<p> Max/Min Golden ratio </p>

b62127e5-e970-4b0a-bdf5-14352910bc06 10 15 12 decimal

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

t1 1 : t2 1.5 : 21 sys

x c 10 x ≤ cases :

t R 1 t - t1 / exp - (1 t - t2 / exp - (* :

inflexion @ Deriv(2) = 0
roots range [0.5 .. 1] x f x R x 2 diff :

X x f x solve : 0.838911

Y X R : 0.243

t S X R X diff t X - (* X R + :

L X X S 1 - X + (1 S - X S + (/ * - : 0.233

Ω x f Y - x 0.5 0.75 solve :

<p> Max/Min Golden ratio </p>

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 :

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

x f x R x 2 diff :

y1 y2 ≤ (min in range ≡

U f 1.5 2 y1 y2 ≤ 1012 -^Extrema 1 el : 1.673

min U f : 0.162 -

x R x c * x R x 2 diff x c * x S X Y + 250 black 1 5 mat L 0 . 14 black 1 5 mat X 0 . 14 black 1 5 mat Ω Y . 14 black 1 5 mat U min + 200 black 1 5 mat 8 1 sys

Plot arguments ... x R x c * x R x 2 diff x c * x S X Y + 250 black 15 mat L 0.14 black 15 mat X 0.14 black 15 mat Ω Y . 14 black 15 mat U min + 200 black 15 mat 81 sys

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 ≡ x c 10 x ≤ (x 5 ≤ (& cases : 71 line

Mach number ... autosolve on canvas 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 3 1 sys

show/hide plot arguments x Subsonic x Supersonic x c * k 1 + 20 black 15 mat 31 sys

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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Mach number

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α β γ t F α β t * - γ t * / ln :

α 1 :

β 1 :

γ 1 :

x sol α β γ tt F x - 0 ≡ tt 1 - 1 solve :

r1 0 4.5 + 15 red 15 mat :

r2 1 4.5 - + 15 red 15 mat :

Scalar inverse on the canvas

x sol 4.5 - 1 + 15 red 1 5 mat 4.5 0 + 15 red 1 5 mat 3 1 sys

Mother function to inverse

α β γ x F r1 r2 3 1 sys

Construct a complete discrete inverser ... typical for this example L H N Inverse U 0 : dx H L - N / : i 1 N 1 + range U i el L dx i * dx - (+ : for i 1 U rows range invXFR i el α β γ t F U i el ≡ t 1 - 1 solve : for U invXFR augment 51 line :

INV_XFR 4.588 - 4.588 100 Inverse :

Discrete inverse

INV_XFR

Solving for a vectorised function

t0 1 time :

x f x 2 π * / 2 π * x / tanh * sqrt :

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.

Design a discrete function inverser  in the L, H span , N points 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 0.02 20 solve : for ω sol augment 51 line :

Sol f 0.06 0.99 100 Inverse :

1 time t0 - 22 s *

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f