.front());
} else
//multiple roots
ans.push_back(b);
return ans;
}
std::vector<double> RS_Math::cubicSolver(const std::vector<double>& ce)
//cubic equation solver
// x^3 + ce[0] x^2 + ce[1] x + ce[2] = 0
{
// std::cout<<"x^3 + ("<<ce[0]<<")*x^2+("<<ce[1]<<")*x+("<<ce[2]<<")==0"<<std::endl;
std::vector<double> ans(0,0.);
if (ce.size() != 3) return ans;
// depressed cubic, Tschirnhaus transformation, x= t - b/(3a)
// t^3 + p t +q =0
double shift=(1./3)*ce[0];
double p=ce[1] -shift*ce[0];
double q=ce[0]*( (2./27)*ce[0]*ce[0]-(1./3)*ce[1])+ce[2];
//Cardano's method,
// t=u+v
// u^3 + v^3 + ( 3 uv + p ) (u+v) + q =0
// select 3uv + p =0, then,
// u^3 + v^3 = -q
// u^3 v^3 = - p^3/27
// so, u^3 and v^3 are roots of equation,
// z^2 + q z - p^3/27 = 0
// and u^3,v^3 are,
// -q/2 \pm sqrt(q^2/4 + p^3/27)
// discriminant= q^2/4 + p^3/27
//std::cout<<"p="<<p<<"\tq="<<q<<std::endl;
double discriminant= (1./27)*p*p*p+(1./4)*q*q;
if ( fabs(p)< 1.0e-75) {
ans.push_back((q>0)?-pow(q,(1./3)):pow(-q,(1./3)));
ans[0] -= shift;
// DEBUG_HEADER
// std::cout<<"cubic: one root: "<<ans[0]<<std::endl;
return ans;
}
//std::cout<<"discriminant="<<discriminant<<std::endl;
if(discriminant>0) {
std::vector<double> ce2(2,0.);
ce2[0]=q;
ce2[1]=-1./27*p*p*p;
auto r=quadraticSolver(ce2);
if ( r.size()==0 ) { //should not happen
std::cerr<<__FILE__<<" : "<<__func__<<" : line"<<__LINE__<<" :cubicSolver()::Error cubicSolver("<<ce[0]<<' '<<ce[1]<<' '<<ce[2]<<")\n";
}
double u,v;
u= (q<=0) ? pow(r[0], 1./3): -pow(-r[1],1./3);
//u=(q<=0)?pow(-0.5*q+sqrt(discriminant),1./3):-pow(0.5*q+sqrt(discriminant),1./3);
v=(-1./3)*p/u;
//std::cout<<"u="<<u<<"\tv="<<v<<std::endl;
//std::cout<<"u^3="<<u*u*u<<"\tv^3="<<v*v*v<<std::endl;
ans.push_back(u+v - shift);
// DEBUG_HEADER
// std::cout<<"cubic: one root: "<<ans[0]<<std::endl;
}else{
std::complex<double> u(q,0),rt[3];
u=std::pow(-0.5*u-sqrt(0.25*u*u+p*p*p/27),1./3);
rt[0]=u-p/(3.*u)-shift;
std::complex<double> w(-0.5,sqrt(3.)/2);
rt[1]=u*w-p/(3.*u*w)-shift;
rt[2]=u/w-p*w/(3.*u)-shift;
// DEBUG_HEADER
// std::cout<<"Roots:\n";
// std::cout<<rt[0]<<std::endl;
// std::cout<<rt[1]<<std::endl;
// std::cout<<rt[2]<<std::endl;
ans.push_back(rt[0].real());
ans.push_back(rt[1].real());
ans.push_back(rt[2].real());
}
// newton-raphson
for(double& x0: ans){
double dx=0.;
for(size_t i=0; i<20; ++i){
double f=( (x0 + ce[0])*x0 + ce[1])*x0 +ce[2];
double df=(3.*x0+2.*ce[0])*x0 +ce[1];
if(fabs(df)>fabs(f)+RS_TOLERANCE){
dx=f/df;
x0 -= dx;
}else
break;
}
}
return ans;
}
/** quartic solver
* x^4 + ce[0] x^3 + ce[1] x^2 + ce[2] x + ce[3] = 0
@ce, a vector of size 4 contains the coefficient in order
@return, a vector contains real roots
