al_nleqsolve

2ce7f30a-561b-433e-a0c6-a024454c1885 39 4 5 decimal

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Split region iterate al_nleqsolve ...

al_nleqsolve

result k 1 x# Ω rows range u# k el x# k el : for x# Jac u# Ω u# Jacobian : StepMax 0 : Eps 1015 -^: u# X0 StepMax Eps x# Ω x# Jac al_nleqsolve : 51 line :

x y a c ff x 2^y 2^+ (2^2 c 2^* x 2^y 2^- (* - a 4^c 4^- (- :

x y f1 x y 2 sqrt 1 ff :

x y f2 x 0.5 - y 0.5 - 22 ff : 1 - 2 x * + (2^1 - 2 y * + (2^+ (2^- 64 x y - (* 1 - 2 x * + 1 - 2 y * + (* + 16 / -

x Ω 1 - 2 x 1 el * + (2^1 - 2 x 2 el * + (2^+ (2^- 64 x 1 el x 2 el - (* 1 - 2 x 1 el * + 1 - 2 x 2 el * + (* + 16 / - 0.5 21 mat :

X0 3 0.5 21 mat :

result 3.32842712474619 0.5 21 mat

X0 2 - 0.5 21 mat :

result 2.32842712474619 - 0.5 21 mat

In this example ▲ , al_nleqsolve is unique candidate solving for datum 0.5 Other datum iterative methods produce accurate solutions. 'Iter' while loop offered ▼

datum 1.375 :

t0 1 time :

al_nleqsolve dependent of 'datum'

result k 1 x# Ω rows range u# k el x# k el : for x# Jac u# Ω u# Jacobian : StepMax 0 : Eps 1015 -^: u# X0 StepMax Eps x# Ω x# Jac al_nleqsolve : 51 line :

x y a c ff x 2^y 2^+ (2^2 c 2^* x 2^y 2^- (* - a 4^c 4^- (- :

x y f1 x y 2 sqrt 1 ff :

x y f2 x 0.5 - y 0.5 - 22 ff : 1 - 2 x * + (2^1 - 2 y * + (2^+ (2^- 64 x y - (* 1 - 2 x * + 1 - 2 y * + (* + 16 / -

x Ω 1 - 2 x 1 el * + (2^1 - 2 x 2 el * + (2^+ (2^- 64 x 1 el x 2 el - (* 1 - 2 x 1 el * + 1 - 2 x 2 el * + (* + 16 / - datum 21 mat :

al_nleqsolve target recursive Iter X0 0.5 - datum 21 mat : j 1 : result 2 el datum - abs 1015 -^> X0 result 1 el datum 21 mat : j j 1 + : 21 line while result j 21 mat 41 line :

Iter Iter eval :

Iter 0.639246823306437 - 1.375 21 mat 11721 mat

x y axes x y / - :

Timing varies wrt 'j iter' 9 iter only 0.2 s

1 time t0 - 24.9 s *

sol Iter 11 el (1 el Iter 11 el (2 el augment :

f1 f2 sol axes 4 1 sys

sol 0.639246823306437 - 1.375 12 mat

Split region PROCEDURE ... Associate wrt red lobes. Associate TOL as needed.

◄  left of max left lobe X0 2 - datum 21 mat :

►  right of max left lobe X0 1 - datum 21 mat :

faster + accurate 'roots'

X x datum f2 x 1 - roots : 0.6392 -

XX x datum f2 x 2 - roots : 1.774 -

sol 11 el datum f2 9.6 109 -^*

X datum f2 XX datum f2 21 sys 6.5 1014 -^* - 4.4 1014 -^* 21 sys

x datum f2 0 ≡ 11 mat 0.1 - 11 mat 1012 -^FindRoot 0.639246824395962 - 11 mat

X 0.639246824395971 -

Revisit Electron function ...

3.141592653589793238

x y axes x y / - :

datum π - :

AlgLib Solver for a system of nonlinear equations 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 :

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 + datum 21 mat :

a 816 π * sqrt * : 351.67021 = x y f 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + : 21 sys

al_nleqsolve target recursive Iter X0 6 datum 21 mat : j 1 : result 2 el datum - abs 1014 -^> X0 result 1 el datum 21 mat : j j 1 + : 21 line while result j 21 mat 41 line :

Iter Iter eval :

Iter 6.0777806822053 3.14159265358979 - 21 mat 3821 mat

π ± 3.14159265358979 3.14159265358979 - 21 sys

xy Iter 11 el (1 el Iter 11 el (2 el augment :

zero Iter 11 el (1 el 0 augment :

From approximate initial X0 ► ...... ...... few iter ... refine X0 ►

7 datum 21 mat

al_nleqsolve is only 5 D accuracy Found no way(s) to improve ?

6 datum 21 mat

faster + accurate 'roots'

Solve intersection

x q 0.25 x * :

X x datum f x 6 roots : 6.07784295549946

scalar intersection ... y = q(x Xq x x q f x 12 roots : 12.816

X datum f 11017 -^* -

zero 11 el datum f 1.3 107 -^*

Yq Xq q : 3.204

X datum 12 mat 6.077842955499 3.14159265359 - 12 mat

f axes xy zero x q Xq Yq 1 2 mat 6 1 sys

Xq Yq 12 mat 12.816 3.204 12 mat

6.0777806822053 3.14159265358979 - 21 mat

Solve an hypersurface isocurve ...

c 1 :

t0 1 time :

f 2 c * y * 3 x 2^* y 2^* y 4^- c 2^x 2^y 2^+ (* - 4 c 4^* + (* x cos * y sinh * 2 c * y * x 4^3 y 2^* x 2^* - c 2^x 2^y 2^+ (* - 4 c 4^* - (* x sin * y cosh * x y * x 2^y 2^+ (y 2^x 2^- 2 c 2^* + (* (* 2 c * sin * 31 mat :

x y f f 1 el f 2 el + f 3 el + :

level_0 0 : level_1 1 : level_2 2 : 13 mat

y x f0 x y f level_0 + : y x f1 x y f level_1 + : y x f2 x y f level_2 + : 31 sys

r 1 : a 1 : b 1 : 13 mat y x circle a x * (2^b y * (2^+ r - : 21 sys

circle axes 2 1 sys

very long expansion symbolic x y f1 eval :

"FindRoot" needs 
the two algorithms
in explicit form. eq_1 symbolic : eq_2 a x * (2^b y * (2^+ r - : 21 sys

roots accuracy ε 1012 -^:

guess wisely ... init 0.25 - 1 - 21 mat 1 - 0.5 - 21 mat 0.25 121 mat 1 0.5 21 mat 14 mat :

roots j 123414 mat : ct 1 : j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el ε FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose stack : 41 line :

roots roots eval : 0.1832 - 0.9831 - 0.8628 - 0.5056 - 0.1832 0.9831 0.8628 0.5056 42 mat

◄ roots piled down ▲

You are at liberty to select oblate/prolate ellipse ... Check initials !

f0 f1 f2 circle roots 5 1 sys

1 time t0 - min 4.2