290 lines
7.6 KiB
C++
290 lines
7.6 KiB
C++
/*
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LA: linear algebra C++ interface library
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Copyright (C) 2021 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
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This program is free software: you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation, either version 3 of the License, or
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(at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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*/
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#include "polynomial.h"
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#include "miscfunc.h"
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#include <stdio.h>
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#include <string.h>
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#include <math.h>
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#include "nonclass.h"
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namespace LA {
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template <typename T>
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void Polynomial<T>::polydiv(const Polynomial &rhs, Polynomial &q, Polynomial &r) const
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{
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if(rhs[rhs.degree()]==(T)0) laerror("division by a polynomial with zero leading coefficient - simplify it first");
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if(rhs.degree()==0) //scalar division
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{
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q= *this/rhs[0];
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r.resize(0,false);
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r[0]=0;
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return;
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}
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int rdegree= rhs.degree();
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int qdegree= degree()-rdegree;
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if(qdegree<0)
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{
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q.resize(0,false);
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q[0]=0;
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r= *this;
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return;
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}
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//general case
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q.resize(qdegree,false);
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r= *this; r.copyonwrite();
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for(int i=degree(); i>=rdegree; --i)
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{
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T tmp= r[i]/rhs[rdegree];
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q[i-rdegree]= tmp;
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r -= rhs.shifted(i-rdegree)*tmp;
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}
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r.resize(rhs.degree()-1,true);
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}
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template <typename T>
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NRMat<T> Sylvester(const Polynomial<T> &p, const Polynomial<T> &q)
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{
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int nm=p.degree()+q.degree();
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NRMat<T> a(nm,nm);
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a.clear();
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for(int i=0; i<q.degree(); ++i)
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for(int j=p.degree(); j>=0; --j)
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a(i,i+p.degree()-j)=p[j];
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for(int i=0; i<p.degree(); ++i)
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for(int j=q.degree(); j>=0; --j)
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a(q.degree()+i,i+q.degree()-j)=q[j];
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return a;
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}
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template <typename T>
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NRMat<T> Polynomial<T>::companion() const
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{
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if((*this)[degree()]==(T)0) laerror("zero coefficient at highest degree - simplify first");
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NRMat<T> a(degree(),degree());
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a.clear();
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for(int i=0; i<degree(); ++i) a(degree()-1,i) = -(*this)[i]/(*this)[degree()];
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for(int i=0; i<degree()-1; ++i) a(i,i+1) = (T)1;
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return a;
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}
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template<>
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NRVec<typename LA_traits<int>::complextype> Polynomial<int>::roots() const
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{
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laerror("roots not implemented for integer polynomials");
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return NRVec<typename LA_traits<int>::complextype>(1);
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}
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template<>
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NRVec<typename LA_traits<unsigned int>::complextype> Polynomial<unsigned int>::roots() const
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{
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laerror("roots not implemented for integer polynomials");
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return NRVec<typename LA_traits<unsigned int>::complextype>(1);
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}
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template <typename T>
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NRVec<typename LA_traits<T>::complextype> Polynomial<T>::roots() const
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{
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NRMat<T> a=this->companion();
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NRVec<typename LA_traits<T>::complextype> r(degree());
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gdiagonalize(a,r,NULL,NULL);
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return r;
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}
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template <typename T>
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NRVec<T> Polynomial<T>::realroots(const typename LA_traits<T>::normtype thr) const
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{
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NRVec<typename LA_traits<T>::complextype> r = roots();
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NRVec<T> rr(degree());
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int nr=0;
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for(int i=0; i<degree(); ++i)
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{
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if(MYABS(r[i].imag())<thr)
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{
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rr[nr++] = r[i].real();
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}
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}
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rr.resize(nr,true);
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rr.sort();
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return rr;
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}
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template <typename T>
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Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y)
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{
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if(x.size()!=y.size()) laerror("vectors of different length in lagrange_interpolation");
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if(x.size()<1) laerror("empty vector in lagrange_interpolation");
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if(x.size()==1)
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{
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Polynomial<T> p(0);
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p[0]=y[0];
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return p;
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}
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int n=x.size()-1;
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Polynomial<T> p(n);
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p.clear();
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for(int i=0; i<=n; ++i)
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{
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T prod=(T)1;
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for(int j=0; j<=n; ++j) if(j!=i) prod *= (x[i]-x[j]);
