Circle rkfixed

2be5105f-27ac-41c1-9f44-ff33b963fb6f 66 4 5 decimal

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

4 appVersion 0.98.6179.21440 ≡

a 1 :

b 1 :

as 5 :

sy o : size 7 : clr black : 13 mat

x y Hyperbola x 2^a 2^/ y 2^b 2^/ - (1 - :

Y 1.9109 - 1.6284 1.9109 1.6284 1.9109 - 1.6284 - 1.9109 1.6284 - 42 mat :

ortho-axes x y XY x y / - :

x y Astroide x 2^y 2^+ as 2^- (3^27 as 2^* x 2^* y 2^* + :

radius r 2.5106 :

Circle rkfixed

Circle definition t u D u 2 el u 1 el - 21 mat :

t0 0 : t1 2 π * : N 200 : 31 line

IC 0 r 21 mat :

Either 'rkfixed' or 'Rkadapt' s IC t0 t1 N t u D rkfixed :

x s 1 col : X s 2 col : Y s 3 col : 31 line

circle X Y augment : (inverse the path Y <- X ≡

Hyperbola Astroide XY circle 1.9109 - 1.6284 1.9109 1.6284 1.9109 - 1.6284 - 1.9109 1.6284 - 4 2 mat 5 1 sys

x H x y Hyperbola 0 ≡ y solve maple :

x A x y Astroide 0 ≡ y solve maple :

X x H 1 el x A 1 el - 0 ≡ x 2 - 2 solve : 1.91 - 1.91 21 mat

x H 1 el x H 2 el x A 1 el x A 2 el circle X 1 el X 1 el H 1 el sy size clr 15 mat X 2 el X 2 el H 1 el sy size clr 15 mat X 1 el X 1 el H 2 el sy size clr 15 mat X 2 el X 2 el H 2 el sy size clr 15 mat 101 sys

x H 1 el x H 2 el x A 1 el x A 2 el circle X 1 el X 1 el H 1 el sy size clr 1 5 mat X 2 el X 2 el H 1 el sy size clr 1 5 mat X 1 el X 1 el H 2 el sy size clr 1 5 mat X 2 el X 2 el H 2 el sy size clr 1 5 mat 10 1 sys

X 1 el 2^X 1 el H 1 el 2^+ sqrt 2.5106

Solve non-linear ... [f, g] courtesy 'overlord'

x y f x 2^y * y 2^x * + 1 + :

x y g x 2^y + 3 - :

Clear the system x y Clear 1

eq_1 x 2^y * y 2^x * + 1 + :

eq_2 x 2^y + 3 - :

eq_1 0 ≡ eq_2 0 ≡ 21 mat x 2 ≡ y 2 - ≡ 21 mat FindRoot 2.2404 2.0194 - 21 mat

x y axes x y / - :

in lexicographic order x y 21 mat 2.2404 2.0194 - 21 mat

guess wisely ... set Options init 2 - 0.5 - 21 mat 2 - 321 mat 0.5 - 321 mat 2 0.5 - 21 mat 22 - 21 mat 15 mat :

collect j 1234515 mat : ct 1 : x y Clear 12 mat j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose sol 3 el transpose sol 4 el transpose sol 5 el transpose stack : 41 line :

f g axes collect 41 sys

f g axes collect 4 1 sys

collect 1.8074 - 0.2667 - 1.1526 - 1.6716 0.1167 - 2.9864 1.8363 0.3719 - 2.2404 2.0194 - 52 mat

r 0.125 - :

x y Clear 1

x y Γ x 2^y 2^+ r + :

x y Ω x 4^2 x 2^* y * + 3 x * y 2^* - y 3^+ :

Ω Γ 0.35 0.05 - 1 2 mat 3 1 sys

Sanity check

x 0.35 :

y 0.05 - :

x y Γ 0.000000000000000

suffers  granularity x y Ω 6.25 106 -^*

x y Clear 1

eq_1 x 2^y 2^+ r + :

eq_2 1 x 4^(* 2 x 2^* y * + 3 x * y 2^* - 1 y 3^(* + :

Brute force  refined by user trial/error from sanity eq_1 0 ≡ eq_2 0 ≡ 21 mat 0.35 0.05 - 21 mat FindRoot 0.350000000000000 0.050000000000000 - 21 mat

No real roots ... option RootsRange ± 25

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 :

a 816 π * sqrt * : 351.67021

x y Clear 1

isolate a 351.67021 :

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

1. The upper limit

al_nleqsolve 1.78568577383018 14.2455293828432 21 mat

x y XY x y / - :

L 14.245 : H 14.245529 : N 19 : 13 mat x L H N xd : 21 line

XY Φ 2 1 sys

n 1 x length range Sols n el y x n el y Φ vectorize y solve at : 11 line for

high x Sols augment :

high high rows row 14.24553 0.00001 12 mat

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

2. The lower limit

find low limit u 1.7857 2 1.7875 range :

Explain . . .

0. The solver is implicit-recursive

implicit: in the sense that it solves y given Φ(x,y)=0.

recursive: as it runs on the vectorized discrete x.

