// cohen, 5.7.2, p. 270
func regulatorRealQuad(poly *poly.IntPolynomial) float64 {
D := Discriminant(poly).Int64()
f := math.Sqrt(float64(D))
c := -poly.Coeff(0).Int64()
// try all of the small elements
for x := int64(1); x < 100; x++ {
for y := int64(1); y < 100; y++ {
z := x*x - c*y*y
if z == 4 || z == -4 {
u := x
v := y
return math.Log((float64(u) + float64(v)*math.Sqrt(float64(c))) / 2.0)
}
}
}
d := int64(math.Floor(f))
var b int64
if d&1 == D&1 {
b = d
} else {
b = d - 1
}
var u, v int64
u1 := -b
u2 := int64(2)
v1 := int64(1)
v2 := int64(0)
p := b
q := int64(2)
for {
A := (p + d) / q
t := p
p = A*q - p
if t == p && v2 != 0 {
u = (u2*u2 + v2*v2*D) / q
v = (2 * u2 * v2) / q
break
}
u1, u2 = u2, A*u2+u1
v1, v2 = v2, A*v2+v1
t = q
q = (D - p*p) / q
if q == t && v2 != 0 {
u = (u1*u2 + D*v1*v2) / q
v = (u1*v2 + u2*v1) / q
break
}
}
return math.Log((math.Abs(float64(u)) + math.Abs(float64(v))*f) / 2.0)
}
// Discriminant returns the discriminant of this polynomial.
func Discriminant(p *poly.IntPolynomial) *mathx.Int {
if p.Degree() == 2 {
c, b, a := p.Coeff(0), p.Coeff(1), p.Coeff(2)
ac4 := mathx.NewInt(4).Mul(a).Mul(c)
return b.Mul(b).Sub(ac4)
}
return nil
}