289 lines
8.8 KiB
C++
289 lines
8.8 KiB
C++
/*
|
|
LA: linear algebra C++ interface library
|
|
Copyright (C) 2021 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
|
|
|
|
This program is free software: you can redistribute it and/or modify
|
|
it under the terms of the GNU General Public License as published by
|
|
the Free Software Foundation, either version 3 of the License, or
|
|
(at your option) any later version.
|
|
|
|
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, see <http://www.gnu.org/licenses/>.
|
|
*/
|
|
|
|
|
|
#ifndef _POLYNOMIAL_H
|
|
#define _POLYNOMIAL_H
|
|
|
|
#include "la_traits.h"
|
|
#include "vec.h"
|
|
#include "nonclass.h"
|
|
#include "matexp.h"
|
|
|
|
namespace LA {
|
|
|
|
|
|
template <typename T>
|
|
class Polynomial : public NRVec<T> {
|
|
private:
|
|
int size() const; //prevent confusion with vector size
|
|
public:
|
|
Polynomial(): NRVec<T>() {};
|
|
template<int SIZE> Polynomial(const T (&a)[SIZE]) : NRVec<T>(a) {};
|
|
Polynomial(const NRVec<T> &v) : NRVec<T>(v) {}; //allow implicit conversion from NRVec
|
|
Polynomial(const int n) : NRVec<T>(n+1) {};
|
|
Polynomial(const T &a, const int n) : NRVec<T>(n+1) {NRVec<T>::clear(); (*this)[0]=a;};
|
|
|
|
int degree() const {return NRVec<T>::size()-1;};
|
|
void resize(const int n, const bool preserve=true) {NRVec<T>::resize(n+1,preserve);}
|
|
|
|
Polynomial& operator+=(const T &a) {NOT_GPU(*this); NRVec<T>::copyonwrite(); (*this)[0]+=a; return *this;}
|
|
Polynomial& operator-=(const T &a) {NOT_GPU(*this); NRVec<T>::copyonwrite(); (*this)[0]-=a; return *this;}
|
|
Polynomial operator+(const T &a) const {return Polynomial(*this) += a;};
|
|
Polynomial operator-(const T &a) const {return Polynomial(*this) -= a;};
|
|
Polynomial operator-() const {return NRVec<T>::operator-();}
|
|
Polynomial operator*(const T &a) const {return NRVec<T>::operator*(a);}
|
|
Polynomial operator/(const T &a) const {return NRVec<T>::operator/(a);}
|
|
Polynomial& operator*=(const T &a) {NRVec<T>::operator*=(a); return *this;}
|
|
Polynomial& operator/=(const T &a) {NRVec<T>::operator/=(a); return *this;}
|
|
|
|
Polynomial& operator+=(const Polynomial &rhs)
|
|
{
|
|
NOT_GPU(*this); NRVec<T>::copyonwrite();
|
|
if(rhs.degree()>degree()) resize(rhs.degree(),true);
|
|
for(int i=0; i<=rhs.degree(); ++i) (*this)[i] += rhs[i];
|
|
return *this;
|
|
}
|
|
|
|
Polynomial& operator-=(const Polynomial &rhs)
|
|
{
|
|
NOT_GPU(*this); NRVec<T>::copyonwrite();
|
|
if(rhs.degree()>degree()) resize(rhs.degree(),true);
|
|
for(int i=0; i<=rhs.degree(); ++i) (*this)[i] -= rhs[i];
|
|
return *this;
|
|
}
|
|
Polynomial operator+(const Polynomial &rhs) const {return Polynomial(*this) += rhs;};
|
|
Polynomial operator-(const Polynomial &rhs) const {return Polynomial(*this) -= rhs;};
|
|
Polynomial operator*(const Polynomial &rhs) const //NOTE: naive implementation! For very long polynomials FFT-based methods should be used
|
|
{
|
|
NOT_GPU(*this);
|
|
Polynomial r(degree()+rhs.degree());
|
|
r.clear();
|
|
for(int i=0; i<=rhs.degree(); ++i) for(int j=0; j<=degree(); ++j) r[i+j] += rhs[i]*(*this)[j];
|
|
return r;
|
|
};
|
|
Polynomial& operator*=(const Polynomial &rhs) {*this = (*this)*rhs; return *this;};
|
|
void simplify(const typename LA_traits<T>::normtype thr=0)
|
|
{
|
|
NOT_GPU(*this);
|
|
this->copyonwrite();
|
|
int n=degree();
|
|
while(n>0 && MYABS((*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;
|
|
if(shift>0)
|
|
{
|
|
Polynomial r(degree()+shift);
|
|
for(int i=0; i<shift; ++i) r[i]=0;
|
|