1. The low limit is searched first on a practical low range.

2. The upper limit is searched over a generously populated

range, starting from a solvable mid - limit.

3. steps [1, 2] are trial/error.

4. Pairwise solutions aren't true @ the limits.

5. The model document is unique for the given Φ(x,y)

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n 1 u length range Sols n el y u n el y Φ vectorize y solve at : 11 line for

low u Sols augment :

low 1 row 1.7857 0.0000625 12 mat

reasonable limits
solving [2] branches xlimits 1.7857 14.245529 at

x y Clear 1

Solving for the green red lobes

within their respective limits.

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solving 6 values u 5 :

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

x y f 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.0025 - :

pair wise u pairwise 5 19.58772 - 4.029687 - 4.029688 19.58772 41 mat 12 mat

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

x y axes x y / - :

axes f Φ u 25 u 25 - 2 2 mat 14.24553 5 14.24553 5 - 2 2 mat 5 1 sys

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

pair wise u pairwise 5 2.597823 - 2.597823 21 mat 12 mat

axes f Φ u 25 u 25 - 22 mat 14.24553 5 14.24553 5 - 22 mat 51 sys

1.78568577383018 14.2455293828432 21 mat

x y Clear 1

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

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 100 : 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 2 π * :

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

Export branch1 branch2 21 mat ≡

pair wise u pairwise 6.283185 3.235341 - 3.235341 21 mat 12 mat

size 100 :

index 1 to explode the nest pair u pairwise 2 col 1 el : 3.2353 - 3.2353 21 mat

1 time t0 - min 0.9

branch1 branch2 branch1 - branch2 - u pair 1 el + size black 1 5 mat u pair 2 el + size black 1 5 mat 6 1 sys

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

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

sanity the limits u 1.78568577383018 14.2455293828432 21 mat :

u limitSolve vectorize 1.785685773830 0.000000000005 12 mat 14.245529382843 0.000000000003 12 mat 21 mat

Descartes folium x y axes x y / - : 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 * - : x y λine x y - : 61 sys

Descartes1 Descartes2 Descartes3 axes λine 5 1 sys

x y Clear 1

a π - :

x q 1 x * 2 sqrt + :

Regular Strophoid

x y axes x y / - :

x y Strophoid x x 2^y 2^+ (* a a cos x 2^y 2^- (* * + 2 a sin * x * y * + :

x f x y Strophoid 0 ≡ y solve maple :

recast maple solve x s x π - (x π + (* - sqrt x * x π - / ± :

x f x π - (x π + (* - sqrt x * x π - / x π - (x π + (* - sqrt x * x π - / - 21 mat :

C x f 1 el x q - 0 ≡ x π - 0 solve x f 2 el x q - 0 ≡ x π - π solve 21 mat : 0.798466 - 2.315192 - 1.699442 21 mat 21 mat

π - 0 Strophoid π - f 21 mat 0.000000000000000 0.000000000000000 0.000000000000000 21 mat 21 mat

c C 1 el C 2 el 1 el C 2 el 2 el 13 mat transpose :

plot x s x q c 1 el c 1 el f 1 el + 20 black 15 mat c 2 el c 2 el f 2 el + 20 black 15 mat c 3 el c 3 el f 2 el + 20 black 15 mat π - 1 π - 1 - 22 mat 61 sys :

plot

c 0.798465770194794 - 2.31519163678993 - 1.69944236423153 31 mat

c 1 el f 1 el c 2 el f 2 el c 3 el f 2 el 31 mat 0.615747809 0.900977558 - 3.113653577 31 mat

solve algo x Y x y Strophoid y - 0 ≡ y π - 0 solve ± :

2 - y Strophoid y - y 1.25 - roots 1.04465342237895 -

x 2 - :

Not too accurate
whatsoever ... x Y 1.04465355915983 - 1.04465355915983 21 sys

True sanity x f 0.942403184848852 0.942403184848852 - 21 mat

absolute error 1.04465342237895 0.942403184848852 - 0.10225

relative error 1.04465342237895 1.04465342237895 0.942403184848852 / - 0.06385 -

Revisit/Export this independent section

a 816 π * sqrt * : 351.67021

Clear the system x y Clear 1

isolate a 351.67021 :

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

x y f 5 - 0.75 x * + 1 y * - :

x y axes x y / - :

eq_1 1 - 3 / x 2^y 2^+ sqrt * exp 2 y 2^* - x 2^+ a / * - 0.005 + :

Observe plot 2D contour straight line eq_2 5 - 0.75 x * + 1 y * - :

guess wisely ... set Options init 5 2.5 - 21 mat 10321 mat 12 mat :

collect j 1212 mat : ct 1 : x y Clear 12 mat j j sol j el eq_1 0 ≡ eq_2 0 ≡ 21 mat init 1 j el FindRoot : ct ct 1 + : 21 line for all sol 1 el transpose sol 2 el transpose stack : 41 line :

collect collect eval :

axes Φ f collect 4 1 sys

3.985 2.011 - 11.793 3.845 22 mat

collect 3.9854 2.011 - 11.7935 3.8451 22 mat