<p> algo style discrete inverser/solver </p>

fd55d695-ecbe-47fe-93c5-36323d840ca9 35 4 5 decimal

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

r 0.25 - : x y f x 2^y 2^+ r + : 21 sys

Solve at(2) . . . didactic

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t0 1 time :

lo 0.25 : hi 0.475 : N 19 : 13 mat

4 appVersion 0.98.6179.21440 ≡

find low limit ζ lo hi lo hi lo - N / + (range :

1 variable ... 2 values # u j el y f y solve at

C1 j 1 ζ length range at j el Ω ζ j el y f y solve 1 el at : 11 line for at 21 line :

C2 j 1 ζ length range at j el Ω ζ j el y f y solve 2 el at : 11 line for at 21 line :

ζ C1 augment ζ C2 augment 2 1 sys

1 time t0 - 5 s *

C1 C2 12 mat 0.433 - 0.426 - 0.4184 - 0.4105 - 0.402 - 0.3929 - 0.3833 - 0.3731 - 0.3622 - 0.3505 - 0.338 - 0.3247 - 0.3103 - 0.2947 - 0.2777 - 0.2591 - 0.2385 - 0.2152 - 0.2 - 191 mat 0.433 0.426 0.4184 0.4104 0.4019 0.3929 0.3833 0.3731 0.3622 0.3505 0.338 0.3247 0.3102 0.2947 0.2777 0.2591 0.2384 0.2152 0.1884 191 mat 12 mat

More profitable "roots" at(2) . . . circle solve

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lo 0.0001 : hi 0.4999 : N 51 : 13 mat

find low limit u lo hi lo hi lo - N / + (range :

t0 1 time :

C3 j 1 u length range rho j el ρ u j el y f y roots at : 11 line for rho 21 line :

u C3 augment u C3 - augment u - C3 augment u - C3 - augment 4 1 sys

1 time t0 - 4.8 s *

C3 1 row C3 C3 rows row 21 mat 0.5 11 mat 0.01 11 mat 21 mat

revisit Electron at(2)

Solve/evaluate contour

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x y Clear 1

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

# # at

x y axes x y / - :

εlectron 816 π * sqrt * : 351.67021

isolate εlectron 351.67021 :

L 1.785686 : H 14.245529 : N 100 : 13 mat x L H N xd : 21 line

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

n 1 x length range Esol n el at2 x n el y Φ vectorize y solve at : for

plot branch_1 x Esol augment : branch_2 x Esol - augment : branch_3 x - Esol augment : branch_4 x - Esol - augment : 41 sys :

round limits:  more decimals solve complains
exact values from al_nlqsolve 1.78568577383018 14.2455293828432 21 mat 6 round vectorize 1.785686 14.245529 21 mat

plot axes 2 1 sys

Local pairwise solutions

L < u < H ... escape end-points u 10 :

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

pair wise u pairwise 1 el 10 4.1749 - 4.1749 21 mat 12 mat

Y u pairwise 2 el : 4.1749 - 4.1749 21 mat

plot u Y 1 el + 40 black 1 5 mat u Y 2 el + 40 black 1 5 mat 3 1 sys

Ko x n R f Ko x n^* R x * x n^1 - x 1 - / * + :

ε 1012 -^:

n 130 :

Ko 1 :

R 10 n * :

λ 0.8164215 - :

x c 1 λ x ≤ (x 1 ε - ≤ (& cases :

Ko x n R f x c * λ 100000 λ 10000 - 2 2 mat 1 100000 1 10000 - 2 2 mat 3 1 sys

L H N xd N+1 pts L...H U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

L 584.3 - : H 4757.73 : N 100 : 15 mat

λ x ≤ (x π 4 / ≤ (&

U L H N xd :

x F Ko x n R f :

Ko π 4 / n R f 4757.73007622302

w sol x F w - 0 ≡ x λ 1 solve :

Sol i 1 N 1 + (range sol i el U i el sol eval : for sol 21 line :

Sol Sol : MySol Sol U augment : 21 sys

u 1500 :

X x F u - 0 ≡ x λ 1 solve : 0.5357

MySol 0.25 - u X u X 1000 - 3 2 mat 2 1 sys

u X X F 2 round 13 mat 1500 0.5357 150013 mat

Sol Min Sol Max 21 mat 0.816404 - 0.785396 21 mat

<p> algo style discrete inverser/solver </p>

Design an algo style discrete function inverser  in the [L,H,N] f L H N lo hi Inverse L 0 : H 0 : N 0 : 13 mat dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x lo hi solve eval : for ω sol augment 61 line :

roots solve a triplet...

x y z Clear 1

V list Collect collect list ct 1 : i list a ct el V i 1 el : ct ct 1 + : 21 line for a 41 line :

