755de259-5578-4cd6-a300-9ed803059ac9 14 4 5 decimal

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

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

roots vs solve [timing 2 min]

f x init Inv u f x - 0 ≡ u init roots :

x f x exp : x sol f x 0 Inv : 21 line

x f x sol 2 1 sys

x f 12 π * sqrt / x 2^- 2 / exp * : x sol_1 f x 2 - Inv : x sol_2 f x 2 Inv : 31 line

x sol_1 x sol_2 5 10 x * f * 3 1 sys

Solving each branch

By including the roots range in the solve block,
we get independence from roots (range) options. x F 12 π * sqrt / x 2^- 2 / exp * :

x sol t F x - 0 ≡ t 5 - 0 solve :

x Sol t F x - 0 ≡ t 05 solve :

maple will not solve any and all functions y t F ≡ t solve maple 2 π * ln 2 y ln * + (- sqrt 2 π * ln 2 y ln * + (- sqrt - 21 mat

Scalar inverse on the canvas

x sol x Sol 0.3989 0 + 30 green 1 5 mat 3 1 sys x F

0 F 0.3989

Maple code "root" generatrix

This dual solver is most precious for Engineering/Scientific works. 1. given the independent, it solves for the dependent. 2. given the dependent, it solves for the independent. Just a matter of coding the respective root range appropriately. The AntColony model was developped before and we use the result. sol(w):=root[f(t,β,colony)-w,t] recast the model in the Maple symbolic executable format that Smath loves crunching. This format is "general" wrt 't', but it can be recast in term of either solving for 't' or 'w' by equating to 0 and specifying the variable to slove for and its respective bracketed root range.

x μ colony f colony μ 1 el 11 μ 2 el / 1 - (μ 3 el x μ 4 el - (* - exp * + / + :

# # root

a native Maple code, not documented

1. The general coding to be adapted for each solver

w sol t μ colony f w - t root :

w sol colony μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* μ 1 el μ 2 el * + w μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* - μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + / t root

evaluate symbolic the numerator to enable generalise the "root solve" sol(w).
Doing this transaction will enable to recast the solver for solving for  ...
... 'w' or for 't' ... after passing in the solve bloc "solve" equated to 0
and respecify for either 'w' or 't' within their respective range. Num colony μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* μ 1 el μ 2 el * + w μ 2 el μ 2 el - 1 + (μ 3 el t μ 4 el - (* - exp * + (* - :

Num 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * +

Verhulst fit to 'ant' data μ 11000 0.76 1.35 5.037 41 mat : colony 500 : 12 mat

2. Solving for 'w' [dependent] given 't' independent

superflous => strictly equivalent to f(x,μ,colony) w t Sol 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / 0 ≡ w 011000 solve :

w 5 Sol 8758

inverse solver

3. Solving for 't' [independent] given 'w' dependent

This is the "inverser" we are interested in ... solving for 'x' given: y:=f(x) w t Sol 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony w - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / 0 ≡ t 010 solve :

8758 t Sol 5

4. At this point, the dual solver is done for use in-situ.

Next, we recast the original model from the symbolic 1. above and make 'w=0', that will plot and interpolates the dependent.

t AntColony 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + (colony 0 - (* μ 1 el μ 2 el * + 1 μ 2 el - (μ 3 el t μ 4 el - (* - exp * μ 2 el + / :

x c 10 x ≤ (x 10 ≤ (& cases :

ant 15002110032050 3.5 38004540058700610550711200811100911900102 mat :

X ant 1 col : Y ant 2 col : data X Y . 12 black plotG : 31 line

w 7000 :

given Y from Y:=f(x) => solve for 'x' w t Sol 4.46

x AntColony x c * w t Sol w + 15 red 1 5 mat data 3 1 sys

How to use:

Calculation is left Auto. Once the 1rst run

is complete, you will see the solver responds

solving for x <= given Y:= f(x)

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