Plot CreateMesh λ γ

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

&[DATE] &[TIME] - &[FILENAME]

Finding Distance of a Point

Plot CreateMesh λ γ

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 - 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 -^:

3D

Example 1

t.o 0 time :

sur Tore eval : A 7 - 712 mat 4 - 412 mat 311 mat rndBox : tmp x 2^y 2^+ z 2^+ 22^+ 12^- (: x y z f tmp 2^422^* x 2^y 2^+ (* - : B 3 A 00013 mat ε N Dist : a δ No s Cross λ 16 mat B : 61 sys

Plot3

a 0.6588 2.3021 - 0.9189 13 mat

A 0.9046 3.1611 - 313 mat

s rows 6

δ 2.2648

Cross 2.3684 109 -^* -

Example 2

sur fcos eval : A 8 - 812 mat 8 - 812 mat 511 mat rndBox : x y z f 2 x 2^7 / y 2^14 / + cos * z - : B 3 A 00013 mat ε N Dist : a δ No s Cross λ 16 mat B : 51 sys

Plot3

a 0.1314 - 2.4844 - 1.8067 13 mat

A 0.2342 - 3.4567 - 513 mat

s rows 6

δ 3.3397

Cross 7.2 1014 -^*

Example 3

sur par eval : A 4 - 412 mat 4 - 412 mat 611 mat rndBox : x y z f x 2^22^/ y 2^12^/ + z - : B 3 A 00013 mat ε N Dist : a δ No s Cross λ 16 mat B : 51 sys

Plot3

a 1.488 2.2372 5.5586 13 mat

A 1.1596 0.2623 613 mat

s rows 8

δ 2.0501

Cross 4.842 1012 -^* -

Example 4

sur fexp eval : A 2 - 212 mat 2 - 212 mat 611 mat rndBox : x y z f 8 x 2^2 y * (2^+ 2 / - exp * z - : B 3 A 00013 mat ε N Dist : a δ No s Cross λ 16 mat B : 51 sys

Plot3

a 0.7502 0.0954 - 5.929 13 mat

A 1.0659 0.2559 - 613 mat

s rows 11

δ 0.3612

Cross 2.2274 109 -^*

0 time t.o - 39.542 s *

2D Examples

Example 1

t.o 0 time :

A 6 - 612 mat 4 - 412 mat rndBox : x y f x 2^y * x y + sin - : B 2 A 0012 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 5.8793 0.0117 - 12 mat

A 5.8649 0.449 12 mat

s rows 9

δ 0.4609

Cross 1.5536 1010 -^*

Example 2

A 4 - 212 mat 1 - 312 mat rndBox : 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 1.9397 - 0.2861 12 mat

A 1.9251 - 0.3111 12 mat

s rows 11

δ 0.029

Cross 6.1553 1011 -^* -

Example 3

A 2 - 212 mat 1.5 - 1.5 12 mat rndBox : αo 0.842 : x y f 1 x 2^y 2^+ αo 2^- (2^2 x * y αo - (* (2^+ / 2 - : B 2 A 0 A 2 el 12 mat ε N Dist : a δ No s Cross λ 16 mat B : 51 sys

Plot

a 0.2052 - 0.868 - 12 mat

A 1.4516 - 1.0923 - 12 mat

s rows 3

δ 1.2664

Cross 3.311 1010 -^*

Example 4

816 π * sqrt * eval 351.6702

Disabled because take a long time.

A 4 - 4 - 12 mat 2 - 212 mat rndBox : x y f e 13 / x 2^y 2^+ sqrt * -^x 2^2 y 2^* - 351.6702 / * 0.005 + : B 2 A 0012 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 2.5758 - 2.4945 12 mat

A 4 - 0.6159 12 mat

s rows 9

δ 2.3574

Cross 1.7952 1010 -^*

Example 5

A 12 - 1212 mat 4 - 1012 mat rndBox : x y f 2 x 2^* 4 x * y * + 4 y 3^* + 26 y * - 22 - : B 2 A 0 A 2 el 12 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 0.2052 - 0.868 - 12 mat

A 7.6661 9.7702 12 mat

s rows 3

δ 1.2664

Cross 3.311 1010 -^*

Example 6

A 1 - 112 mat 1 - 112 mat rndBox : x y f x 2^y 2^+ (2^x 3^3 x * y 2^* - (- : B 2 A 0 A 2 el 12 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 0.5561 - 0.6603 - 12 mat

A 0.9082 - 0.5955 - 12 mat

s rows 8

δ 0.358

Cross 6.4132 108 -^* -

Example 7

A 4 - 412 mat 4 - 412 mat rndBox : x y f x 6^y 6^+ 26^- : B 2 A 0012 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 2 - 0.1793 12 mat

A 1.4941 - 0.1793 12 mat

s rows 8

δ 0.5059

Cross 5.959 1012 -^* -

Example 8

A 1 - 612 mat 2 - 212 mat rndBox : x y f x 2 - (2^5 y 2^* + x 2^y 10^* + 3 x y * sin * - 9 - : B 2 A 0012 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 0.0545 - 0.9938 - 12 mat

A 0.0487 - 0.9474 - 12 mat

s rows 11

δ 0.0468

Cross 3.2088 1010 -^* -

Example 9

A 1 - 112 mat 1 - 112 mat rndBox : x y f x 4^x 2^y * - y 3^+ : B 2 A A 1 el 012 mat ε N Dist : a δ No s Cross λ 16 mat B : 41 sys

Plot

a 0.0545 - 0.9938 - 12 mat

A 0.9741 0.6283 12 mat

s rows 11

δ 0.0468

Cross 3.2088 1010 -^* -

Example 10

A 1.5 - 1.5 12 mat 1.5 - 1.5 12 mat rndBox : a b g a 0.125 10 b * sin * + (4^: x y f x y g y x g + 0.4 4^- : B 2 A A 1 el 012 mat ε N Dist : a δ No s Cross λ 16 mat B : 51 sys

Plot

a 0.4456 0.4711 12 mat

A 0.7613 1.1704 12 mat

s rows 5

δ 0.7672

Cross 2.4839 109 -^*

0 time t.o - 103.246 s *