LGS x 2 y * - z + 1 + 2 x * - 4 y * + z - 5 + x y + z - 4 + x - 3 y * - z + e - x y - z + 2 sqrt + 51 sys :

collect  xyz triplet list 34513 mat :

assign symbolic x y z LGS LGS list Collect :

recover the system
to be solved . x y z LGS 4 x + y + z - x z - e + 3 y * + (- x y - z + 2 sqrt + 31 mat

assign/isolate x y z LGS x y z LGS :

system v 000 LGS : M1 100 LGS v - : M2 010 LGS v - : M3 001 LGS v - : 31 line u M1 M2 M3 augment : u v 12 mat 41 line :

U V 12 mat system : 111 - 1 - 3 - 111 - 133 mat 4 2.71828182845905 - 1.41421356237309 31 mat 12 mat

Don't understand
negate  -(U^{-1}*V) x y z 31 mat U 1 -^V * (- : 2.70710678118654 - 0.640859085770477 1.93375230458393 31 mat

What's that freak ??? sanity x y z LGS : 31015 -^* 31015 -^* - 31015 -^* - 31 mat

================== Implicit solve "roots" ===================

x y z Clear 1

x y z 31 mat x y z LGS x y z 31 mat roots (: 2.70710678118654 - 0.640859085770477 1.93375230458394 31 mat

sanity x y z LGS : 41015 -^* - 41015 -^* 11014 -^* 31 mat

x y z Clear 1

'f' dummy algo style f Solve f x solve :

x f x 2^5 x * - 3 + :

x Φ x 3^5 x * - 3 + :

x Φ Solve 2.49086436843124 - 0.656617068589391 1.83423994642952 31 mat

x f Solve 0.697226753670473 4.30277372262774 21 mat

2.49086436843124 - 0.656617068589391 1.83423994642952 31 mat Φ vectorize 1105 -^* - 1105 -^* 2105 -^* - 31 mat

0.697226753670473 4.30277372262774 21 mat f vectorize 9106 -^* - 7106 -^* - 21 mat

x f x Φ 2 1 sys

algo style solve near parallel intersection(s) f Φ x L H Intersection f Φ - 0 ≡ x L H solve :

x f x Φ x 0.5 - 1.5 Intersection 0121 mat

More accurate "roots" algo ... solves the closest to '0'

f Roots f x roots : Φ Roots Φ x roots : 21 sys

R x f Roots x Φ Roots 21 mat : 0.697224362268005 0.65662043104711 21 mat

R 1 el f R 2 el Φ 21 mat 91016 -^* 11015 -^* 21 mat

more accurate solve by shrinking the range sanity x f x 0.697 0.6975 solve x Φ x 1.834 1.835 solve 21 sys : 0.697224362268335 1.83424318430767 21 sys

sanity 1 el f sanity 2 el Φ 21 mat 11012 -^* - 31011 -^* - 21 mat

The ROBUST-ACCURATE iterative root(s) Intersection Solve scanner f q x0 x1 TOL Solve ρ x f x q - 0 ≡ x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : n 0 : xn f xn q - abs TOL > xn f xn q - (a f a q - (* 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 51 line X i el xn : 21 line for X 31 line :

x f x 2 π * / 2 π * x / tanh * sqrt :

x q 0.5 :

z f q 2 - 21014 -^Solve :

x f x q z 1 el z 1 el f + 20 black z 2 el z 2 el f + 20 black 2 5 mat 3 1 sys

z 1 el z 2 el 21 mat f vectorize 0.500000000000009 0.500000000000001 21 mat

z 1.5719 - 1.5719 21 mat

α 1.25 :

TOL 1014 -^:

x f x tan :

x q α x * :

x f x q - x 0.5 1 solve 0.7593

range wisely to avoid numerical instability r f q 1 - 1 TOL Solve :

x f x q 2 1 sys

r f α r * - vectorize 51015 -^* 011014 -^* - 31 mat

r 0.759307689030624 - 0 0.759307689030617 31 mat

Solving for a vectorised function

x f x 2 π * / 2 π * x / tanh * sqrt :

t0 1 time :

In Mathematics, an InverseFunction is the solution of this function. f(x) is the model of difficulty, documented in Mathematica 4.0. The seeded solve scanner is of no gain versus the direct "Sol". The 'roots' can't be applied wrt the derivative that does not take discrete vector of values. In lieu, the 'solve' block concludes pretty well.. a bit slow. _____________________________ Note: the function starts existing around 'x = 0.02, f(0.02) = 0.0564' our choice to solve is wise , down to lower possible limit. The upper limit 0.99 is practical. In fact, at 1, the solution is infinite. Pushing a bit more, 'solve' cries to murder. _____________________________ This exercise is purely didactic/demo. No further exploration will be applied. It aims at setting up the solver wisely. The f(x) does not plot part of the low region, this default is due to Smath too granular in this region.

