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newton solver for polynomial

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This commit is contained in:
Jiri Pittner 2021-06-12 17:49:43 +02:00
parent f03953ba2d
commit 73aed62650
3 changed files with 51 additions and 12 deletions
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16
polynomial.cc
View File

@ -142,6 +142,22 @@ return p;
}
template <typename T>
T Polynomial<T>::newton(const T x0, const typename LA_traits<T>::normtype thr, const int maxit) const
{
Polynomial<T> d=derivative(1);
T x=x0;
for(int i=0; i<maxit; ++i)
{
T v=value(*this,x);
if(abs(v)<thr) break;
x -= v/value(d,x);
}
return x;
}
/***************************************************************************//**

35
polynomial.h
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@ -72,11 +72,11 @@ public:
for(int i=0; i<=rhs.degree(); ++i) for(int j=0; j<=degree(); ++j) r[i+j] += rhs[i]*(*this)[j];
return r;
};
void simplify(const typename LA_traits<T>::normtype thr)
void simplify(const typename LA_traits<T>::normtype thr=0)
{
NOT_GPU(*this);
int n=degree();
while(n>0 && abs((*this)[n])<thr) --n;
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()];};
@ -103,27 +103,41 @@ public:
return r;
}
}
Polynomial derivative() const
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==0)
if(n-d<0)
{
Polynomial r(0);
r[0]=0;
return r;
}
Polynomial r(n-1);
for(int i=1; i<=n; ++i) r[i-1] = (*this)[i]* ((T)i);
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
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+1);
r[0]=0;
for(int i=0; i<=n; ++i) r[i+1] = (*this)[i]/((T)(i+1));
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;
}
void polydiv(const Polynomial &rhs, Polynomial &q, Polynomial &r) const;
@ -132,6 +146,7 @@ public:
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
//@@@gcd, lcm euler and svd

12
t.cc
View File

@ -2203,20 +2203,25 @@ cout << (value(p,u)*value(q,u) -value(r,u)).norm()<<endl;
}
if(0)
if(1)
{
int n;
cin >>n ;
NRVec<double> r(n);
r.randomize(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);
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]);
cout<<"test newton "<<xdif<<endl;
}
if(1)
if(0)
{
int n;
cin >>n ;
@ -2227,6 +2232,9 @@ Polynomial<double> p=lagrange_interpolation(x,y);
cout <<x<<y<<p;
NRVec<double> yy=values(p,x);
cout<<"interpolation error= "<<(y-yy).norm()<<endl;
Polynomial<double>q=p.integral(2);
Polynomial<double>pp=q.derivative(2);
cout<<"test deriv. "<<(pp-p).norm()<<endl;
}
}