Doublet visit

4c449c12-f50e-4069-9bc9-d158a85db1d4 91 4 5 decimal

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

x f 101 x + (* 2 x ln * exp - :

solve & roots ... wisely

x f 2 5 x - ( * 2 1 sys

x Φ x abs f :

X x abs f x solve : 10.9161 - 10.9161 21 mat

X Φ vectorize 0.000000959 0.000000485 21 mat

Exact r x f 0 ≡ x 12 roots : 10.9160797830995

Exact r x f 0 ≡ x 6 roots : 10.9160797830995

r Φ 0.000000000000000

x f x diff maple 25 x - (*

x f 1 x / e - :

r x f x 0.25 roots : 0.367879441171443

r x f x 0.5 roots : 0.367879441171442

r x f x f x diff - x 1 roots : 0.817748257906414

r f 1.4954 -

x f x f x diff r r f + 20 black 1 5 mat 3 1 sys

Reference: Butler,Carbon Dioxide Equilibria, Prob. 2-11, pg 41 A solution containing 10^-3 M sodium bicarbonate [NaHCO3] is treated with 5*10^4 M sodium carbonate [Na2CO2] What is the alkalinity of the final solution ?

Coefficients pertaining to the "Equilibrium solution" k1 10 6.3 -^: (1rst dissociation constant carbonic acid ≡ k2 10 10.3 -^: (2nd dissociation constant carbonic acid ≡ kw 1014 -^: kh 10 1.5 -^: PCO2 10 3.5 -^: 13 mat 31 line

balance H OH * kw ≡ CO2 kh PCO2 * ≡ H HCO3 * k1 CO2 * ≡ H CO3 * k2 HCO3 * ≡ H HCO3 2 CO3 * + OH + ≡ 51 mat :

ionisation of water: equilibrium expression Henry's law CO2: equilibrium gas/liquid equilibrium expression for bicarbonate HCO3 equilibrium expression for carbonate CO3 charge balance 51 mat

Alkalinity is a measure of a solution's ability to resist a change in pH, is often used by environmental Engineers. In this example, the alkalinity is zero because alkalinity is defined to begin at the CO2 equivalence point. Adding or removing CO2 by changing the PCO2 does not affect alkalinity, it remains zero.

H CO2 OH HCO3 CO3 51 mat balance H CO2 OH HCO3 CO3 51 mat roots : 2.241 106 -^* 1105 -^* 4.462 109 -^* 2.236 106 -^* 5.002 1011 -^* 51 mat

sanity H OH * kw - CO2 kh PCO2 * - H HCO3 * k1 CO2 * - H CO3 * k2 HCO3 * - H HCO3 2 CO3 * + OH + (- 51 mat : 61022 -^* 51021 -^* 11021 -^* 71022 -^* 11015 -^* - 51 mat

H 2.241 106 -^*

pH alkalinity H log10 - 5.65

Solving the scalen triangle inscribed circle

Cartesian scalen triangle in-circle [Xc,Yc,R)
The solving system is borrowed from Mathcad . x (2^y (2^+ 2 x * Xc * - 2 y * Yc * - Xc 2^Yc 2^+ R 2^- (+ 0 ≡

x 1 - 011 - 41 mat :

y 1 - 0.5 01 - 41 mat :

system init ε GivenFind system init ε FindRoot :

system x 1 el (2^y 1 el (2^+ 2 x 1 el * Xc * - 2 y 1 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ 0 ≡ x 2 el (2^y 2 el (2^+ 2 x 2 el * Xc * - 2 y 2 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ 0 ≡ x 3 el (2^y 3 el (2^+ 2 x 3 el * Xc * - 2 y 3 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ 0 ≡ 31 mat :

ε 1012 -^:

refine solutions
observe result init Xc 0.05 ≡ Yc 0.5 - ≡ R 1 ≡ 31 mat :

NOT in conventional
lexicographic order
wrt [Xc,Yc,R] ... ? R Xc Yc 31 mat system init ε GivenFind : 1.1267347735825 - 0.0625000000000 0.6250000000000 - 31 mat