Design a discrete function inverser  in the L, H span , N points f L H N lo hi Inverse dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x lo hi solve : for ω sol augment 51 line :

Sol f 0.06 0.99 100 0.02 20 Inverse :

1 time t0 - 17 s *

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f

β 0.0009996 12.4435128 228.3303460 1.7276022 41 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

eit90 259.3467 - 0.0011907 248.5939 - 0.00844974 218.7916 - 0.09171804 189.3442 - 0.21585975 38.8344 - 0.84414211 0.01 1.0000000 29.7646 1.11813889 156.5985 1.60980185 231.928 1.89279768 419.527 2.56891730 660.323 3.37600860 961.78 4.28642053 122 mat :

The "roots' discrete function inverse  in the [L,H,N] f L H N Inverse dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x roots : for ω sol 1 round vectorize augment 51 line :

Pt100 XFR
Thermometer x f β 1 el β 2 el 1 β 4 el β 3 el β 4 el - (/ x 259.3467 + (- β 4 el β 2 el * / exp * + β 3 el β 3 el β 4 el - (/ x 259.3467 + (- β 3 el β 2 el * / exp * - (* + :

in-situ °C from measured Ω x f 0.84414211 - 0 ≡ x roots 38.421 -

Sol f 0.0011907 4.5 1200 Inverse : XY eit90 2 col eit90 1 col augment : data XY o 7 black plot : 31 sys

data Sol 2 1 sys

°C Ω fixed points 259.3467 - 0.00119007 TP Hydrogen 248.5939 - 0.00844974 TP Neon 218.7916 - 0.09171804 TP Oxygen 189.3442 - 0.21585975 TP Argon 38.8344 - 0.84414211 TP Mercury 0.01 1 TP Water 29.7646 1.11813889 MP Gallium 156.5985 1.60980185 FP Indium 231.928 1.89279768 FP Tin 419.527 2.5689173 FP Zinc 660.323 3.3760086 FP Aluminum 961.78 4.28642053 FP Silver 133 mat

L H N xd N+1 pts L...H U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

x f x tan 11 x cos sqrt / + / 0 - :

L 0.001 : H 1.57 : N 50 : 13 mat U L H N xd : 21 sys

i 1 U rows range sol i el x f U i el - 0 ≡ x 0 1.75 solve : for

x f U sol augment π 2 / 10 π 2 / 4 - 2 2 mat 3 1 sys

NO real roots x f U i el - 0 ≡ x roots

Solve/Inverse paiwise ...

0.25 - f 4.25 f 2 f 31 mat 0.127 - 0.619 0.927 i * + 0.642 - 0.995 i * - 31 mat

x f 5953 / 11 x 2 / - (2^- (* :

max 2 f : 198.333

Two branches  iterative discrete solver/inverser ... explode/collect the 2 nests f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x Lo Hi solve eval : for sol 61 line :

N 200 :

range wisely the domains sol f 50 - max N 60 - 5 solveDiscrete :

j 1 sol rows range sol_1 j el sol j el 1 el : sol_2 j el sol j el 2 el : idx j el j : sol_12 idx sol_1 sol_2 augment : 41 line for

sol_1 sol_12 1 col sol_12 2 col augment : sol_2 sol_12 1 col sol_12 3 col augment : 21 line

Solutions are collected pairwise from the solve bloc. They must be exploded [Unested] and collected alternately for each branch, done via sub indexed [sol(j,1), sol(j,2)]

sol_1 sol_2 N 2.16 + 40 green 1 5 mat 41 0 + 20 red 1 5 mat 4 1 sys x f 2 250 2 50 - 2 2 mat 2 1 sys

Unest/collect solutions

x f x abs sin 0.5 * x 2^(* :

u 3 π * - x ≤ (x 3 π * ≤ (* :

maple will not solve any and all functions. y branches y x f u / ≡ x solve maple :

x f u / π 16 π 16 - 2 2 mat 2 1 sys

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

One branch discrete solver/inverser export f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ dx H L - N / : i 1 N range ω i el L dx - (dx i * + : for j 1 ω rows range sol j el x f ω j el - 0 ≡ x Lo Hi solve eval : for sol 61 line :

sol_1 0 :

sol_2 0 :

N 100 :

set upper solver a bit higher than in x ≤ 3*π solveHi f 0 π N 010 solveDiscrete :

V solveHi Unest : i 1 V rows 5 range : j 2 V rows 6 range : branches V i el V j el augment : idx 1 branches rows range : sols idx branches augment : sol_1 sols 1 col sols 2 col augment : sol_2 sols 1 col sols 3 col augment : 81 line

sol_1 sol_2 0 π o 7 black 1 5 mat 3 1 sys

I'm obsessed by Electron as much as by the Babies Fairy !

x y Clear 1

About the Electron function . . .

