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continueing on polynomials

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This commit is contained in:
Jiri Pittner 2021-06-11 17:44:20 +02:00
parent e57d8fde1c
commit f03953ba2d
3 changed files with 158 additions and 5 deletions
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95
polynomial.cc
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@ -56,6 +56,94 @@ r.resize(rhs.degree()-1,true);
}
template <typename T>
NRMat<T> Sylvester(const Polynomial<T> &p, const Polynomial<T> &q)
{
int nm=p.degree()+q.degree();
NRMat<T> a(nm,nm);
a.clear();
for(int i=0; i<q.degree(); ++i)
for(int j=p.degree(); j>=0; --j)
a(i,i+p.degree()-j)=p[j];
for(int i=0; i<p.degree(); ++i)
for(int j=q.degree(); j>=0; --j)
a(q.degree()+i,i+q.degree()-j)=q[j];
return a;
}
template <typename T>
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()];
return a;
}
template<>
NRVec<typename LA_traits<int>::complextype> Polynomial<int>::roots() const
{
laerror("roots not implemented for integer polynomials");
return NRVec<typename LA_traits<int>::complextype>(1);
}
template <typename T>
NRVec<typename LA_traits<T>::complextype> Polynomial<T>::roots() const
{
NRMat<T> a=this->companion();
NRVec<typename LA_traits<T>::complextype> r(degree());
gdiagonalize(a,r,NULL,NULL);
return r;
}
template <typename T>
NRVec<T> Polynomial<T>::realroots(const typename LA_traits<T>::normtype thr) const
{
NRVec<typename LA_traits<T>::complextype> r = roots();
NRVec<T> rr(degree());
int nr=0;
for(int i=0; i<degree(); ++i)
{
if(abs(r[i].imag())<thr)
{
rr[nr++] = r[i].real();
}
}
rr.resize(nr,true);
rr.sort();
return rr;
}
template <typename T>
Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y)
{
if(x.size()!=y.size()) laerror("vectors of different length in lagrange_interpolation");
if(x.size()<1) laerror("empty vector in lagrange_interpolation");
if(x.size()==1)
{
Polynomial<T> p(0);
p[0]=y[0];
return p;
}
int n=x.size()-1;
Polynomial<T> p(n);
p.clear();
for(int i=0; i<=n; ++i)
{
T prod=(T)1;
for(int j=0; j<=n; ++j) if(j!=i) prod *= (x[i]-x[j]);
if(prod==(T)0) laerror("repeated x-value in lagrange_interpolation");
Polynomial<T> tmp=polyfromroots(x,i);
p += tmp * y[i] / prod;
}
return p;
}
/***************************************************************************//**
* forced instantization in the corresponding object file
******************************************************************************/
@ -64,10 +152,13 @@ template class Polynomial<double>;
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); \
//INSTANTIZE(double)
INSTANTIZE(int)
INSTANTIZE(double)
INSTANTIZE(std::complex<double>)

39
polynomial.h
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@ -22,6 +22,7 @@
#include "la_traits.h"
#include "vec.h"
#include "nonclass.h"
namespace LA {
@ -78,6 +79,7 @@ public:
while(n>0 && abs((*this)[n])<thr) --n;
resize(n,true);
};
void normalize() {if((*this)[degree()]==(T)0) laerror("zero coefficient at highest degree - simplify first"); *this /= (*this)[degree()];};
Polynomial shifted(const int shift) const
{
if(shift==0) return *this;
@ -127,9 +129,11 @@ public:
void polydiv(const Polynomial &rhs, Polynomial &q, Polynomial &r) const;
Polynomial operator/(const Polynomial &rhs) const {Polynomial q,r; polydiv(rhs,q,r); return q;};
Polynomial operator%(const Polynomial &rhs) const {Polynomial q,r; polydiv(rhs,q,r); return r;};
NRMat<T> companion() const; //matrix which has this characteristic polynomial
NRVec<typename LA_traits<T>::complextype> roots() const; //implemented for complex<double> and double only
NRVec<T> realroots(const typename LA_traits<T>::normtype thr) const;
//gcd, lcm
//roots, interpolation ... special only for real->complex - declare only and implent only template specialization in .cc
//@@@gcd, lcm euler and svd
};
@ -148,12 +152,43 @@ sum += p[0];
return sum;
}
template <typename T, typename C>
NRVec<C> values(const Polynomial<T> &p, const NRVec<C> &x)
{
NRVec<C> r(x.size());
for(int i=0; i<x.size(); ++i) r[i]=value(p,x[i]);
return r;
}
//scalar+-polynomial
template <typename T>
inline Polynomial<T> operator+(const T &a, const Polynomial<T> &rhs) {return Polynomial<T>(rhs)+=a;}
template <typename T>
inline Polynomial<T> operator-(const T &a, const Polynomial<T> &rhs) {return Polynomial<T>(rhs)-=a;}
//Sylvester matrix
template <typename T>
extern NRMat<T> Sylvester(const Polynomial<T> &p, const Polynomial<T> &q);
//polynomial from given roots
template <typename T>
Polynomial<T> polyfromroots(const NRVec<T> &roots, const int skip= -1)
{
Polynomial<T> p(0);
p[0]=(T)1;
for(int i=0; i<roots.size(); ++i)
if(i!=skip)
p = p.shifted(1) - p * roots[i];
return p;
}
template <typename T>
extern Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y);
}//namespace
#endif

29
t.cc
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@ -2167,7 +2167,7 @@ cout <<Sn;
if(!Sn.is_valid()) laerror("internal error in Sn character calculation");
}
if(1)
if(0)
{
int n,m;
double x;
@ -2187,6 +2187,8 @@ Polynomial<double> z=value(p,q); //p(q(x))
Polynomial<double> y=value(q,p);
cout <<p;
cout <<q;
cout <<q.companion();
cout<<Sylvester(p,q);
cout <<a;
cout <<b;
cout <<r;
@ -2201,5 +2203,30 @@ cout << (value(p,u)*value(q,u) -value(r,u)).norm()<<endl;
}
if(0)
{
int n;
cin >>n ;
NRVec<double> r(n);
r.randomize(1.);
r.sort(0);
Polynomial<double> p=polyfromroots(r);
cout <<p;
cout <<r;
cout <<p.realroots(1e-10);
}
if(1)
{
int n;
cin >>n ;
NRVec<double> x(n+1),y(n+1);
x.randomize(1.);
y.randomize(1.);
Polynomial<double> p=lagrange_interpolation(x,y);
cout <<x<<y<<p;
NRVec<double> yy=values(p,x);
cout<<"interpolation error= "<<(y-yy).norm()<<endl;
}
}