Dual solver algo style ... xd(L,H,N)

25fd0342-dbd1-4eb1-8ae8-26e3bf3fb9ca 52 15 12 decimal

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

U # # f Solve i 1 U rows range data 1 i el U i el y f y solve eval : for v data 1 el : j 2 data cols range v v data j el augment : for v transpose 41 line :

Incremental discretizer ...
Mesh N  within the bounds L, H L H N xd X 0 : i 1 N range X i el L H L - N / i * + : for X 31 line :

The Electron function has no symbolic branch solution otherwise than RootOf which indirectly requires "a solver" from recast RootOf.

t0 1 time :

L 1.7857 : H 14.245529 : N 200 : 13 mat U L H N xd : 21 sys

3 appVersion 0.98.6179 ≡

Optimiz Numeric ... solving the Electron function a 816 π * sqrt * : x y Φ 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + : sols U x y Φ Solve : 31 sys

collect/export both branches new sols : branch1 U new 1 col augment : branch2 U new 2 col augment : 31 sys

L < u < H ... escape end-points u π :

branch1 branch2 branch1 - branch2 - 4 1 sys

x pairwise user x y Φ y solve : x user 12 mat 21 line :

pair wise u pairwise 3.141593 1.464026 - 1.464026 21 mat 12 mat

1 time t0 - min 3.9

roots rejects vectorize
less accurate than solve y x y Φ y roots at

OK recursive solve Δ xlimits : 1.7857 14.245529 at

NFG recursive solve Δ xlimits : 1.78568578 14.245529381 at

collected al_nleqsolve 1.78568577383018 14.2455293828432 21 sys

======================================

x limitSolve sol y x y Φ vectorize y solve at : x sol 12 mat 21 line :

sanity the limits u 1.78568577383018 14.2455293828432 21 mat :

u limitSolve vectorize 1.785685773830 0.000000000621 12 mat 14.245529382843 0.000000000358 12 mat 21 mat

AlgLib implicit solve(x) given y(0)  in  f(x)  (Viacheslav) result k 1 x f rows range u k el x k el : for x Jac u f u Jacobian : StepMax 0 : Eps 1014 -^: u X0 StepMax Eps x f x Jac al_nleqsolve : 51 line :

L H 21 mat 1.78568577383018 14.2455293828432 21 sys ≡

a 816 π * sqrt * : 351.67021

solving for 'x' in vector function
wrt x1 'x' , x2 'y' ... y=0 in the
given vector function. x f 1 - 3 / x 1 el (2^x 2 el (2^+ sqrt * exp 2 x 2 el (2^* - x 1 el (2^+ a / * - 0.005 + 021 mat :

X0 1.8 021 mat :

result 1.78568577383018 021 mat

X0 14021 mat :

result 14.2455293828432 021 mat

al_nleqsolve

AlgLib implicit solve(x) given y(0)  in  f(x)  (Viacheslav) result k 1 x f rows range u k el x k el : for x Jac u f u Jacobian : StepMax 0 : Eps 1015 -^: u X0 StepMax Eps x f x Jac al_nleqsolve : 51 line :

L H 21 mat 1.78568577383018 14.2455293828432 21 sys ≡

a 816 π * sqrt * : 351.67021

solving for 'x' in vector function
wrt x1 'x' , x2 'y' ... y=0 in the
given vector function. x f 1 - 3 / x 1 el (2^x 2 el (2^+ sqrt * exp 2 x 2 el (2^* - x 1 el (2^+ a / * - 0.005 + 021 mat :

X0 1.78 021 mat :

result 1.78568577382904 021 mat

X0 14.25 021 mat :

result 14.2455293828433 021 mat

X0 1.8 021 mat :

result 1.78568577382905 021 mat

X0 14021 mat :

result 14.2455293828432 021 mat

x f 1 - 3 / x (2^y 2^+ sqrt * exp 2 y 2^* - x (2^+ a / * - 0.005 + :

y 0 :

