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Solving the Electron function

4 appVersion 0.98.6179.21440 ≡

AlgLib Solver for a system of nonlinear equations(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 :

Electron has no symbolic expansion. For solving, we just need the exact end points. We get them from al_nleqsolve ... further adapted wrt the convenient solve bloc(s).

a 816 π * sqrt * : 351.67021

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 :

x y axes x y / - :

Solve floating values

X0 10421 mat :

result 10.0047954317284 4.1748940017124 21 mat

result f 11016 -^* - 021 mat

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

f result 1 el result 2 el result 1 el 0 2 2 mat axes 3 1 sys

L 1.78568577383018 : H 14.2455293828432 : 21 mat

Local pairwise solutions

L<u<H ... escape end-points u 14.245 :

x pairwise set Roots range ± 5 user x y f y solve : x user 12 mat 31 line :

pair wise u pairwise 14.245 0.0688 - 0.0688 21 mat 12 mat

Solving accurately for both end ▼ points

X0 1.8 021 mat :

result 1.78568577383018 021 mat

X0 14021 mat :

result 14.2455293828432 021 mat

a 816 π * sqrt * : 351.67021

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

OK recursive solve Δ xlimits : 1.7857 14.2455 at

exact al_nleqsolve 1.78568577383018 14.2455293828432 21 sys

Bracket wisely the implicit pair-wise iterative "solve"

populate the solutions L 1.7857 : H 14.2455 : N 100 : 13 mat U L H L H L - N / + (range : 21 sys

sanity the
bracket range U 1 row U U rows row 21 mat 1.7857 11 mat 14.2455 11 mat 21 mat

sols set Roots range ± 5 i 1 U rows range data 1 i el U i el y Φ y solve : for v data 1 el : j 2 data cols range v v data j el augment : for v 51 line :

t0 1 time :

isolate the project new sols eval :

1 time t0 - 55 s *

blue U new 2 row transpose augment : red U new 1 row transpose augment : 21 sys

blue red 1.7857 0 + 100 black 1 5 mat 14.2455294 0 + 100 black 1 5 mat 10.009 4.175 + 100 red 1 5 mat 5 1 sys

very accurate solve ... ~ 11 D's x userSolve user y x y Φ vectorize y solve at : x user augment 21 line :

1.7857 U ≤ (U 14.2455 ≤ (&

u L H 21 mat :

u userSolve vectorize 1.7857 0.003358 12 mat 14.2455 0.016142 12 mat 21 mat

R U reverse new 2 row transpose reverse augment : B U new 1 row transpose augment : circular B R stack : 31 sys

NUMERIC

blue 67 row 10.009 4.175 12 mat

sub circular 1175 range 12 range el :

xx vx RowIdx ct 1 : i 1 vx rows range xx vx i el ≡ idx ct el i : continue if ct ct 1 + : 21 line for idx 31 line :

sub 10 4.175 + 40 red 1 5 mat 2 1 sys

The duplicate row 'stack' is ignored. The first/last points are inaccurate.

vx blue 2 col :

vx Max vx RowIdx 6711 mat

Electron circular :

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

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

Perimeter  Polygon wrt vertices coordinates P vx i 1 + el vx i el - (2^vy i 1 + el vy i el - (2^+ sqrt i 1 vx rows 1 - sum :

Center of gravity Xo 16 A * / vx i el vx i 1 + el + (vx i el vy i 1 + el * vx i 1 + el vy i el * - (* i 1 vx rows 1 - sum * : Yo 16 A * / vy i el vy i 1 + el + (vx i el vy i 1 + el * vx i 1 + el vy i el * - (* i 1 vx rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 74.416 32.312 21 mat

Center of gravity Xo abs Yo abs 21 mat 8.716 0.000 21 mat