b62127e5-e970-4b0a-bdf5-14352910bc06 5 3 5 decimal

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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 : 0.838817

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

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

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

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.

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 - 15 s *

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f