**/
std::vector<double> RS_Math::quarticSolver(const std::vector<double>& ce)
{
std::vector<double> ans(0,0.);
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<"expected array size=4, got "<<ce.size()<<std::endl;
}
if(ce.size() != 4) return ans;
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
std::cout<<"x^4+("<<ce[0]<<")*x^3+("<<ce[1]<<")*x^2+("<<ce[2]<<")*x+("<<ce[3]<<")==0"<<std::endl;
}
// x^4 + a x^3 + b x^2 +c x + d = 0
// depressed quartic, x= t - a/4
// t^4 + ( b - 3/8 a^2 ) t^2 + (c - a b/2 + a^3/8) t + d - a c /4 + a^2 b/16 - 3 a^4/256 =0
// t^4 + p t^2 + q t + r =0
// p= b - (3./8)*a*a;
// q= c - 0.5*a*b+(1./8)*a*a*a;
// r= d - 0.25*a*c+(1./16)*a*a*b-(3./256)*a^4
double shift=0.25*ce[0];
double shift2=shift*shift;
double a2=ce[0]*ce[0];
double p= ce[1] - (3./8)*a2;
double q= ce[2] + ce[0]*((1./8)*a2 - 0.5*ce[1]);
double r= ce[3] - shift*ce[2] + (ce[1] - 3.*shift2)*shift2;
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<"x^4+("<<p<<")*x^2+("<<q<<")*x+("<<r<<")==0"<<std::endl;
}
if (q*q <= 1.e-4*RS_TOLERANCE*fabs(p*r)) {// Biquadratic equations
double discriminant= 0.25*p*p -r;
if (discriminant < -1.e3*RS_TOLERANCE) {
// DEBUG_HEADER
// std::cout<<"discriminant="<<discriminant<<"\tno root"<<std::endl;
return ans;
}
double t2[2];
t2[0]=-0.5*p-sqrt(fabs(discriminant));
t2[1]= -p - t2[0];
// std::cout<<"t2[0]="<<t2[0]<<std::endl;
// std::cout<<"t2[1]="<<t2[1]<<std::endl;
if ( t2[1] >= 0.) { // two real roots
ans.push_back(sqrt(t2[1])-shift);
ans.push_back(-sqrt(t2[1])-shift);
}
if ( t2[0] >= 0. ) {// four real roots
ans.push_back(sqrt(t2[0])-shift);
ans.push_back(-sqrt(t2[0])-shift);
}
// DEBUG_HEADER
// for(int i=0;i<ans.size();i++){
// std::cout<<"root x: "<<ans[i]<<std::endl;
// }
return ans;
}
if ( fabs(r)< 1.0e-75 ) {
std::vector<double> cubic(3,0.);
cubic[1]=p;
cubic[2]=q;
ans.push_back(0.);
auto r=cubicSolver(cubic);
std::copy(r.begin(),r.end(), std::back_inserter(ans));
for(size_t i=0; i<ans.size(); i++) ans[i] -= shift;
return ans;
}
// depressed quartic to two quadratic equations
// t^4 + p t^2 + q t + r = ( t^2 + u t + v) ( t^2 - u t + w)
// so,
// p + u^2= w+v
// q/u= w-v
// r= wv
// so,
// (p+u^2)^2 - (q/u)^2 = 4 r
// y=u^2,
// y^3 + 2 p y^2 + ( p^2 - 4 r) y - q^2 =0
//
std::vector<double> cubic(3,0.);
cubic[0]=2.*p;
cubic[1]=p*p-4.*r;
cubic[2]=-q*q;
auto r3= cubicSolver(cubic);
//std::cout<<"quartic_solver:: real roots from cubic: "<<ret<<std::endl;
//for(unsigned int i=0; i<ret; i++)
// std::cout<<"cubic["<<i<<"]="<<cubic[i]<<" x= "<<croots[i]<<std::endl;
//newton-raphson
if (r3.size()==1) { //one real root from cubic
if (r3[0]< 0.) {//this should not happen
DEBUG_HEADER