1.1267347735825 - 0.0625000000000 0.6250000000000 - 31 mat

x 1 el (2^y 1 el (2^+ 2 x 1 el * Xc * - 2 y 1 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ x 2 el (2^y 2 el (2^+ 2 x 2 el * Xc * - 2 y 2 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ x 3 el (2^y 3 el (2^+ 2 x 3 el * Xc * - 2 y 3 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ 31 mat 011015 -^* 031 mat

Xc 0.0625 : Yc 0.625 - : R 1.1267347735825 - : 31 mat

sanity x 1 el (2^y 1 el (2^+ 2 x 1 el * Xc * - 2 y 1 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ x 2 el (2^y 2 el (2^+ 2 x 2 el * Xc * - 2 y 2 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ x 3 el (2^y 3 el (2^+ 2 x 3 el * Xc * - 2 y 3 el * Yc * - Xc 2^Yc 2^+ R 2^- (+ 31 mat : 0.00000000000001 - 0.00000000000001 - 0.00000000000001 - 31 mat

Triangle x y augment :

start 0 : end 1 : N 100 : 13 mat

N N : u v N var u v u v u - N / + (range : v start end N var : 31 sys

a R : b R : xc Xc : yc Yc : 14 mat

circle j 1 v rows range y j el i 2 * π * v j el * exp eval : for a y Re * xc + b y Im * yc + augment 21 line :

Triangle circle xc yc + 20 red 1 5 mat 0 0.1667 - . 14 black 1 5 mat 4 1 sys

Triangle circle xc yc + 20 red 15 mat 0 0.1667 - . 14 black 15 mat 41 sys

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

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

Center of gravity x y Xo 16 x y 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 * : x y Yo 16 x y 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

Area
Perimeter x y A x y P 21 mat 1 5.1569 21 mat

Xo x y Xo : Yo x y Yo - : 12 mat 0 0.1667 - 12 mat

"roots" ... Intersection

x R Clear 1

x f x 2^4 - 2 x * 4 x 2^* - 21 mat :

R x f 1 el x f 2 el - x 1 - roots x f 1 el x f 2 el - x 1 roots 21 mat :

x f 1 el x f 2 el R 1 el R 1 el f 1 el + 15 black R 2 el R 2 el f 1 el + 15 black 2 5 mat 3 1 sys

R 0.716515138991168 - 1.11651513899117 21 mat

'roots init' => high accuracy R 1 el f 1 el R 1 el f 2 el - R 2 el f 1 el R 2 el f 2 el - 21 mat 021014 -^* 21 mat

"roots" ... Solving a system

x y Clear 1

U y 3 ≡ 3 x * y + 5 ≡ 21 mat :

v x y 21 mat :

U v roots 0.666666666666667 321 mat

A 013122 mat :

v 3521 mat :

A 1 -^v * 0.666666666666667 321 mat

Solving a chemical balance

k1 10.052 : k2 14.836 : k3 4.982 : 31 line

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot 19.818 10.052 21 mat

γ1 γ2 21 mat 19.818 10.052 21 mat

k1 k2 + k3 + 29.87

γ1 γ2 + 29.87

k1 k2 * k1 k3 * + 199.2105

γ1 γ2 * 199.2106

γ1 γ2 Clear 1

k1 0.5 : k2 0.25 : k3 0.1 : 31 line

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot 0.5006 0.3494 21 mat

γ1 γ2 21 mat 0.501 0.349 21 mat

x f 2 - 7 x 2^* - 2 x 3^* - x 4^+ : x g x : 21 line

x Clear 1

solve root(s) X x f x 3 - 5 solve : 1.9225 - 3.8522 21 mat

solve intersection(s) C x f x g - x 3 - 5 solve : 1.837 - 3.8963 21 mat

plot x f x g X 1 el X 1 el f . 15 red 15 mat X 2 el X 2 el f . 15 red 15 mat C 1 el C 1 el f . 15 black 15 mat C 2 el C 2 el f . 15 bkack 15 mat 61 sys :

plot

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

Solving a function, i.e: inverse

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1 f 3 f 2 f 1.75 f 41 mat 148.75 148.75 198.333 195.234 41 mat

x f 0 ≡ x 35 solve 4

x c 10 x ≤ (x 4 ≤ (& cases :

2 f 198.3333

algo style "iterative inverser" on range [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 augment 41 line :

Inv f 0 198.3333 100 Inverse :

x freak Inv 1 col Inv 2 col x cinterp :

ξ 0.75 :

ξ f ξ f freak 21 mat 120.859 0.75 21 mat

ξ 148.75 :

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

N Inv Inv 1 col reverse Inv 2 col reverse augment stack :

x f x c * 0 ξ ξ freak ξ ξ freak 0 3 2 mat 2 256 2 64 - 2 2 mat 3 1 sys N

50 freak 148.75 freak 195.234 freak 198.3333 freak 41 mat 0.27 1 1.751 1.9 41 mat

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 - :

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.

