<p> Inverse(f,L,H,N) </p>

da084a54-8a21-446c-85e8-dfd6401c3bdb 10 3 5 decimal

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

Inverse/Iterate a scalar cumulative integration

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 05 solve eval : for ω sol augment 41 line :

Setup the companion system

Note:

For this nice parabolic F(x), set integration accuracy 50.

Inversing a function is solving for 'x' given 'y' ...

a most common task in Engineering/Scientific works.

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5953 / 11 x 2 / - (2^- (* 0 ≡ x 25 solve 4

Hi 4 :

z ∫ u F u 0 x int : x ∫ z ∫ : 21 line

x F 5953 / 11 x 2 / - (2^- (* 0 x ≤ (x Hi ≤ (& cases :

1 F 2 F 21 mat 148.75 198.333 21 mat

1 ∫ 2 ∫ 4 ∫ 31 mat 82.63889 264.44444 528.88889 31 mat

x Int x ∫ 0 x ≤ (x Hi ≤ (& cases :

x f x Int :

X 256 : Y x Int X - 0 ≡ x 05 solve : 21 line

X Y 21 mat 256 1.957 21 mat

Inv f 0 528.88888 50 Inverse :

1. X => the seeked value of cumul Σ 2. Y => 'x' at which Σcumul occurs

Inv X Y + 250 red 1 5 mat 2 1 sys

1. blue => the function 2. red => its cumulative integration

x F x Int 2 1 sys

Maple classical/direct inverser

x a b q 1 x b / - exp (a^- :

y x a b q ≡ x solve maple 1 y - ln b * a / -

y a b sol 1 y - ln b * a / - :

a 2 : b 1 : 12 mat

x middle b x * :

x a b q x a b sol x middle 3 1 sys

An inverse function: 1. may exist simply, this example. 2. may exist via "horrible maths" !!!. 3. may not exist at all [Lambert W]. 4. may be evaluated via the iterative "solve bloc". For this last case, it is unlikely to find a good model function [or maybe picewise]. In that case, it is possible to find an excellent interpolation function of the rational type ... easy to conclude from Mathcad 11 and earlier via GenfitMatrix ... sometimes managable in Smath sometimes very difficult or let say "not possible".