qDebug()<<"Quartic Error:: Found one real root for cubic, but negative\n";
return ans;
}
double sqrtz0=sqrt(r3[0]);
std::vector<double> ce2(2,0.);
ce2[0]= -sqrtz0;
ce2[1]=0.5*(p+r3[0])+0.5*q/sqrtz0;
auto r1=quadraticSolver(ce2);
if (r1.size()==0 ) {
ce2[0]= sqrtz0;
ce2[1]=0.5*(p+r3[0])-0.5*q/sqrtz0;
r1=quadraticSolver(ce2);
}
for(auto& x: r1){
x -= shift;
}
return r1;
}
if ( r3[0]> 0. && r3[1] > 0. ) {
double sqrtz0=sqrt(r3[0]);
std::vector<double> ce2(2,0.);
ce2[0]= -sqrtz0;
ce2[1]=0.5*(p+r3[0])+0.5*q/sqrtz0;
ans=quadraticSolver(ce2);
ce2[0]= sqrtz0;
ce2[1]=0.5*(p+r3[0])-0.5*q/sqrtz0;
auto r1=quadraticSolver(ce2);
std::copy(r1.begin(),r1.end(),std::back_inserter(ans));
for(auto& x: ans){
x -= shift;
}
}
// newton-raphson
for(double& x0: ans){
double dx=0.;
for(size_t i=0; i<20; ++i){
double f=(( (x0 + ce[0])*x0 + ce[1])*x0 +ce[2])*x0 + ce[3] ;
double df=((4.*x0+3.*ce[0])*x0 +2.*ce[1])*x0+ce[2];
// DEBUG_HEADER
// qDebug()<<"i="<<i<<"\tx0="<<x0<<"\tf="<<f<<"\tdf="<<df;
if(fabs(df)>RS_TOLERANCE2){
dx=f/df;
x0 -= dx;
}else
break;
}
}
return ans;
}
/** quartic solver
* ce[4] x^4 + ce[3] x^3 + ce[2] x^2 + ce[1] x + ce[0] = 0
@ce, a vector of size 5 contains the coefficient in order
@return, a vector contains real roots
*ToDo, need a robust algorithm to locate zero terms, better handling of tolerances
**/
std::vector<double> RS_Math::quarticSolverFull(const std::vector<double>& ce)
{
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<ce[4]<<"*y^4+("<<ce[3]<<")*y^3+("<<ce[2]<<"*y^2+("<<ce[1]<<")*y+("<<ce[0]<<")==0"<<std::endl;
}
std::vector<double> roots(0,0.);
if(ce.size()!=5) return roots;
std::vector<double> ce2(4,0.);
if ( fabs(ce[4]) < 1.0e-14) { // this should not happen
if ( fabs(ce[3]) < 1.0e-14) { // this should not happen
if ( fabs(ce[2]) < 1.0e-14) { // this should not happen
if( fabs(ce[1]) > 1.0e-14) {
roots.push_back(-ce[0]/ce[1]);
} else { // can not determine y. this means overlapped, but overlap should have been detected before, therefore return empty set
return roots;
}
} else {
ce2.resize(2);
ce2[0]=ce[1]/ce[2];
ce2[1]=ce[0]/ce[2];
//std::cout<<"ce2[2]={ "<<ce2[0]<<' '<<ce2[1]<<" }\n";
roots=RS_Math::quadraticSolver(ce2);
}
} else {
ce2.resize(3);
ce2[0]=ce[2]/ce[3];
ce2[1]=ce[1]/ce[3];
ce2[2]=ce[0]/ce[3];
//std::cout<<"ce2[3]={ "<<ce2[0]<<' '<<ce2[1]<<' '<<ce2[2]<<" }\n";
roots=RS_Math::cubicSolver(ce2);
}
} else {
ce2[0]=ce[3]/ce[4];
ce2[1]=ce[2]/ce[4];
ce2[2]=ce[1]/ce[4];
ce2[3]=ce[0]/ce[4];
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<"ce2[4]={ "<<ce2[0]<<' '<<ce2[1]<<' '<<ce2[2]<<' '<<ce2[3]<<" }\n";
}
if(fabs(ce2[3])<= RS_TOLERANCE15) {
//constant term is zero, factor 0 out, solve a cubic equation
ce2.resize(3);
roots=RS_Math::cubicSolver(ce2);