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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

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 0.3536 0.0525 - 1 2 mat 3 1 sys

Intersection limit

t0 1 time :

lo 0.325 : 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 :

data data :

"real root" exists in the region of possible junction. On the other side, the differential region is 'on error' ... 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 3

Yjunction 0.04526 -

junction junction rows row 0.35353 0.05543 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)

Circle/line intersection

C 6 : Circle ≡

L 0 : Line ≡

Φ vector equalities Φ x 2^y 2^+ C - 0 ≡ x y + L - 0 ≡ 21 mat :

init x 1 ≡ y 1 - ≡ 21 mat :

curious TOL but profitable TOL 1013 -^:

Does not support assignment Φ init TOL FindRoot 1.732051 1.732051 - 21 mat

Assign the solutions sol x y 21 mat : 1.73205080756888 1.73205080756888 - 21 mat

C x 2 ^ - sqrt ± L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 3 1 sys

assign the solutons [x,y] ... for sanity x y find Φ init TOL FindRoot : 1.7321 1.7321 - 21 mat

sanity x 2^y 2^+ x y + 21 mat 6.000000000000020 0.000000000000000 21 mat

x y i * + arg ° 45.000000000000000 -

Solving with LevenbergMarquardt

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x y u Clear 1

Solving the ellipse/line intersection [Smath Designers]

============================================== solving 1rst intersection

Ellipse in context x y E 2 x 2^* 3 y 2^* + 59 - :

ζ λine ζ 8 + 4 / :

x y axes x y / - :

determined condition
determined condition f 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Guess roughly x.0 x 2 ≡ y 2 ≡ 21 mat :

Accuracy TOL 1015 -^:

most robust for this example sol_1 f x.0 TOL LevenbergMarquardt :

sol_1 4321 mat

============================================== solving 2nd intersection

x y Clear 1

Guess wisely y.0 x 6 - ≡ y 0.5 ≡ 21 mat :

Accuracy TOL 1015 -^:

most robust for this example sol_2 f y.0 TOL LevenbergMarquardt :

sol_2 5.371429 - 0.657143 21 mat

E ζ λine sol_1 1 el sol_1 2 el 1 2 mat axes sol_2 1 el sol_2 2 el 1 2 mat 5 1 sys

E ζ λine sol_1 1 el sol_1 2 el 12 mat axes sol_2 1 el sol_2 2 el 12 mat 51 sys

sol_1 sol_2 21 mat 4321 mat 5.3714 - 0.6571 21 mat 21 mat

strophoid a 1 : b 1.5 : xo 1 : X 0.75 : Y 1 : 15 mat

strophoid x f 1 b / 1 x xo - (2^a 4^/ - sqrt x xo - (/ * a 4^* x X - (* Y + :

x q x :

true
extraterrestrial
true sol x f x q - x 0.75 1.75 solve : 0.807 1.6746 21 mat

x f x f - 2 + a 5 a 10 - 2 2 mat 1 x q 0.807 0.807 f . 14 black 1 5 mat 1.6746 1.6746 f . 14 black 1 5 mat 7 1 sys

x f x f - 2 + a 5 a 10 - 22 mat 1 x q 0.807 0.807 f . 14 black 15 mat 1.6746 1.6746 f . 14 black 15 mat 71 sys

x y Clear 1

Strophoid

a π - :

x q x 1 + :

u v Strophoid u u 2^v 2^+ (* a a cos u 2^v 2^- (* * + 2 a sin * u * v * + :

x f x y Strophoid 0 ≡ y solve maple :

