<p> Arrow </p>

d268f242-c463-46b6-997f-e0c4e60162ba 5 4 5 decimal

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

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 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 - 18 s *

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f

x f 1 x 2^- :

3 appVersion 0.98.6179 ≡

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

t0 1 time :

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

<p> Arrow </p>

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 1 x - sqrt 1 x - sqrt - 2 1 sys

1 x - sqrt

Y 0.625

X 1 el 0.6124

sanity <= X 1 el f 0.625

1 x - sqrt -

revisit Lambert W(x)

Solving a DE

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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 :

Lambert two principal branches x W0 w w exp * x - 0 ≡ w 1 e / - roots : x W1 w w exp * x - 0 ≡ w 2 sqrt - roots : 21 line

L 0 : H 10 : N 200 : 13 mat U L H N xd : 21 line

Lambert complementary branches

x W0 x W1 1 e / - 1 1 e / - 5 - 2 2 mat 3 1 sys

Example: solving a DE

t C t diff Vm t C * - Km t C + / ≡

C0 1 :

Vm 2 :

Km 3 :

solution borrowed from Mathematica "ProductLog" DE i 1 N range p i el U i el Vm Km / * (- C0 C0 Km / exp * ln Km / + exp eval : d i el Km p i el W0 * : 21 line for d 21 line :

DEsol U DE augment :

Title 1.5 1.85 DE solution in Pharmacology 12 red 15 mat :

DEsol Title 2 1 sys

t C t diff Vm t C * - Km t C + / ≡