roots.push_back(0.);
}else
roots=RS_Math::quarticSolver(ce2);
}
return roots;
}
//linear Equation solver by Gauss-Jordan
/**
* Solve linear equation set
*@ mt holds the augmented matrix
*@ sn holds the solution
*@ return true, if the equation set has a unique solution, return false otherwise
*
*@Author: Dongxu Li
*/
bool RS_Math::linearSolver(const std::vector<std::vector<double> >& mt, std::vector<double>& sn){
//verify the matrix size
size_t mSize(mt.size()); //rows
size_t aSize(mSize+1); //columns of augmented matrix
if(std::any_of(mt.begin(), mt.end(), [&aSize](const std::vector<double>& v)->bool{
return v.size() != aSize;
}))
return false;
sn.resize(mSize);//to hold the solution
#if false
boost::numeric::ublas::matrix<double> bm (mSize, mSize);
boost::numeric::ublas::vector<double> bs(mSize);
for(int i=0;i<mSize;i++) {
for(int j=0;j<mSize;j++) {
bm(i,j)=mt[i][j];
}
bs(i)=mt[i][mSize];
}
//solve the linear equation set by LU decomposition in boost ublas
if ( boost::numeric::ublas::lu_factorize<boost::numeric::ublas::matrix<double> >(bm) ) {
std::cout<<__FILE__<<" : "<<__func__<<" : line "<<__LINE__<<std::endl;
std::cout<<" linear solver failed"<<std::endl;
// RS_DEBUG->print(RS_Debug::D_WARNING, "linear solver failed");
return false;
}
boost::numeric::ublas:: triangular_matrix<double, boost::numeric::ublas::unit_lower>
lm = boost::numeric::ublas::triangular_adaptor< boost::numeric::ublas::matrix<double>, boost::numeric::ublas::unit_lower>(bm);
boost::numeric::ublas:: triangular_matrix<double, boost::numeric::ublas::upper>
um = boost::numeric::ublas::triangular_adaptor< boost::numeric::ublas::matrix<double>, boost::numeric::ublas::upper>(bm);
;
boost::numeric::ublas::inplace_solve(lm,bs, boost::numeric::ublas::lower_tag());
boost::numeric::ublas::inplace_solve(um,bs, boost::numeric::ublas::upper_tag());
for(int i=0;i<mSize;i++){
sn[i]=bs(i);
}
// std::cout<<"dn="<<dn<<std::endl;
// data.center.set(-0.5*dn(1)/dn(0),-0.5*dn(3)/dn(2)); // center
// double d(1.+0.25*(dn(1)*dn(1)/dn(0)+dn(3)*dn(3)/dn(2)));
// if(fabs(dn(0))<RS_TOLERANCE2
// ||fabs(dn(2))<RS_TOLERANCE2
// ||d/dn(0)<RS_TOLERANCE2
// ||d/dn(2)<RS_TOLERANCE2
// ) {
// //ellipse not defined
// return false;
// }
// d=sqrt(d/dn(0));
// data.majorP.set(d,0.);
// data.ratio=sqrt(dn(0)/dn(2));
#else
// solve the linear equation by Gauss-Jordan elimination
std::vector<std::vector<double> > mt0(mt); //copy the matrix;
for(size_t i=0;i<mSize;++i){
size_t imax(i);
double cmax(fabs(mt0[i][i]));
for(size_t j=i+1;j<mSize;++j) {
if(fabs(mt0[j][i]) > cmax ) {
imax=j;
cmax=fabs(mt0[j][i]);
}
}
if(cmax<RS_TOLERANCE2) return false; //singular matrix