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

x f 2 el x q - x 2.5 - 2 solve 1.9434 - 1.487 21 mat

x π - ( x π + ( * - sqrt x * x π - / ± π - 2 π - 2 - 2 2 mat x q 3 1 sys

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

x π - :

x f 0.000000000000000 0.000000000000000 21 mat

Solving Ellipse otherwise

NOT needed on new document x y Clear 1

determined condition
determined condition system 2 x 2^* 3 y 2^* + 59 - 0 ≡ x - 4 y * + 8 - 0 ≡ 21 mat :

Accuracy TOL 109 -^:

Guess roughly init x 1.75 ≡ y 0.75 ≡ 21 mat :

sol_1 system init TOL FindRoot : 4.000000000000000 3.000000000000000 21 mat

clear from previous use x y Clear 1

Guess roughly init x 7 - ≡ y 0.75 ≡ 21 mat :

sol_2 system init TOL FindRoot : 5.371428571428890 - 0.657142857142777 21 mat

Ellipse in context x y E 2 x 2^* 3 y 2^* + 59 - :

xx λine xx 8 + 4 / :

x y axes x y / - :

E xx λine sol_1 1 el sol_1 2 el 1 2 mat axes sol_2 1 el sol_2 2 el 1 2 mat 5 1 sys

E xx λine sol_1 1 el sol_1 2 el 12 mat axes sol_2 1 el sol_2 2 el 12 mat 51 sys

simply on new document x λine x 8 + 4 / :

Solving . . . seeded scanner

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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 :

60000 NASA 25000 NASA 11000 NASA 0 NASA 41 mat 0.0279 2.5223 22.7074 101.4009 41 mat

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 :

Simply demo ... the symbolic inverse exists for greater convenience.

L 60000 NASA : H 101.4009 : N 200 : 13 mat u L H N xd : 21 sys

altitude i 1 u rows range altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 4 round vectorize altitude 0 round vectorize augment 21 line :

t0 1 time :

NASA(h) solved [m_altitude] from measured kPa

altitude 101.325 0 + 40 red 1 5 mat 11000 NASA 11000 + 100 red 1 5 mat 25000 NASA 25000 + 175 red 1 5 mat 4 1 sys

1 time t0 - min 0.9

MACH ... roots 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

1 time t0 - 66 s *

supersonic

subsonic

Otherwise profitable Discrete "roots" at(2) . . .

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x y u Clear 1

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

lo 0.0001 : hi 0.4999 : N 99 : 13 mat

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

t0 1 time :

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

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

1 time t0 - 9.2 s *

u C augment u C - augment u - C augment u - C - augment 41 sys

x y u Clear 1

x y Φ3 x 2^y 2^+ (2^1 x 3^3 x * y 2^* - (* - : x C 1 x 2^- sqrt ± : 21 sys

x map x y Φ3 0 ≡ y solve maple :

junction 0.5625 - 0.5625 - map 1 el + 15 black 15 mat :

plot x map 1 el x map 2 el x map 3 el x map 4 el 41 sys :

plot x C junction 3 1 sys

Maple  symbolic x q1 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / : x q2 x 32 x * + 916 x * + sqrt - (* - sqrt 2 sqrt / - : 21 sys

lo 0.56245 - : hi 0.99 : N 99 : 13 mat

Hi 0.001 - :

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

find low limit v lo Hi lo Hi lo - N / + (range :

t0 1 time :

T j 1 u length range rho j el dum u j el y Φ3 y roots at : 11 line for rho 21 line :

Trifolium u T augment u T - augment v v q1 vectorize augment v v q2 vectorize augment stack :

Trifolium x C 2 1 sys

1 time t0 - 28 s *

Golden-Ratio

x f x 4^2 x 3^* - 7 x 2^* - 2 - :

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

TOL 1012 -^:

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1 ≥ y2 ≡ "max in range" y1 ≤ y 2≡ "min in range"

X f 1.5 - 1 - y1 y2 ≤ TOL Extrema 1 el f 0.1 - 0.1 y1 y2 ≥ TOL Extrema 1 el f 2.5 3 y1 y2 ≤ TOL Extrema 1 el 31 mat :

X 1 el X 1 el f 12 mat X 2 el X 2 el f 12 mat X 3 el X 3 el f 12 mat 31 sys 1.2656 - 6.5923 - 12 mat 2.8161 108 -^* - 2 - 12 mat 2.7656 39.3452 - 12 mat 31 sys

x f X 1 el X 1 el f + 15 red X 2 el X 2 el f + 15 red X 3 el X 3 el f + 15 red 3 5 mat 2 1 sys

