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quartic-equation.go
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quartic-equation.go
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// https://en.wikipedia.org/wiki/Quartic_equation
package main
import (
"fmt"
"math/cmplx"
)
func main() {
fmt.Println(quartic([]float64{7, 0, 0, 0, 5}))
fmt.Println(quartic([]float64{7, 50, 49, 6, 5}))
fmt.Println(quartic([]float64{10, 4, 0, 0, 0}))
fmt.Println(quartic([]float64{2, 4, 6, 8, 10}))
fmt.Println(quartic([]float64{-6, -4, 190, 45, 19}))
}
func linear(v []float64) (z []complex128) {
a := complex(v[0], 0)
b := complex(v[1], 0)
switch {
case a == 0 && b == 0: // one real root
z = []complex128{0}
case a == 0: // no solution
default: // one real root
z = []complex128{-b / a}
}
return
}
func quadratic(v []float64) (z []complex128) {
a := complex(v[0], 0)
b := complex(v[1], 0)
c := complex(v[2], 0)
d := b*b - 4*a*c
switch {
case a == 0: // equation collapsed to linear
z = linear(v[1:])
case d == 0: // one real root
z = []complex128{-b / (2 * a)}
default: // two complex roots
d = cmplx.Sqrt(d)
z = []complex128{
(-b + d) / (2 * a),
(-b - d) / (2 * a),
}
}
return
}
func cubic(v []float64) (z []complex128) {
a := complex(v[0], 0)
b := complex(v[1], 0)
c := complex(v[2], 0)
d := complex(v[3], 0)
d0 := b*b - 3*a*c
d1 := 2*b*b*b - 9*a*b*c + 27*a*a*d
d2 := cmplx.Sqrt(d1*d1 - 4*d0*d0*d0)
Z := (-1 + cmplx.Sqrt(-3)) / 2
C := (d1 + d2) / 2
if C == 0 {
C = (d1 - d2) / 2
}
switch {
case a == 0: // equation collapsed to quadratic
z = quadratic(v[1:])
case C == 0: // only real root
z = []complex128{-1 / (3 * a) * b}
default: // one real root, two complex roots
C = cmplx.Pow(C, 1.0/3)
z = []complex128{
-1 / (3 * a) * (b + C + d0/C),
-1 / (3 * a) * (b + Z*C + d0/(Z*C)),
-1 / (3 * a) * (b + Z*Z*C + d0/(Z*Z*C)),
}
}
return
}
func quartic(v []float64) (z []complex128) {
a := complex(v[0], 0)
b := complex(v[1], 0)
c := complex(v[2], 0)
d := complex(v[3], 0)
e := complex(v[4], 0)
// collapsed to cubic
if a == 0 {
return cubic(v[1:])
}
p := (8*a*c - 3*b*b) / (8 * a * a)
q := (b*b*b - 4*a*b*c + 8*a*a*d) / (8 * a * a * a)
d0 := c*c - 3*b*d + 12*a*e
d1 := 2*c*c*c - 9*b*c*d + 27*b*b*e + 27*a*d*d - 72*a*c*e
dm := d1*d1 - 4*d0*d0*d0
// three common roots, one simple root
if dm == 0 && d0 == 0 {
r := quadratic([]float64{12 * v[0], 6 * v[1], v[2]})
r0 := r[0]
r1 := r[1]
m0 := cmplx.Abs(a*r0*r0*r0*r0 + b*r0*r0*r0 + c*r0*r0 + d*r0 + e)
m1 := cmplx.Abs(a*r1*r1*r1*r1 + b*r1*r1*r1 + c*r1*r1 + d*r1 + e)
x0 := r[0]
if m1 < m0 {
x0 = r[1]
}
x1 := -b/a - 3*x0
return []complex128{x1, x0, x0, x0}
}
// four roots
dq := d1 * d1
if dm != 0 && d0 == 0 {
dq = -dq
}
Q := cmplx.Pow(0.5*(d1+cmplx.Sqrt(dq-4*d0*d0*d0)), 1/3.0)
S := 0.5 * cmplx.Sqrt(-2*p/3+(Q+d0/Q)/(3*a))
y0 := 0.5 * cmplx.Sqrt(-4*S*S-2*p+q/S)
y1 := 0.5 * cmplx.Sqrt(-4*S*S-2*p-q/S)
x := -b / (4 * a)
z = []complex128{
x - S + y0,
x - S - y0,
x + S + y1,
x + S - y1,
}
return
}