if(imax != i) {//move the line with largest absolute value at column i to row i, to avoid division by zero
std::swap(mt0[i],mt0[imax]);
}
for(size_t k=i+1;k<=mSize;++k) { //normalize the i-th row
mt0[i][k] /= mt0[i][i];
}
mt0[i][i]=1.;
for(size_t j=0;j<mSize;++j) {//Gauss-Jordan
if(j != i ) {
double& a = mt0[j][i];
for(size_t k=i+1;k<=mSize;++k) {
mt0[j][k] -= mt0[i][k]*a;
}
a=0.;
}
}
//output gauss-jordan results for debugging
// std::cout<<"========"<<i<<"==========\n";
// for(auto v0: mt0){
// for(auto v1:v0)
// std::cout<<v1<<'\t';
// std::cout<<std::endl;
// }
}
for(size_t i=0;i<mSize;++i) {
sn[i]=mt0[i][mSize];
}
#endif
return true;
}
/**
* wrapper of elliptic integral of the second type, Legendre form
* @param k the elliptic modulus or eccentricity
* @param phi elliptic angle, must be within range of [0, M_PI]
*
* @author: Dongxu Li
*/
double RS_Math::ellipticIntegral_2(const double& k, const double& phi)
{
double a= remainder(phi-M_PI_2,M_PI);
if(a>0.) {
return boost::math::ellint_2<double,double>(k,a);
} else {
return - boost::math::ellint_2<double,double>(k,fabs(a));
}
}
/** solver quadratic simultaneous equations of set two **/
/* solve the following quadratic simultaneous equations,
* ma000 x^2 + ma011 y^2 - 1 =0
* ma100 x^2 + 2 ma101 xy + ma111 y^2 + mb10 x + mb11 y +mc1 =0
*
*@m, a vector of size 8 contains coefficients in the strict order of:
ma000 ma011 ma100 ma101 ma111 mb10 mb11 mc1
* m[0] m[1] must be positive
*@return a vector contains real roots
*/
RS_VectorSolutions RS_Math::simultaneousQuadraticSolver(const std::vector<double>& m)
{
RS_VectorSolutions ret(0);
if(m.size() != 8 ) return ret; // valid m should contain exact 8 elements
std::vector< double> c1(0,0.);
std::vector< std::vector<double> > m1(0,c1);
c1.resize(6);
c1[0]=m[0];
c1[1]=0.;
c1[2]=m[1];
c1[3]=0.;
c1[4]=0.;
c1[5]=-1.;
m1.push_back(c1);
c1[0]=m[2];
c1[1]=2.*m[3];
c1[2]=m[4];
c1[3]=m[5];
c1[4]=m[6];
c1[5]=m[7];
m1.push_back(c1);
return simultaneousQuadraticSolverFull(m1);
}
/** solver quadratic simultaneous equations of a set of two **/
/* solve the following quadratic simultaneous equations,
* ma000 x^2 + ma001 xy + ma011 y^2 + mb00 x + mb01 y + mc0 =0
* ma100 x^2 + ma101 xy + ma111 y^2 + mb10 x + mb11 y + mc1 =0
*
*@m, a vector of size 2 each contains a vector of size 6 coefficients in the strict order of:
ma000 ma001 ma011 mb00 mb01 mc0
ma100 ma101 ma111 mb10 mb11 mc1
*@return a RS_VectorSolutions contains real roots (x,y)
*/
RS_VectorSolutions RS_Math::simultaneousQuadraticSolverFull(const std::vector<std::vector<double> >& m)
{
RS_VectorSolutions ret;
if(m.size()!=2) return ret;
if( m[0].size() ==3 || m[1].size()==3 ){
return simultaneousQuadraticSolverMixed(m);
