static SBasis my_inverse(SBasis f, int order){
double a0 = f[0][0];
if(a0 != 0) {
f -= a0;
}
double a1 = f[0][1];
if(a1 == 0)
THROW_NOTINVERTIBLE();
//assert(a1 != 0);// not invertible.
if(a1 != 1) {
f /= a1;
}
SBasis g=SBasis(order, Linear());
g[0] = Linear(0,1);
double df0=derivative(f)(0);
double df1=derivative(f)(1);
SBasis r = Linear(0,1)-g(f);
for(int i=1; i<order; i++){
//std::cout<<"i: "<<i<<std::endl;
r=Linear(0,1)-g(f);
//std::cout<<"t-gof="<<r<<std::endl;
r.normalize();
if (r.size()==0) return(g);
double a=r[i][0]/pow(df0,i);
double b=r[i][1]/pow(df1,i);
g[i] = Linear(a,b);
}
return(g);
}
SBasis integral(SBasis const &c) {
SBasis a;
a.resize(c.size() + 1, Linear(0,0));
a[0] = Linear(0,0);
for(unsigned k = 1; k < c.size() + 1; k++) {
double ahat = -c[k-1].tri()/(2*k);
a[k][0] = a[k][1] = ahat;
}
double aTri = 0;
for(int k = c.size()-1; k >= 0; k--) {
aTri = (c[k].hat() + (k+1)*aTri/2)/(2*k+1);
a[k][0] -= aTri/2;
a[k][1] += aTri/2;
}
a.normalize();
return a;
}