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if(prod==(T)0) laerror("repeated x-value in lagrange_interpolation");
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Polynomial<T> tmp=polyfromroots(x,i);
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p += tmp * y[i] / prod;
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}
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return p;
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}
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template <typename T>
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T Polynomial<T>::newton(const T x0, const typename LA_traits<T>::normtype thr, const int maxit) const
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{
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Polynomial<T> d=derivative(1);
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T x=x0;
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for(int i=0; i<maxit; ++i)
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{
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T v=value(*this,x);
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if(MYABS(v)<thr) break;
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x -= v/value(d,x);
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}
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return x;
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}
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template <typename T>
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Polynomial<T> poly_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr, const int d)
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{
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Polynomial<T> big,small;
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if(p.degree() < q.degree()) {big=q; small=p;} else {big=p; small=q;}
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small.simplify(thr);
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if(small.degree()==0) return small;
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Polynomial<T> help;
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do {
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help=small;
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small= big%small;
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big=help;
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small.simplify(thr);
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}
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while((d<0 && small.degree() != 0) || (d>=0 && small.degree()>=d));
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return big;
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}
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template <>
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Polynomial<int> svd_gcd(const Polynomial<int> &p, const Polynomial<int> &q, const typename LA_traits<int>::normtype thr)
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{
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laerror("SVD gcd only for floating point numbers");
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return p;
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}
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template <>
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Polynomial<unsigned int> svd_gcd(const Polynomial<unsigned int> &p, const Polynomial<unsigned int> &q, const typename LA_traits<unsigned int>::normtype thr)
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{
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laerror("SVD gcd only for floating point numbers");
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return p;
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}
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template <typename T>
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Polynomial<T> svd_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr)
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{
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Polynomial<T> big,small;
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if(p.degree() < q.degree()) {big=q; small=p;} else {big=p; small=q;}
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small.simplify(thr);
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if(small.degree()==0) return small;
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NRMat<T> s = Sylvester(p,q);
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NRMat<T> u(s.nrows(),s.nrows()),v(s.ncols(),s.ncols());
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NRVec<typename LA_traits<T>::normtype> w(s.nrows());
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singular_decomposition(s,&u,w,&v,0);
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int rank=0;
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for(int i=0; i<w.size(); ++i) if(w[i]>thr*::sqrt((double)big.degree()*small.degree())) ++rank;
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int d=big.degree()+small.degree()-rank;
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return poly_gcd(big,small,thr,d);
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}
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template <typename T>
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Polynomial<T> Polynomial<T>::pow(const int n) const
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{
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if(n<0) laerror("negative exponent in polynomial::pow");
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if(n==0) {Polynomial<T> r(0); r[0]=(T)1; return r;}
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return positive_power(*this,n);
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}
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template <typename T>
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Polynomial<T> Polynomial<T>::powx(const int i) const
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{
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if(i<0) laerror("negative exponent in polynomial::pow");
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if(i==0) {Polynomial<T> r(0); r[0]= this->sum(); return r;}
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if(i==1) return *this;
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Polynomial<T> r(i*this->degree());
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r.clear();
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for(int j=0; j<=this->degree(); ++j) r[j*i] = (*this)[j];
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return r;
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}
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template <typename T>
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void Polynomial<T>::binomial(const int n)
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{
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resize(n,false);
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for(int i=0; i<=n; ++i) (*this)[i] = (T) binom(n,i);
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}
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template <typename T>
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Polynomial<T> Polynomial<T>::composition(const Polynomial &rhs) const
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{
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return value(*this,rhs);
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}
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/***************************************************************************//**
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* forced instantization in the corresponding object file
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******************************************************************************/
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template class Polynomial<int>;
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template class Polynomial<unsigned int>;
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template class Polynomial<double>;
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template class Polynomial<std::complex<double> >;
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#define INSTANTIZE(T) \
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template NRMat<T> Sylvester(const Polynomial<T> &p, const Polynomial<T> &q); \
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template Polynomial<T> lagrange_interpolation(const NRVec<T> &x, const NRVec<T> &y); \
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template Polynomial<T> poly_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr,const int d); \
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template Polynomial<T> svd_gcd(const Polynomial<T> &p, const Polynomial<T> &q, const typename LA_traits<T>::normtype thr); \
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INSTANTIZE(int)
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INSTANTIZE(unsigned int)
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INSTANTIZE(double)
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INSTANTIZE(std::complex<double>)
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#define INSTANTIZE2(T) \
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template Polynomial<T> hermite_polynomial(int); \
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INSTANTIZE2(int)
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INSTANTIZE2(double)
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}//namespace
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