}
if(m[0].size()!=6 || m[1].size()!=6) return ret;
/** eliminate x, quartic equation of y **/
auto& a=m[0][0];
auto& b=m[0][1];
auto& c=m[0][2];
auto& d=m[0][3];
auto& e=m[0][4];
auto& f=m[0][5];
auto& g=m[1][0];
auto& h=m[1][1];
auto& i=m[1][2];
auto& j=m[1][3];
auto& k=m[1][4];
auto& l=m[1][5];
/**
Collect[Eliminate[{ a*x^2 + b*x*y+c*y^2+d*x+e*y+f==0,g*x^2+h*x*y+i*y^2+j*x+k*y+l==0},x],y]
**/
/*
f^2 g^2 - d f g j + a f j^2 - 2 a f g l + (2 e f g^2 - d f g h - b f g j + 2 a f h j - 2 a f g k) y + (2 c f g^2 - b f g h + a f h^2 - 2 a f g i) y^2
==
-(d^2 g l) + a d j l - a^2 l^2
+
(d e g j - a e j^2 - d^2 g k + a d j k - 2 b d g l + 2 a e g l + a d h l + a b j l - 2 a^2 k l) y
+
(-(e^2 g^2) + d e g h - d^2 g i + c d g j + b e g j - 2 a e h j + a d i j - a c j^2 - 2 b d g k + 2 a e g k + a d h k + a b j k - a^2 k^2 - b^2 g l + 2 a c g l + a b h l - 2 a^2 i l) y^2
+
(-2 c e g^2 + c d g h + b e g h - a e h^2 - 2 b d g i + 2 a e g i + a d h i + b c g j - 2 a c h j + a b i j - b^2 g k + 2 a c g k + a b h k - 2 a^2 i k) y^3
+
(-(c^2 g^2) + b c g h - a c h^2 - b^2 g i + 2 a c g i + a b h i - a^2 i^2) y^4
*/
double a2=a*a;
double b2=b*b;
double c2=c*c;
double d2=d*d;
double e2=e*e;
double f2=f*f;
double g2=g*g;
double h2=h*h;
double i2=i*i;
double j2=j*j;
double k2=k*k;
double l2=l*l;
std::vector<double> qy(5,0.);
//y^4
qy[4]=-c2*g2 + b*c*g*h - a*c*h2 - b2*g*i + 2.*a*c*g*i + a*b*h*i - a2*i2;
//y^3
qy[3]=-2.*c*e*g2 + c*d*g*h + b*e*g*h - a*e*h2 - 2.*b*d*g*i + 2.*a*e*g*i + a*d*h*i +
b*c*g*j - 2.*a*c*h*j + a*b*i*j - b2*g*k + 2.*a*c*g*k + a*b*h*k - 2.*a2*i*k;
//y^2
qy[2]=(-e2*g2 + d*e*g*h - d2*g*i + c*d*g*j + b*e*g*j - 2.*a*e*h*j + a*d*i*j - a*c*j2 -
2.*b*d*g*k + 2.*a*e*g*k + a*d*h*k + a*b*j*k - a2*k2 - b2*g*l + 2.*a*c*g*l + a*b*h*l - 2.*a2*i*l)
- (2.*c*f*g2 - b*f*g*h + a*f*h2 - 2.*a*f*g*i);
//y
qy[1]=(d*e*g*j - a*e*j2 - d2*g*k + a*d*j*k - 2.*b*d*g*l + 2.*a*e*g*l + a*d*h*l + a*b*j*l - 2.*a2*k*l)
-(2.*e*f*g2 - d*f*g*h - b*f*g*j + 2.*a*f*h*j - 2.*a*f*g*k);
//y^0
qy[0]=-d2*g*l + a*d*j*l - a2*l2
- ( f2*g2 - d*f*g*j + a*f*j2 - 2.*a*f*g*l);
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<qy[4]<<"*y^4 +("<<qy[3]<<")*y^3+("<<qy[2]<<")*y^2+("<<qy[1]<<")*y+("<<qy[0]<<")==0"<<std::endl;
}
//quarticSolver
auto roots=quarticSolverFull(qy);
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
std::cout<<"roots.size()= "<<roots.size()<<std::endl;
}
if (roots.size()==0 ) { // no intersection found
return ret;
}
std::vector<double> ce(0,0.);
for(size_t i0=0;i0<roots.size();i0++){
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<"y="<<roots[i0]<<std::endl;
}
/*
Collect[Eliminate[{ a*x^2 + b*x*y+c*y^2+d*x+e*y+f==0,g*x^2+h*x*y+i*y^2+j*x+k*y+l==0},x],y]