X 1.265564 - 2.816051 108 -^* - 2.765564 31 mat

Solving numeric a user system ... NO symbolic found

b 1 : c π : 21 line

f a0 0 ≡ a1 c ≡ a1 a2 + a3 + 0 ≡ a1 2 a2 * + 3 a3 * + b ≡ 41 mat :

init a0 1 ≡ a1 1 ≡ a2 1 ≡ a3 1 ≡ 41 mat :

a0 a1 a2 a3 sol f init FindRoot :

solutions in lexicographic order of "init" solutions a0 a1 a2 a3 sol vectorize : 0 3.1416 7.2832 - 4.1416 41 mat

Solving quaternion ======================================

α 0.5 :

β 1.5 :

p q r Clear 1

The Quaternion system f p q + r - 1012 -^≡ p 2^q 2^+ r 2^+ 1 ≡ β p * α r * + 1012 -^≡ 31 mat :

init p 1012 -^≡ q 1012 -^≡ r 1012 -^≡ 31 mat :

Sanity check vs Mathcad 11 f init FindRoot 0.1961 - 0.7845 0.5883 31 mat

p q r 31 mat 0.1961 - 0.7845 0.5883 31 mat

Quaternion p α - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / q 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* r 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 33 mat :

Next: evaluate numeric

modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt :

α 0.5 :

β 1.5 :

Quaternion 2 col 0.1961 - 0.7845 0.5883 31 mat

V α - modulo / α modulo / α β + (modulo / α β + (- modulo / α modulo / α - modulo / 32 mat :

V 0.1961 - 0.1961 0.7845 0.7845 - 0.1961 0.1961 - 32 mat

Next: stack a 'p' value

p 0.6 :

q Quaternion 2 col p stack :

q 0.1961 - 0.7845 0.5883 0.6 41 mat

Nex: the unit vector

q 0.1961 - 0.7845 0.5883 0.6 41 mat :

q q q norme / : 0.1682 - 0.6727 0.5045 0.5145 41 mat

Doublet visit

R 1 :

x y Doublet y R 2^- x 2^+ y 2^+ (* x 2^y 2^+ / :

x y Clear 1

level 0.25 :

α 2 :

x y Doublet_1 y R 2^- x 2^+ y 2^+ (* x 2^y 2^+ / level + :

x y Doublet_3 y R 2^- x 2^+ y 2^+ (* x 2^y 2^+ / α level * + :

x y Doublet_2 y R 2^- x 2^+ y 2^+ (* x 2^y 2^+ / level - :

x y Doublet_4 y R 2^- x 2^+ y 2^+ (* x 2^y 2^+ / α level * - :

Doublet Doublet_1 Doublet_2 Doublet_3 Doublet_4 5 1 sys

R 1.5 :

x y Doublet y R 2^- x 2^+ y 2^+ (* x 2^y 2^+ / :

y F x y Doublet 0 ≡ x solve maple :

x C R 2^x 2^- sqrt ± :

y F x y Doublet R ≡ x solve maple :

x C

y F 3 - 2 y * + (y * 332 y * + (* - 4 y 2^* + (* - sqrt 2 sqrt 3 - 2 y * + (* / 3 - 2 y * + (y * 332 y * + (* - 4 y 2^* + (* - sqrt 2 sqrt 3 - 2 y * + (* / - 21 mat

x Φ 3 - 2 x * + (x * 332 x * + (* - 4 x 2^* + (* - sqrt 2 sqrt 3 - 2 x * + (* / ± :

x Φ 1 5 1 5 - 2 2 mat 2 1 sys

x F x y Doublet 0.5 ≡ x solve maple :

x F 1 - 2 y * + (y * 9 - 2 y * 1 - 2 y * + (* + (* - sqrt 2 sqrt 1 - 2 y * + (* / 1 - 2 y * + (y * 9 - 2 y * 1 - 2 y * + (* + (* - sqrt 2 sqrt 1 - 2 y * + (* / - 21 mat

x Ω 1 - 2 x * + (x * 9 - 2 x * 1 - 2 x * + (* + (* - sqrt 2 sqrt 1 - 2 x * + (* / - (± :

x Ω

accurate RootSecant ... RootDichotomy

x Clear 1

x f x 0.5 1 -^x - exp * :

x q π sqrt x sqrt erf * :

