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| (appendix_solvecubic)= | ||
| # Решение кубического уравнения | ||
|
|
||
| ```{proof:function} solve_cubic | ||
| **Решение кубического уравнения** | ||
| :::julia | ||
| """ | ||
| solve_cubic(a, b, c, d) -> Tuple{T,T,T} | ||
| Решает в действительных числах кубическое уравнение вида | ||
| a x³ + b x² + c x + d = 0 | ||
| Вовзвращает кортеж из трёх корней, где первые корни действительные, | ||
| а комплексные корни представлены как not a number. | ||
| """ | ||
| function solve_cubic(a, b, c, d) | ||
| A, B, C, D = promote(float(a), float(b), float(c), float(d)) ./ (1, 3, 3, 1) | ||
| δ₁ = A * C - B * B | ||
| δ₂ = A * D - B * C | ||
| δ₃ = B * D - C * C | ||
| d13 = δ₁ * δ₃ | ||
| d22 = δ₂ * δ₂ | ||
| Δ = 4 * d13 - d22 | ||
| nanvalue = zero(A) / zero(A) | ||
| if Δ < 0 | ||
| At = Cb = Db = zero(A) # A-tilde, C-bar, D-bar | ||
| if B^3 * D ≥ A * C^3 | ||
| At, Cb, Db = A, δ₁, -2 * B * δ₁ + A * δ₂ | ||
| else | ||
| At, Cb, Db = D, δ₃, -D * δ₂ + 2 * C * δ₃ | ||
| end | ||
| T₀ = -copysign(At, Db) * sqrt(-Δ) | ||
| T₁ = -Db + T₀ | ||
| p = cbrt(T₁ / 2) | ||
| q = T₁ == T₀ ? -p : -Cb / p | ||
| x₁ = Cb ≤ 0 ? p + q : -Db / (p^2 + q^2 + Cb) | ||
| x, w = B^3 * D ≥ A * C^3 ? (x₁ - B, A) : (-D, x₁ + C) | ||
| return (x/w, nanvalue, nanvalue) | ||
| else | ||
| δ₁ == δ₂ == δ₃ == 0 && return (-B/A, -B/A, -B/A) | ||
| sΔ = sqrt(Δ) | ||
| θA, θD = abs.(atan.((A*sΔ, 2*B*δ₁ - A*δ₂, D*sΔ, D*δ₂ - 2*C*δ₃)) ./ 3) | ||
| sCA, sCD = sqrt.(.-min.((δ₁, δ₃), 0)) | ||
| x₁A, x₁D = 2 .* (sCA, sCD) .* cos.((θA, θD)) | ||
| x₃A, x₃D = .-((sCA, sCD)) .* (cos.((θA, θD)) .+ sqrt(3) .* sin.((θA, θD))) | ||
| xlt = (x₁A + x₃A > 2 * B) ? x₁A : x₃A | ||
| xst = (x₁D + x₃D < 2 * C) ? x₁D : x₃D | ||
| xl, wl = xlt - B, A | ||
| xs, ws = -D, xst + C | ||
| E = wl * ws | ||
| F = -xl * ws - wl * xs | ||
| G = xl * xs | ||
| xm, wm = C * F - B * G, C * E - B * F | ||
| return (xs/ws, xm/wm, xl/wl) | ||
| end | ||
| end | ||
| ::: | ||
| >Оригинальный алгоритм изложен в работе: | ||
| > | ||
| >J. F. Blinn. | ||
| >How to Solve a Cubic Equation, Part 5: Back to Numerics. | ||
| >IEEE Computer Graphics and Applications, V. 27, no. 3, pp. 78-89, 2007. | ||
| >https://doi.org/10.1109/MCG.2007.60 | ||
| > | ||
| >Имплементация взята из пакета [CubicEoS.jl](https://github.com/vvpisarev/CubicEoS.jl). | ||
| ``` |
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