*/
ce.resize(3);
ce[0]=a;
ce[1]=b*roots[i0]+d;
ce[2]=c*roots[i0]*roots[i0]+e*roots[i0]+f;
// DEBUG_HEADER
// std::cout<<"("<<ce[0]<<")*x^2 + ("<<ce[1]<<")*x + ("<<ce[2]<<") == 0"<<std::endl;
if(fabs(ce[0])<1e-75 && fabs(ce[1])<1e-75) {
ce[0]=g;
ce[1]=h*roots[i0]+j;
ce[2]=i*roots[i0]*roots[i0]+k*roots[i0]+f;
// DEBUG_HEADER
// std::cout<<"("<<ce[0]<<")*x^2 + ("<<ce[1]<<")*x + ("<<ce[2]<<") == 0"<<std::endl;
}
if(fabs(ce[0])<1e-75 && fabs(ce[1])<1e-75) continue;
if(fabs(a)>1e-75){
std::vector<double> ce2(2,0.);
ce2[0]=ce[1]/ce[0];
ce2[1]=ce[2]/ce[0];
// DEBUG_HEADER
// std::cout<<"x^2 +("<<ce2[0]<<")*x+("<<ce2[1]<<")==0"<<std::endl;
auto xRoots=quadraticSolver(ce2);
for(size_t j0=0;j0<xRoots.size();j0++){
// DEBUG_HEADER
// std::cout<<"x="<<xRoots[j0]<<std::endl;
RS_Vector vp(xRoots[j0],roots[i0]);
if(simultaneousQuadraticVerify(m,vp)) ret.push_back(vp);
}
continue;
}
RS_Vector vp(-ce[2]/ce[1],roots[i0]);
if(simultaneousQuadraticVerify(m,vp)) ret.push_back(vp);
}
if(RS_DEBUG->getLevel()>=RS_Debug::D_INFORMATIONAL){
DEBUG_HEADER
std::cout<<"ret="<<ret<<std::endl;
}
return ret;
}
RS_VectorSolutions RS_Math::simultaneousQuadraticSolverMixed(const std::vector<std::vector<double> >& m)
{
RS_VectorSolutions ret;
auto p0=& (m[0]);
auto p1=& (m[1]);
if(p1->size()==3){
std::swap(p0,p1);
}
if(p1->size()==3) {
//linear
std::vector<double> sn(2,0.);
std::vector<std::vector<double> > ce;
ce.push_back(m[0]);
ce.push_back(m[1]);
ce[0][2]=-ce[0][2];
ce[1][2]=-ce[1][2];
if( RS_Math::linearSolver(ce,sn)) ret.push_back(RS_Vector(sn[0],sn[1]));
return ret;
}
// DEBUG_HEADER
// std::cout<<"p0: size="<<p0->size()<<"\n Solve[{("<< p0->at(0)<<")*x + ("<<p0->at(1)<<")*y + ("<<p0->at(2)<<")==0,";
// std::cout<<"("<< p1->at(0)<<")*x^2 + ("<<p1->at(1)<<")*x*y + ("<<p1->at(2)<<")*y^2 + ("<<p1->at(3)<<")*x +("<<p1->at(4)<<")*y+("
// <<p1->at(5)<<")==0},{x,y}]"<<std::endl;
const double& a=p0->at(0);
const double& b=p0->at(1);
const double& c=p0->at(2);
const double& d=p1->at(0);
const double& e=p1->at(1);
const double& f=p1->at(2);
const double& g=p1->at(3);
const double& h=p1->at(4);
const double& i=p1->at(5);
/**
y (2 b c d-a c e)-a c g+c^2 d = y^2 (a^2 (-f)+a b e-b^2 d)+y (a b g-a^2 h)+a^2 (-i)
*/
std::vector<double> ce(3,0.);
const double& a2=a*a;
const double& b2=b*b;
const double& c2=c*c;
ce[0]= -f*a2+a*b*e-b2*d;
ce[1]=a*b*g-a2*h- (2*b*c*d-a*c*e);
ce[2]=a*c*g-c2*d-a2*i;
// DEBUG_HEADER
// std::cout<<"("<<ce[0]<<") y^2 + ("<<ce[1]<<") y + ("<<ce[2]<<")==0"<<std::endl;
std::vector<double> roots(0,0.);
if( fabs(ce[1])>RS_TOLERANCE15 && fabs(ce[0]/ce[1])<RS_TOLERANCE15){
roots.push_back( - ce[2]/ce[1]);
}else{
std::vector<double> ce2(2,0.);
ce2[0]=ce[1]/ce[0];