"RootSecant" is viable  => guess wisely tial/error Φ a b ε X Root Secant us 1 el ε : us 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : us i el us i 1 - el - abs ε ≥ us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : i i 1 + : 21 line while us 71 line :

x Φ x f x q - eval :

a 0.25 : b 0.5 : ε 1015 -^: 13 mat

sol Φ a b ε X :

sol 11015 -^* 0.410800236270318 0.410800235082865 0.383337807794288 0.384527302650352 0.384477763318508 0.384477669224028 0.384477669231789 0.38447766923179 0.384477669231792 0.384477669231791 0.384477669231791 121 mat

index 1 to explode X sol sol rows row 1 el : 0.3845

x f x q X X f + 20 black 1 5 mat 3 1 sys

X f X q 21 mat 1.098 1.098 21 mat

Hypergeom ... bracket wisely wrt 'user'

x Γ x Gamma :

N n p x hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 39 :

n 12 :

p 20 :

user 0.125 :

x hypergeom N n p x hypergeom :

x f user :

x Φ x hypergeom x f - eval :

a 2.5 : b 3 : ε 109 -^: 13 mat

sol Φ a b ε X :

sol 1109 -^* 2.8064589330141 2.82494681593557 2.81422230007792 2.81423201555926 2.81423202762452 2.81423202762468 71 mat

X sol sol rows row 1 el : 2.8142

Continuous PDF Hypergeometric

0 x ≤ x hypergeom out hypergeometric if X user + 20 red 1 5 mat 2 1 sys

X hypergeom 0.125

Γ(x) [15 D] Alvaro Same otherwise

Γ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh z 0.5 + : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ 1 el Γ κ 1 + el z κ + / κ 114 sum + (* zp 2^* zg - exp * eval 21 line :

Valid for Re(ζ)>0

x Γ x Γ :

x g 2 :

x Γ x g 2 1 sys

COMMENTS: 1. Plot first the system to enable the bracketing. 2. Sometimes, "solve" is hard to refine .... ...get higher accuracy from "RootDichotomy"

"RootDichotomy"  is a robust companion. # Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε ≥ xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

x Φ x Γ x g - eval :

ε 1015 -^:

R x Φ 2.875 3.125 ε D : 3.00000000000002

Sanity check R Φ 0.000000000000000

r 0 x mwbesselj x 030 solve :

0 x mwbesselj

r 2.404826 5.520078 8.653728 11.791534 14.930918 18.071064 21.211637 24.352472 27.493479 91 mat

sanity checked up to mwbesselj(0,40) 040 mwbesselj 0.007366890584237

"roots" fails ... mwbesselj(0,x) does NOT
have symbolic derivative(s) diff(▪,▪) exact 0 x mwbesselj x 1 roots :

r Clear 1

ε 1015 -^:

x Φ 0 x mwbesselj :

braket wisely
rootDichotomy
... etc. r x Φ 2.3 2.5 ε D x Φ 5.4 5.6 ε D x Φ 8.6 8.7 ε D x Φ 11.7 11.9 ε D x Φ 14.5 15.5 ε D 51 mat : 2.40482555769577 5.52007811028631 8.65372791291101 11.7915344390143 14.9309177084878 51 mat

x f 0 x mwbesselj :

0 r mwbesselj vectorize 081016 -^* 61016 -^* - 71016 -^* 51016 -^* - 51 mat

x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

typical h 0.1 :

Recomfort yourself about mwbesselj(0,x) 1rst derivative

x f x h f D1 2 1 sys

http://mathworld.wolfram.com/GoatProblem.html

Tieing a goat to a point on the interior of the fence with radius a=1 using a chain of length L=r, consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection ... which cannot be solved exactly, but which has approximate solution: L= 1.15872847 Smath outputs 12 certified decimals ... 4 more decimals than Mathematica !

