<p>Plot CreateMesh λ γ</p>

9e0e925d-186b-49dc-99b9-5d0a9c29a453 55 4 5 false false 0 decimal

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Finding Distance of a Point

γ 0.8232 - 0.4194 - 0.3827 0.5677 0.6187 - 0.543 0.009 0.6643 0.7474 33 mat :

θ 0 2.01 π * (0.1 range :

λ 1 - (10 random -^eval : λ1 1 - (10 random -^eval : λ2 1 - (10 random -^eval : 31 sys

x rndBox x length 1 ≡ x 1 el a b 12 mat x min x max 12 mat eval : a b a - (109 -^* 109^random * + eval 21 line if 11 line :

a rndBoxS a 1 - (10 random -^* 11 mat eval 11 line :

x y rndBox x rndBox y rndBox 12 mat eval 11 line :

x y z rndBox x rndBox y rndBox z rndBox 13 mat eval 11 line :

2D Plot Plot x y f No a 1 el a 2 el a 15 black 15 mat a 1 el a 2 el . 5 black 15 mat A 1 el A 2 el . 5 red 15 mat A 1 el A 2 el A 11 black 15 mat A a stack s δ θ cos vectorize * A 1 el + δ θ sin vectorize * A 2 el + augment 91 sys :

u v F2 21 u cos * + (v cos * 21 u cos * + (v sin * 1 u sin * 31 mat :

u v F1 u v 8 u 2^v 2^+ 2 / - exp * 31 mat :

u v F3 u v 2 u 2^7 / v 2^14 / + cos * 31 mat :

u v f4 2 u sqrt * v cos * 1 u sqrt * v sin * u 31 mat :

NN 24 :

fexp u v F1 3 - 33 - 3 NN NN CreateMesh 11 line :

fcos u v F3 7 - 78 - 8 NN NN CreateMesh 11 line :

Tore u v F2 02 π * 02 π * NN NN CreateMesh 11 line :

par u v f4 0902 π * 0.5 NN * NN CreateMesh 11 line :

3D Plot Plot3 Σγ sur γ * eval : Lγ s γ * eval : aAγ A a stack γ * eval : 13 mat A γ * (1 el A γ * (2 el . 6 red 15 mat No γ * 1 col No γ * 2 col augment a γ * (1 el a γ * (2 el a 16 black 15 mat a γ * (1 el a γ * (2 el . 6 black 15 mat Σγ 1 col Σγ 2 col augment A γ * (1 el A γ * (2 el A 12 black 15 mat aAγ 1 col aAγ 2 col augment Lγ 1 col Lγ 2 col augment s 113 range el γ * (1 el s 113 range el γ * (2 el . 6 green 15 mat 91 sys 21 line :

C A x.o ε N Dist a x.o transpose : c 1 C range : g 0 : r 0 : s x.o : NA A x.o - norme eval : 16 mat x L x κ el A κ el - (2^κ κ Clear C sum x C 1 + el x F * + : C 2 ≡ x F x 1 el x 2 el f : x Φ x L x 1 el diff x L x 2 el diff x L x 3 el diff 13 mat transpose : 21 line x F x 1 el x 2 el x 3 el f : x Φ x L x 1 el diff x L x 2 el diff x L x 3 el diff x L x 4 el diff 14 mat transpose : 21 line if x ϕ x A transpose - norme NA 1 k N / - (* - x F 21 mat : g c el x F x c el diff : x G g : 13 mat k 0 N range a a x ϕ al_nleqsolve eval : a F abs ε ≤ r r 1 + : s r c el a c el eval : 12 mat a c el s s rows c el : See note 12 mat if 21 line for α s r c el : α α transpose 0 stack x Φ al_nleqsolve eval : 12 mat a α c el transpose : G a G : δ A a - norme : eN G G norme / : D A a - transpose δ / : 15 mat a δ a a δ 2 / eN transpose * + stack s a stack eN 1 el D 2 el * eN 2 el D 1 el * - α C el 16 mat eval 81 line :

Changes to your code are: - Use row vectors as points - You give h as parameter, I convert it to a number of steps (guess that is correct, and must to give the same results) - Use nomrme(A-xo) instead norme(A) as the length frome where try to solve This is the actual major change and I don't know if it works better. - Add the Laplace multipliers as the final refinament

Sometimes can't adress any point to s, but this modification seems that can't correct this issue:

r 0 ≡ (N 1000 < (& N trace C A x.o ε + eval ε 2 N * Dist 21 line α s r c el : α α transpose 0 stack x Φ al_nleqsolve eval : 12 mat a α c el transpose : G a G : δ A a - norme : eN G G norme / : D A a - transpose δ / : 15 mat a δ a a δ 2 / eN transpose * + stack s a stack eN 1 el D 2 el * eN 2 el D 1 el * - α C el 16 mat eval 31 line if

Note: I'm not sure if we need to take the s-value, this is, the best root, or leave a as the last obtained root, but which don't verify abs(f(a))<ε

For the cross Product:

Cross eN D † (3 el ≡

then

Cross eN 1 el D 2 el * eN 2 el D 1 el * - ≡

for cover 2D and 3D cases.

Default iterations and error:

N 10 :

ε 103 -^:

Example 2

Shows a bad initial guess and a good one. Cross value is similar in both cases.

A 2 - 212 mat : x y f 11 x + (3^+ 3 y * 1 x + (* - y 3^+ abs ln : B 2 A 0 1.5 12 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

A 4 - 212 mat 1 - 312 mat rndBox :

a 0.6527 - 112 mat

A 2 - 212 mat

s rows 3

δ 1.6779

Cross 5.2998 1010 -^* -

A 2 - 212 mat : x y f 11 x + (3^+ 3 y * 1 x + (* - y 3^+ abs ln : B 2 A 2 - 0.5 12 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 2.8794 - 112 mat

A 2 - 212 mat

s rows 3

δ 1.3317

Cross 7.9915 1010 -^*