1.78568577382904 f 14.2455293828433 f 21 mat 2.813887664239 1017 -^* 1.600789426767 1017 -^* 21 mat

x y f 1 - 3 / x (2^y 2^+ sqrt * exp 2 y 2^* - x (2^+ a / * - 0.005 + :

x y axes x y / - :

f axes 1.7857 2 1.7857 2 - 2 2 mat 3 1 sys

x y f x 3^y 3^- 31 0.4 - (* x * y * - :

y F x y f 0 ≡ x solve maple :

maple(solve(f(x,y)≡0,x))

x F 203 nthroot (2^25 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot (2^* 60 x * + 10203 nthroot * 25 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot * / 203 nthroot (2^25 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot (2^3 x * 203 nthroot * + (203 nthroot (2^* - i 3 sqrt * 203 nthroot (2^25 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot (2^* 60 x * - (* + (* 40025 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot * / 203 nthroot (2^25 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot (2^3 x * 203 nthroot * + (203 nthroot (2^* i 3 sqrt * 203 nthroot (2^25 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot (2^* 60 x * - (* + (* 40025 x 3^* 5 sqrt x 3^108 - 125 x 3^* + (* sqrt * + 3 nthroot * / - 31 sys :

x q1 x F 1 el Re eval : x q2 x F 2 el Re eval : x q3 x F 3 el Re eval : 31 sys

u 0.95245 :

X 0.755957587291973 - :

x y axes x y / - :

u q2 u q3 - u q3 21 mat 0.000000000000000 0.755957587291973 - 21 mat

a 1 0.4 - (:

x c2 1 x u ≤ (0 x ≤ (& cases :

Area loop 3 a 2^* 2 / 0.54

ε 1012 -^:

x q1 x q2 x c2 * x q3 x c2 * u X + 15 black 1 5 mat 4 1 sys 1 - ( x q1 * 1 - ( x q2 * x c2 * 1 - ( x q3 * x c2 * u X - + 15 black 1 5 mat 4 1 sys

x q3 x ε u int x q2 x ε u int - 0.540

x q2 x ε u int x q3 x ε u int - 0.540 -

Φ axes 2 1 sys f axes 2 1 sys

x y Descartes1 x 3^y 3^+ 1 x * y * - : x y Descartes2 x 3^y 3^+ 2 x * y * - : x y Descartes3 x 3^y 3^+ 3 x * y * - : 31 sys

Descartes1 Descartes2 Descartes3 axes 4 1 sys

L'aire de la boucle est égale à celle du domaine situé entre la courbe et son asymptote (d'équation x + y = –a) ; valeur commune : 3a^2/2. https://www.mathcurve.com/courbes2d/courbes2d.shtml

In closed agreement ... can't verdict

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y 1 round vectorize augment 61 line :

Cumulative integration over data set. Last value is the total area under data.

branch2 Σ

x branch2 1 col : y branch2 2 col : 12 mat

branch2 Σ branch2 Σ rows row 14.24553 37.2 12 mat

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

NOT accurate A 37.4

vx branch2 1 col : vy branch2 2 col : 12 mat

x c 1 1.7857 x ≤ (x 14.245529 ≤ (& cases :

x electron vx vy x ainterp :

Int(f,a,b,n) ... Simpson/Romberg Kernel

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : α 0 : β 1 : h β α - n / : 14 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

Shade area

'N' random in critical region N 1000 :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

size/define critical region.
Plot first before filling xmin 0 : xmax 15 : ymin 0 : ymax 5 : 22 mat space_x xmax xmin - (: space_y ymax ymin - (: 21 line 21 sys

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x electron ≤ (:

A routine that finds an array of N random 
pairs in the critical region. Pts i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 21 line :

a b c spec . 5 #808080 13 mat : fill Pts a b c spec pts : 21 line

x f x electron :

a 1.7857 : b 14.245529 : n 100 : 13 mat

f a b n Int 37.2143

X vx : Y vy : A 37.2143 : 13 mat

Perimeter ... Center of Gravity

Perimeter  Polygon wrt vertices coordinates P X i 1 + el X i el - (2^Y i 1 + el Y i el - (2^+ sqrt i 1 X rows 1 - sum : 15.914

Center of gravity wrt vertices coordinates Xo 16 A * / X i el X i 1 + el + (X i el Y i 1 + el * X i 1 + el Y i el * - (* i 1 X rows 1 - sum * : Yo 16 A * / Y i el Y i 1 + el + (X i el Y i 1 + el * X i 1 + el Y i el * - (* i 1 X rows 1 - sum * : 21 mat

Xo Xo abs : Yo Yo abs : 21 mat

a 2 a 2 - 2 2 mat b 3 b 3 - 2 2 mat branch2 fill Xo Yo . 15 blue 1 5 mat 5 1 sys

Center of gravity Xo Yo 21 mat 8.72 1.69 21 mat

Perimeter
Area P A 21 mat 15.914 37.214 21 mat