void unipoly::normal_base(INT p, matrix& F, matrix& N, INT verbose_level)
// compare Lueneburg~\cite{Lueneburg87a} p. 106.
{
INT f_v = (verbose_level >= 1);
domain d(p);
with ww(&d);
INT i, f;
Vector v, V;
unipoly x, mue;
f = degree();
F.Frobenius(*this, p, FALSE);
if (f_v) {
cout << "unipoly::normal_base(): Frobenius:\n" << F << endl;
}
F.KX_cyclic_module_generator(v, mue, verbose_level - 1);
V.m_l(f);
V[0] = v;
x.x();
if (f_v) {
cout << "unipoly::normal_base(): V[0]=" << v << endl;
}
for (i = 1; i < f; i++) {
F.KX_module_apply(x, v);
V[i] = v;
if (f_v) {
cout << "unipoly::normal_base(): V["<<i<<"]=" << v << endl;
}
}
N.from_vector_of_columns(V);
if (f_v) {
cout << "unipoly::normal_base(): N=\n" << N << endl;
}
#if 0
matrix T, P, Pv, Q, Qv;
integer a1;
Vector v, vv, b, bb;
T = F;
T.elements_to_unipoly();
T.minus_X_times_id();
if (f_v) {
cout << "F - x * Id=\n" << T << endl;
}
T.smith_normal_form(P, Pv, Q, Qv, f_vv, FALSE);
if (f_v) {
cout << "Q=\n" << Q << endl;
}
a1.m_i_i(1);
Q.evaluate_at(a1);
if (f_v) {
cout << "Q(1)=\n" << Q << endl;
cout << "F=\n" << F << endl;
}
Q.to_vector_of_columns(v);
b = v.s_i(d - 1);
vv.m_l(d);
vv[1] = b;
if (f_v) {
cout << "N[0]=" << b << endl;
}
for (i = 1; i < d; i++) {
bb.mult(F, b);
b = bb;
vv[i] = b;
if (f_v) {
cout << "N[" << i << "]=" << b << endl;
}
}
N.from_vector_of_columns(vv);
if (f_v) {
cout << "N=" << N << endl;
}
#endif
}