R 1 :

L Clear 1

This construction
courtesy Carlos . f 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + π R 2^* 2 / ≡ :

ε 1012 -^:

init L 1 ≡ :

L f init ε FindRoot : 1.158728473018

R L Goat 212 / R 2^* 2 L 2 R * / asin * 2 L 2 R * / asin * sin - (* (* L 2^L 2 R * / acos * + :

x c 10 x ≤ (x 2 R * ≤ (& cases :

track 0 π R 2^* 2 / L π R 2^* 2 / L 032 mat :

R x Goat x c * track 2 1 sys

π R 2^* 2 /

sanity check R L Goat π 2 / - 0.000000000000093 -

L

Solving the Hyperbolic function, i.e: inverse

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n 7.5 : x f n x / : 21 line

y Inverse y x f ≡ x solve maple : 152 y * /

X x f x - x n roots :

u 0.3125 :

X 2.73861278752583

n sqrt 2.73861278752583

x f u - x n u / roots 24

n u / 24

track 0 u n u / u n u / 032 mat :

x f track 2 1 sys

x f 1 x / :

y Inverse y x f ≡ x solve maple : 1 y /

X x f x - x 2 roots :

X 1.000000000000000

u 4 :

x f u - x 1 u / roots 0.25

1 u / 0.25

track 0 u 1 u / u 1 u / 032 mat :

x f track 2 1 sys

x f 1 x 2^- :

X Clear 1

y Inverse y x f ≡ x solve maple : 1 y - sqrt 1 y - sqrt - 21 mat

Inversing a function is solving for 'x', given the Cartesian 'y <= f(x)'. A most advanced maths work that Smath/Maple solves. Given 'y' we recover the two "x's"

y > 1 pure imaginary Y 0.625 :

target 0 Y X 1 el Y X 1 el 032 mat :

X 1 Y - sqrt 1 Y - sqrt - 21 mat : 0.61237 0.61237 - 21 mat

X f vectorize 0.625 0.625 21 mat

Arrow

xo 0 :

size 0.125 :

sub 0.0125 :

arrow1 xo - 1.5 size * sub i * 1.5 size * sub i * - 15 mat transpose : arrow2 xo - size 2 sub * i * size 2 sub * i * - 15 mat transpose : 21 sys

α rot α cos α sin α sin - α cos 22 mat :

Numeric A1 arrow1 Re 1.25 size * - arrow1 Im X 1 el + augment : A2 arrow2 Re size - X 1 el + 0 + arrow2 Im Y + augment : 21 sys

Arrow1 A1 90 ° * - rot * : Arrow2 A2 0 ° * - rot * : 21 sys

x f Arrow2 Arrow1 target 5 1 sys

Y 0.625

Y 1.5 :

1 Y - sqrt 1 Y - sqrt - 21 mat 0.7071 i * 0.7071 i * - 21 mat

X 1 el 0.6124

x f Arrow2 Arrow1 target 51 sys

sanity <= X 1 el f 0.625

So 0.175 : a 1 : P 1 : 13 mat x Eiffel So ln P * x a * So * + P / exp : 21 line

u 4096 :

set above visual sol

Eif x Eiffel u - x 100 roots : 57.4899

track 0 u Eif u Eif 032 mat :

x Eiffel track 2 1 sys

Provisional clear the all system. x a P So S u ζ Clear 1

So a P x f 1 u / u So S int : (a So * P / u 0 x int ≡ S solve maple So ln P * a So * x * + P / exp

So 0.1 : a 1 : P 1 : 13 mat

x Eiffel So ln P * x a * So * + P / exp :

x y axes x y / - :

ζ 24 :

u 0.375 :

E x Eiffel u - x 1 roots ζ - : 10.7824 -

track 40 - u E u E 032 mat :

xx ζ + Eiffel xx - ζ + Eiffel axes track 4 1 sys

xx ζ + Eiffel xx - ζ + Eiffel axes track 41 sys

U 3636 - 1231 mat :

suggestive ... solving cross product

n 0.25 :

A 12036 - 31 mat :

system A x y z 31 mat † n U * - 0 ≡ :

init x y z 31 mat 11 - 131 mat ≡ :

ε 1012 -^:

x y z 31 mat system init ε FindRoot : 1.106 - 0.25 4.068 31 mat

A x y z 31 mat † 8.99999999999996 8.99999999999988 - 2.99999999999999 31 mat