ce2[1]=ce[2]/ce[0];
roots=quadraticSolver(ce2);
}
// for(size_t i=0;i<roots.size();i++){
// std::cout<<"x="<<roots.at(i)<<std::endl;
// }
if(roots.size()==0) {
return RS_VectorSolutions();
}
for(size_t i=0;i<roots.size();i++){
ret.push_back(RS_Vector(-(b*roots.at(i)+c)/a,roots.at(i)));
// std::cout<<ret.at(ret.size()-1).x<<", "<<ret.at(ret.size()-1).y<<std::endl;
}
return ret;
}
/** verify a solution for simultaneousQuadratic
*@m the coefficient matrix
*@v, a candidate to verify
*@return true, for a valid solution
**/
bool RS_Math::simultaneousQuadraticVerify(const std::vector<std::vector<double> >& m, RS_Vector& v)
{
RS_Vector v0=v;
auto& a=m[0][0];
auto& b=m[0][1];
auto& c=m[0][2];
auto& d=m[0][3];
auto& e=m[0][4];
auto& f=m[0][5];
auto& g=m[1][0];
auto& h=m[1][1];
auto& i=m[1][2];
auto& j=m[1][3];
auto& k=m[1][4];
auto& l=m[1][5];
/**
* tolerance test for bug#3606099
* verifying the equations to floating point tolerance by terms
*/
double sum0=0., sum1=0.;
double f00=0.,f01=0.;
double amax0, amax1;
for(size_t i0=0; i0<20; ++i0){
double& x=v.x;
double& y=v.y;
double x2=x*x;
double y2=y*y;
double const terms0[12]={ a*x2, b*x*y, c*y2, d*x, e*y, f, g*x2, h*x*y, i*y2, j*x, k*y, l};
amax0=fabs(terms0[0]), amax1=fabs(terms0[6]);
double px=2.*a*x+b*y+d;
double py=b*x+2.*c*y+e;
sum0=0.;
for(int i=0; i<6; i++) {
if(amax0<fabs(terms0[i])) amax0=fabs(terms0[i]);
sum0 += terms0[i];
}
std::vector<std::vector<double>> nrCe;
nrCe.push_back(std::vector<double>{px, py, sum0});
px=2.*g*x+h*y+j;
py=h*x+2.*i*y+k;
sum1=0.;
for(int i=6; i<12; i++) {
if(amax1<fabs(terms0[i])) amax1=fabs(terms0[i]);
sum1 += terms0[i];
}
nrCe.push_back(std::vector<double>{px, py, sum1});
std::vector<double> dn;
bool ret=linearSolver(nrCe, dn);
// DEBUG_HEADER
// qDebug()<<"i0="<<i0<<"\tf=("<<sum0<<','<<sum1<<")\tdn=("<<dn[0]<<","<<dn[1]<<")";
if(!i0){
f00=sum0;
f01=sum1;
}
if(!ret) break;
v -= RS_Vector(dn[0], dn[1]);
}
if( fabs(sum0)> fabs(f00) && fabs(sum1)>fabs(f01)){
v=v0;
sum0=f00;
sum1=f01;
}
// DEBUG_HEADER
// std::cout<<"verifying: x="<<x<<"\ty="<<y<<std::endl;
// std::cout<<"0: maxterm: "<<amax0<<std::endl;
// std::cout<<"verifying: fabs(a*x2 + b*x*y+c*y2+d*x+e*y+f)/maxterm="<<fabs(sum0)/amax0<<" required to be smaller than "<<sqrt(6.)*sqrt(DBL_EPSILON)<<std::endl;
// std::cout<<"1: maxterm: "<<amax1<<std::endl;
// std::cout<<"verifying: fabs(g*x2+h*x*y+i*y2+j*x+k*y+l)/maxterm="<< fabs(sum1)/amax1<<std::endl;
const double tols=2.*sqrt(6.)*sqrt(DBL_EPSILON); //experimental tolerances to verify simultaneous quadratic
return (amax0<=tols || fabs(sum0)/amax0<tols) && (amax1<=tols || fabs(sum1)/amax1<tols);
}
//EOF
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