A closed lobe that has no symbolic expression

of the ± branches. It complicates more familiar

expansion of f(x,y) but delights Smath.

Similarly to above example, can be collected

unested plotted each lobe for 2D quick plot.

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a 816 π * sqrt * : 351.67021

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

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

x y axes x y / - :

solving 6 values u 7.5 :

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

axes rd gn u 25 u 25 - 2 2 mat 14.2455293 5 14.2455293 5 - 2 2 mat 5 1 sys

x green user x y gn y solve : x user 12 mat 21 line :

axes rd gn u 25 u 25 - 22 mat 14.24553 5 14.24553 5 - 22 mat 51 sys

pair wise u green 7.5 18.063615 - 6.294379 - 6.294379 18.063615 41 mat 12 mat

x red user x y rd y solve : x user 12 mat 21 line :

pair wise u red 7.5 3.716206 - 3.716206 21 mat 12 mat

Otherwise profitable Discrete-Difference "roots" at(3) . . .

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r 0.125 - :

x y u Clear 1

x y Γ x 2^y 2^+ r + :

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

Ω Γ 0.35355 0.04526 - 1 2 mat 3 1 sys

Intersection limit

t0 1 time :

lo 0.35 : hi 0.36 : N 1000 : 13 mat

find low limit u lo hi lo hi lo - N / + (range :

last value "on error" ... NO real root. data j 1 u length range rho j el at3 u j el y Ω y roots u j el y Γ y roots at : break try for rho 21 line :

1 time t0 - 0.8 s *

data data :

"real root" exists in the positive region left of 0.35355. On the other side, the differential region is negative ... thus => "NO real root".

k 1 data rows range : junction u k el data augment : junction junction : 31 sys

u 0.35355 : Yjunction u y Γ u y Ω ≡ y roots : 21 sys

1 time t0 - min 0.8

Yjunction 0.04526 -

junction junction rows row 0.35355 0.0525 12 mat

junction 0.35355 0.5 0.35355 0.05 - 2 2 mat 0.35 0.0525 0.36 0.0525 2 2 mat 3 1 sys

real root

NO real root at(3)

Solve-plot canvas . . .

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NASA(h) is kPa vs h[m] ... we solve [m] actual altitude vs measured [kPa] h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

roots does not support equalities x altitude h NASA x - 0 ≡ h 0100000 solve :

x altitude 1 1 sys

kPa 30 :

kPa altitude 9183

x λ 10 x ≤ (x 60000 ≤ (& cases :

x NASA x λ * 5000 - kPa kPa altitude kPa kPa altitude 10 - 3 2 mat 2 1 sys

"roots" otherwise ... MACH on canvas

x U Clear 1

γ 1.4 : (Air isentropic ≡ k 0.88128485 : (Pitot constant ≡ p static pressure ≡ qc dynamic pressure ≡ x Pitot dynamic ratio qc/P ≡ x qc p / ≡ (supersonic Pitot ratio ≡ 61 line

Observe the canvas solutions:

as 'x' in Supersonic(x) runs on the plot canvas,

the 'roots' solves and plots each canvas pixel result ...

each result on the basis of 96 ppi [pixel per inch].

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Plot limit x c 10 x ≤ (x 5 ≤ (& cases :

t0 1 time :

x Subsonic 2 γ 1 - / (x 1 + (γ 1 - γ /^1 - (* sqrt 0 x ≤ (x k ≤ (& Supersonic cases : x Supersonic U x 1 + (117 U 2^* / - (2.5^* sqrt / k - 0 ≡ U 1 roots k x ≤ Subsonic cases : 21 line

x Subsonic x Supersonic x c * k 1 + 20 black 1 5 mat 2 0 Pitot-Mach 12 red 1 5 mat 4 1 sys

supersonic

1 time t0 - 71 s *

x y u Clear 1

r 0.25 - : x y c x 2^y 2^+ r + : lo 0.0001 : hi 0.4999 : N 99 : 13 mat u lo hi lo hi lo - N / + (range : 41 sys

C j 1 u length range rho j el ρ u j el y c y roots at : 11 line for rho 21 line :

t0 1 time :

u C augment u C - augment u - C augment u - C - augment 4 1 sys

1 time t0 - 9.5 s *

r 0.5 : x f r x 2^- sqrt ± : 21 sys

x f r sqrt 0 + 75 black 1 5 mat 2 1 sys