e02ac5bb-37b1-417e-8517-ad2bcf03219c 11 6 5 decimal

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Lagrange elliptical multipliers ...

1:: Define/Install the system

Given some numerical values ... a b c 13 mat 25 sqrt 513 mat :

... for any candidate points x 2^y 2^+ z 2^+ λ - ≡

Minimize & Maximize the parameter 'λ'

2:: Subject the Lagrange system

φ x 2^a 2^/ y 2^b 2^/ + z 2^c 2^/ + 1 - : Ψ x y + z - : 21 sys

subject to Φ x 2^y 2^+ z 2^+ :

The formal "Lagrangian" L Φ λ φ * + μ Ψ * + :

The Lagrange gradients
wrt intrants [x,y,z,λ,μ]
From other artefacts μ=1 fx L x diff : fy L y diff : fz L z diff : fλ L λ diff : fμ L μ diff : 51 sys x 4 λ + (* 2 μ * + 2 / 2 y * 5 λ + (* 5 μ * + 5 / 2 z * 25 λ + (* 25 μ * - 25 / 554 - x 2^+ (* 4 y 2^* + (* 4 z 2^* + 100 / x y + z - 51 sys

3:: Numerical Maximize/Minimize 'λ'

V X 15 range el vectorize 11 line :

Credit and thanks to Alvaro [20181023] for the dressing the roots solver

Numerical Minimize 'λ' X F x X 1 el : y X 2 el : z X 3 el : λ X 4 el : μ X 5 el : 15 mat fx fy fz fλ fμ 15 mat transpose 21 line :

Guess 'λ' values λ 21 - 04 - 151 mat :

λ.1 X F V λ roots : 1.5738 1.3771 - 0.1967 4.4118 - 0.324 51 mat

Numerical Maximize 'λ'
The maximum of F is the minimum of G = -F X G x X 1 el : y X 2 el : z X 3 el : λ X 4 el : μ X 5 el : 15 mat fx - fy - fz - fλ fμ 15 mat transpose 21 line :

Guess 'λ' values λ 11310 - 351 mat :

λ.2 X G V λ roots : 1.026 1.539 2.5649 10 - 3.0779 51 mat

4:: Recover 'λ' minor/major elliptical radii

λ.1 λ.1 13 range el : λ.2 λ.2 13 range el : 21 sys

λ λ.1 norme λ.2 norme 21 sys : 2.10042 3.162278 21 sys