Quartic Equation

Nov 3 2016
def quarticFormula := qtF

def qtF $f $x :=
  match coefficients f x as list mathExpr with
    | $a_0 :: $a_1 :: $a_2 :: $a_3 :: $a_4 :: [] -> qtF' a_4 a_3 a_2 a_1 a_0

def qtF' $a $b $c $d $e :=
  match (a, b, c, d, e) as
    (mathExpr, mathExpr, mathExpr, mathExpr, mathExpr) with
    | (#1, #0, $p, #0, $q) ->
      let (s1, s2) := qF' 1 p q
          (r1, r2) := qF' 1 0 (- s1)
          (r3, r4) := qF' 1 0 (- s2)
       in (r1, r2, r3, r4)
    | (#1, #0, $p, $q, $r) ->
      let u := (3)#%1
                 (withSymbols [u]
                   cF (u * (p + u) ^ 2 + (-4) * r * u + (- (q ^ 2))) u)
          (r1, r2) := qF (y ^ 2 + (p + u) / 2 + sqrt u * (y - q / (2 * u))) y
          (r3, r4) := qF
                        (y ^ 2 + (p + u) / 2 + (- sqrt u) * (y - q / (2 * u)))
                        y
       in (r1, r2, r3, r4)
    | (#1, _, _, _, _) ->
      (4)#(%1 - b / 4, %2 - b / 4, %3 - b / 4, %4 - b / 4)
        (withSymbols [x, y]
          qtF
            (substitute
               [(x, y - b / 4)]
               (x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e))
            y)
    | (_, _, _, _, _) -> qtF' 1 (b / a) (c / a) (d / a) (e / a)

def w := ((-1) + i * sqrt 3) / 2

qtF ((x - 1) * (x - 2) * (x - 3) * (x - 4)) x
-- (4, 1, 3, 2)
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