Equation Solver Module

[Kieran Brown]

I think a set of equation solvers from linear to quartic should be implemented and optimized for getting exact solutions as well as approximations. I have made a simple example for a cubicSolve function

/// # Cubic Equation Solver
///
/// Solves the cubic equation of the form `ax^3 + bx^2 + cx + d = 0` - **eq: 1**
/// Should follow some variation of the following algorithm
/// - important:  a, b, c, d should all be real numbers.
/// - reference:   [Cubic Equation](https://en.wikipedia.org/wiki/Cubic_equation)
///
/// Procedure:
/// 1.  Divide both sides of equation by a.
///     i. In the case that a == 0 fallback onto the `quadraticSolve`.
///     ii. other wise move on to step 2.
///     The equation should now look as follows x^3 + (b/a)x^2 + (c/a)x + d/a = 0.
/// 2.  Calculate the discriminant D
///     i. Let r_1 = b/a ,  r_2 = c/a , and r_3 = d/a.
///     ii. Let `G = (3*r_2 - r_1^2)/9` and `F = (9*r_1*r_2 - 27*r_3 - 2*r_1^3)/54`
///     iii. Then `D = G^3 + F^2`
///   a. If `D > 0` , one real root with 2 complex conjugate roots.
///   b. if `D = 0` , all roots are real and atleast 2 are repeated.
///   c. If `D < 0`, all roots are real and unequal.
/// A.  `D > 0`,  where `cbrt` and `sqrt`  are the cuberoot and squareroot respectively.
///     1. The only real solution: ` x_1 = cbrt(F + sqrt(D)) + cbrt(F - sqrt(D)) - (1/3)*r_1`
///     2.  First Complex  solution :  `x_2 = Complex(-(1/2)(cbrt(F + sqrt(D)) + cbrt(F - sqrt(D))) - (1/3)*r_1 , (cbrt(3)/2)*cbrt(F + sqrt(D)) - cbrt(F - sqrt(D)))`
///     3.  Second Complex  solution:  `x_3 = Complex(-(1/2)(cbrt(F + sqrt(D)) + cbrt(F - sqrt(D))) - (1/3)*r_1 ,  -(cbrt(3)/2)*cbrt(F + sqrt(D)) - cbrt(F - sqrt(D)))`
/// B.  For `D = 0`  and `D < 0 ` Reduce Equation to Depressed Cubic
///     1. Make the substitution of `x = t - b/(3a)` in **eq: 1**  resulting in:
///              `t^3 + p*t + q = 0` -  **eq: 2** where:
///              `p = (3*a*c - b^2)/(3*a^2)` and `q =  (2b^3 - 9*a*b*c + 27*a^2*d)/(27*a^3)`
///              i. If `p = q = 0` then all roots `t_1 = t_2 = t_3 = 0`
///              ii. If `4*p^3 + 27*q^2 = 0` and  `p != 0` then  `t_1 = 3*q/p` and ` t_2 = t_3 =  -3*q/(2*p)`
///              iii. If `D < 0`  then `t(k) = 2*sqrt(-p/3)*cos((1/3)*acos((3*q/(2*p)*sqrt(-3/p))) - 2*pi*k/3)` for  `k = 1,2,3`
///     2. Convert back to x using `x(k) = t(k) - b/(3*a)`
///
///   - note: General Optimizations can be performed for trigonometric identities or for making approximations based upon values  of a ,b , c, and d
///
///
///
///
func cubicSolve<R: Real, C: Complex>(a: R, b: R, c: R, d: R) -> [C] {
    if a == 0 { return quadraticSolve(a: b, b: c, c: d) }

    let r_1 = b/a
    let r_2 = c/a
    let r_3 = d/a

    let g = (3.0*r_2 - r_1^2.0)/9.0 // G
    let f = (9*r_1*r_2 - 27*r_3 - 2*r_1^3)/54 // F


    let d = g^3 + f^2 // discriminant
    if d > 0 {
        let x_1 = cbrt(f + sqrt(d)) + cbrt(f - sqrt(d)) - (1/3)*r_1
        let x_2 = Complex(-(1/2)(cbrt(f + sqrt(d)) + cbrt(f - sqrt(d))) - (1/3)*r_1 , (cbrt(3)/2)*cbrt(f + sqrt(d)) - cbrt(f - sqrt(d)))
        let x_3 = Complex(-(1/2)(cbrt(f + sqrt(d)) + cbrt(f - sqrt(d))) - (1/3)*r_1 ,  -(cbrt(3)/2)*cbrt(f + sqrt(d)) - cbrt(f - sqrt(d)))
        return [x_1, x_2, x_3]

    } else if d = 0 {
        let p = (3*a*c - b^2)/(3*a^2)
        let q =  (2b^3 - 9*a*b*c + 27*a^2*d)/(27*a^3)
        let conversionFactor = -(-b/(3*a))

        if p == 0 && q == 0 {
            return [0, 0, 0]
        } else if p != 0 && 4*p^3 + 27*q^2 == 0 {

            let t_1 = 3*q/p
            let t_2 = -3*q/(2*p)
            let t_3 = -3*q/(2*p)
            return [t_1 + conversionFactor, t_2 + conversionFactor, t_3 + conversionFactor]
        }

    } else if d < 0 {
        return (1...3).map {
            2*sqrt(-p/3)*cos((1/3)*acos((3*q/(2*p)*sqrt(-3/p))) - 2*pi*$0/3) + conversionFactor
        }
    }
}

[Susan Cheng]

Checking discriminant == 0 is hard.
However, it always requires a solver version that only returns the real root.

For example, I don't need the points of interaction of two bezier curve which come from imaginary world. :)