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implemented polynomial gcd

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This commit is contained in:
Jiri Pittner 2021-06-12 21:42:31 +02:00
parent 73aed62650
commit cc65575536
3 changed files with 96 additions and 7 deletions
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51
polynomial.cc
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@ -19,6 +19,7 @@
#include "polynomial.h"
#include <stdio.h>
#include <string.h>
#include <math.h>
namespace LA {
@ -77,8 +78,8 @@ NRMat<T> Polynomial<T>::companion() const
if((*this)[degree()]==(T)0) laerror("zero coefficient at highest degree - simplify first");
NRMat<T> a(degree(),degree());
a.clear();
for(int i=0; i<degree(); ++i) a(degree()-1,i) = -(*this)[i];
for(int i=0; i<degree()-1; ++i) a(i,i+1) = (*this)[degree()];
for(int i=0; i<degree(); ++i) a(degree()-1,i) = -(*this)[i]/(*this)[degree()];
for(int i=0; i<degree()-1; ++i) a(i,i+1) = (T)1;
return a;
}
@ -158,6 +159,49 @@ return x;
}
template <typename T>
Polynomial<T> poly_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr, const int d)
{
Polynomial<T> big,small;
if(p.degree() < q.degree()) {big=q; small=p;} else {big=p; small=q;}
small.simplify(thr);
if(small.degree()==0) return small;
Polynomial<T> help;
do {
help=small;
small= big%small;
big=help;
small.simplify(thr);
}
while((d<0 && small.degree() != 0) || (d>=0 && small.degree()>=d));
return big;
}
template <>
Polynomial<int> svd_gcd(const Polynomial<int> &p, const Polynomial<int> &q, const typename LA_traits<int>::normtype thr)
{
laerror("SVD gcd only for floating point numbers");
return p;
}
template <typename T>
Polynomial<T> svd_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr)
{
Polynomial<T> big,small;
if(p.degree() < q.degree()) {big=q; small=p;} else {big=p; small=q;}
small.simplify(thr);
if(small.degree()==0) return small;
NRMat<T> s = Sylvester(p,q);
NRMat<T> u(s.nrows(),s.nrows()),v(s.ncols(),s.ncols());
NRVec<typename LA_traits<T>::normtype> w(s.nrows());
singular_decomposition(s,&u,w,&v,0);
int rank=0;
for(int i=0; i<w.size(); ++i) if(w[i]>thr*::sqrt((double)big.degree()*small.degree())) ++rank;
int d=big.degree()+small.degree()-rank;
return poly_gcd(big,small,thr,d);
}
/***************************************************************************//**
@ -170,6 +214,9 @@ template class Polynomial<std::complex<double> >;
#define INSTANTIZE(T) \
template NRMat<T> Sylvester(const Polynomial<T> &p, const Polynomial<T> &q); \
template Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y); \
template Polynomial<T> poly_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr,const int d); \
template Polynomial<T> svd_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr); \
INSTANTIZE(int)

14
polynomial.h
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@ -75,6 +75,7 @@ public:
void simplify(const typename LA_traits<T>::normtype thr=0)
{
NOT_GPU(*this);
this->copyonwrite();
int n=degree();
while(n>0 && abs((*this)[n])<=thr) --n;
resize(n,true);
@ -148,8 +149,6 @@ public:
NRVec<T> realroots(const typename LA_traits<T>::normtype thr) const;
T newton(const T x0, const typename LA_traits<T>::normtype thr=1e-14, const int maxit=1000) const; //solve root from the guess
//@@@gcd, lcm euler and svd
};
//this is very general, can be used also for nesting polynomials
@ -203,6 +202,17 @@ template <typename T>
extern Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y);
template <typename T>
extern Polynomial<T> poly_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr=0, const int d= -1);
template <typename T>
extern Polynomial<T> svd_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr=0);
template <typename T>
Polynomial<T> poly_lcm(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr=0)
{
return p*q/poly_gcd(p,q,thr);
}
}//namespace

38
t.cc
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@ -2203,18 +2203,23 @@ cout << (value(p,u)*value(q,u) -value(r,u)).norm()<<endl;
}
if(1)
if(0)
{
int n;
cin >>n ;
NRVec<double> r(n);
r.randomize(1.);
//r.randomize(1.);
//wilkinson's ill-conditionel polynomial
for(int i=0; i<n;++i) r[i]=i+1;
double x0=r[0]*0.8+r[1]*0.2;
r.sort(0);
Polynomial<double> p=polyfromroots(r);
cout <<p;
cout <<r;
cout <<p.realroots(1e-10);
NRVec<double> rr= p.realroots(1e-10);
rr.resize(n,true);
cout <<rr;
cout <<"root error = "<<(r-rr).norm()<<endl;
double x=p.newton(x0);
double xdif=1e10;
for(int i=0; i<n; ++i) if(abs(x-r[i])<xdif) xdif=abs(x-r[i]);
@ -2237,4 +2242,31 @@ Polynomial<double>pp=q.derivative(2);
cout<<"test deriv. "<<(pp-p).norm()<<endl;
}
if(1)
{
int n;
cin >>n;
NRVec<double> rr(n);
rr.randomize(1.);
rr.sort(0);
if(rr.size()>2) rr[1]=rr[0];//make a degenerate root
NRVec<double> pr(2*n);
NRVec<double> qr(2*n);
pr.randomize(1.);
qr.randomize(1.);
for(int i=0; i<n; ++i) {pr[i]=qr[i]=rr[i];}
Polynomial<double> p=polyfromroots(pr);
Polynomial<double> q=polyfromroots(qr);
Polynomial<double> g=poly_gcd(p,q,1e-8);
cout <<"GCD ="<<g;
Polynomial<double> gg=svd_gcd(p,q,1e-13);
cout <<"SVDGCD ="<<gg;
NRVec<double> rrr=g.realroots(1e-5);
NRVec<double> rrrr=gg.realroots(1e-5);
cout <<rr<<rrr<<rrrr;
cout <<"test gcd "<<(rr-rrr).norm()<<endl;
cout <<"test svdgcd "<<(rr-rrrr).norm()<<endl;
}
}