for(int i=0; i<=degree(); ++i) r[shift+i] = (*this)[i];
|
|
return r;
|
|
}
|
|
else
|
|
{
|
|
if(shift+degree()<0)
|
|
{
|
|
Polynomial r(0);
|
|
r[0]=0;
|
|
return r;
|
|
}
|
|
Polynomial r(shift+degree());
|
|
for(int i= -shift; i<=degree(); ++i) r[shift+i] = (*this)[i];
|
|
return r;
|
|
}
|
|
}
|
|
Polynomial derivative(const int d=1) const
|
|
{
|
|
if(d==0) return *this;
|
|
if(d<0) return integral(-d);
|
|
NOT_GPU(*this);
|
|
int n=degree();
|
|
if(n-d<0)
|
|
{
|
|
Polynomial r(0);
|
|
r[0]=0;
|
|
return r;
|
|
}
|
|
Polynomial r(n-d);
|
|
for(int i=d; i<=n; ++i)
|
|
{
|
|
int prod=1;
|
|
for(int j=i; j>=i-d+1; --j) prod *= j;
|
|
r[i-d] = (*this)[i]* ((T)prod);
|
|
}
|
|
return r;
|
|
};
|
|
Polynomial integral(const int d=1) const
|
|
{
|
|
if(d==0) return *this;
|
|
if(d<0) return derivative(-d);
|
|
NOT_GPU(*this);
|
|
int n=degree();
|
|
Polynomial r(n+d);
|
|
for(int i=0; i<d; ++i) r[i]=0;
|
|
for(int i=0; i<=n; ++i)
|
|
{
|
|
int prod=1;
|
|
for(int j=i+1; j<=i+d;++j) prod *= j;
|
|
r[i+d] = (*this)[i]/((T)prod);
|
|
}
|
|
return r;
|
|
}
|
|
Polynomial composition(const Polynomial &rhs) const;
|
|
Polynomial even_powers() const {int d=degree()/2; Polynomial r(d); for(int i=0; i<=degree(); i+=2) r[i/2] = (*this)[i]; return r;};
|
|
Polynomial odd_powers() const {int d=(degree()-1)/2; Polynomial r(d); if(degree()==0) {r[0]=0; return r;} for(int i=1; i<=degree(); i+=2) r[(i-1)/2] = (*this)[i]; return r;};
|
|
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;
|
|
T newton(const T x0, const typename LA_traits<T>::normtype thr=1e-14, const int maxit=1000) const; //solve root from the guess
|
|
Polynomial pow(const int i) const; //integer power
|
|
Polynomial powx(const int i) const; //substitute x^i for x in the polynomial
|
|
void binomial(const int n); //(1+x)^n
|
|
|
|
};
|
|
|
|
//this is very general, can be used also for composition (nesting) of polynomials
|
|
//for an alternative algorithm which minimizes number of multiplications cf. also matexp.h
|
|
template <typename T, typename C>
|
|
C value(const Polynomial<T> &p, const C &x)
|
|
{
|
|
C sum(x);
|
|
sum=0; //get matrix dimension if C is a matrix
|
|
for(int i=p.degree(); i>0; --i)
|
|
{
|
|
sum+= p[i];
|
|
sum= sum*x; //not *= for matrices
|
|
}
|
|
sum += p[0];
|
|
return sum;
|
|
}
|
|
|
|
//evaluate only even powers
|
|
template <typename T, typename C>
|
|
C even_value(const Polynomial<T> &p, const C &x)
|
|
{
|
|
C sum(x);
|
|
sum=0; //get matrix dimension if C is a matrix
|
|
int d=p.degree();
|
|
if(d&1) --d;
|
|
C x2 = x*x;
|
|
for(int i=d; i>0; i-=2)
|
|
{
|
|
sum+= p[i];
|
|
sum= sum*x2; //not *= for matrices
|
|
}
|
|
sum += p[0];
|
|
return sum;
|
|
}
|
|
|
|
//evaluate only odd powers
|
|
template <typename T, typename C>
|
|
C odd_value(const Polynomial<T> &p, const C &x)
|
|
{
|
|
C sum(x);
|
|
sum=0; //get matrix dimension if C is a matrix
|
|
int d=p.degree();
|
|
if(d==0) return sum;
|
|
if((d&1)==0) --d;
|
|
C x2 = x*x;
|
|
for(int i=d; i>2; i-=2)
|
|
{
|
|
sum+= p[i];
|
|
sum= sum*x2; //not *= for matrices
|
|
}
|
|
sum += p[1];
|
|
sum *= x;
|
|
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);
|
|
|
|
|
|
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);
|
|
}
|
|
|
|
|
|
template <typename T>
|
|
Polynomial<T> hermite_polynomial(int n) //physicists definition
|
|
{
|
|
Polynomial<T> h(n);
|
|
h.clear();
|
|
h[n]=ipow((T)2,n);
|
|
for(int m=1; n-2*m>=0; m+=1)
|
|
{
|
|
int i=n-2*m;
|
|
h[i] = (-h[i+2] *(i+2)*(i+1)) /(4*m);
|
|
}
|
|
return h;
|
|
}
|
|
|
|
|
|
}//namespace
|
|
#endif
|