abc16478-fe5f-473e-baea-2708953408e8 21 4 5 decimal

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

The Smath "solve"

f q Y a b solveGEN x f x q - Y ≡ x a b solve :

f L H N a b 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 a b solve eval : for ω sol augment 41 line :

f x init Inv u f x - 0 ≡ u 0 roots :

x f x 20 - :

x q 5 :

x f 5 - 0 ≡ x 2030 solve 25

x f x q - 0 ≡ x 2030 solve 25

x f 5 2 1 sys

x f x sqrt :

x q 0.75 x * :

x0 f q 0 1.5 2 solveGEN : x1 f q 0.25 0.75 1.5 solveGEN : x2 f q 0.125 - 13 solveGEN : 31 mat 1.7778 1 2.0979 31 mat

Sanity check x1 f x1 q - 0.25

plot x f x q x1 x1 f x1 x1 q 22 mat x0 x0 f . 14 black 15 mat x1 x1 f . 14 black 15 mat x1 x1 q . 14 black 15 mat 61 sys :

plot

Maple code inverser

The task of inversing a function is generally difficult or simply not possible. Many inverse exist as built-in from approximation [asin, acos, atan ... probit]

x c d Q 1 c x d^* - exp - :

y c d Inverse y x c d Q ≡ x solve maple :

y c d Inverse 1 y - ln c / ln d / exp

x φ x x 2^x 3^31 mat :

b 2.6543 - 34709.6547 5785.1904 - 31 mat :

x f b x φ * :

2 f 92552

x f 2 200000 2 0 2 2 mat 2 1 sys

Numerical inverser Inv f 092552500 2.05 Inverse :

Inv 2 1 sys

p 2 π * k * t * t T t diff * - :

q 2 π * h * u * T1 u T - (* - :

Solve DE [simple] p q ≡ t T t diff solve maple h u * T1 - u T + (* k t * / -

Roots solve on the canvas

x F x tan x - :

x sol xx F x - 0 ≡ xx 1 roots :

x Sol xx F x - 0 ≡ xx 1 - roots :

Won't solve: sol(x) is NOT a function x F x sol - 0 ≡ x 1 1.17 solve #

roots solve for quick plot

x F x sol x Sol 3 1 sys

0.5 sol 0.75 - Sol 21 mat 0.975 1.0679 - 21 mat

u 1.165561185207215 :

u F u sol - 5.4688 1015 -^* -

x q xx F x ≡ xx 0 π 2 / solve :

x Q xx F x ≡ xx π 2 / - 0 solve :

x q x Q 2 1 sys

Implicit solve

x f x 2^4 - 2 x * 4 x 2^* - 21 mat :

x f 1 el x f 2 el - x 1 - 2 solve implicit of ≡ x f 1 el x f 2 el - ≡ 0 ≡

x f 1 el x f 2 el 2 1 sys

r x f 1 el x f 2 el - x 1 - 1.25 solve : 0.716515193517882 - 1.11651509667259 21 mat

Alternately, more accurate roots

R x f 1 el x f 2 el - x 1 - roots x f 1 el x f 2 el - x 1 roots 21 mat :

R 0.716515138991168 - 1.11651513899117 21 mat

A more difficult example, that you need to instruct wisely wrt the initial(s).

x F 12 π * sqrt / x 2^- 2 / exp * :

The code reads: for F(xx) running over xx find 'x' that makes F(xx)-x=0 wrt to initial ± 2 assign scalar 'x' to a function for recovery in in the canvas scalar plot. It goes this way only at time it plots on the canvas, it does not create a function internally.

x Inverse xx F x - 0 ≡ xx 2 - roots :

x inverse xx F x - 0 ≡ xx 2 roots :

x f x F x - 0 ≡ 2 - 2 roots :

Two complementary branches

x Inverse x inverse 0 3 1 sys x F 0 2 1 sys x f

Solving each branch

x F 12 π * sqrt / x 2^- 2 / exp * :

By including the roots range in the solve block, we get independence from roots (range) options.

x sol t F x - 0 ≡ t 5 - 0 solve :

x Sol t F x - 0 ≡ t 05 solve :

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 + 20 green 1 5 mat 3 1 sys

0 F 0.3989

More example

α β γ t F α β t * - γ t * / ln :

α 1 :

β 1 :

γ 1 :

Bracketed Roots range 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

Rework this example with Maple

t α β γ F α β t * - γ t * / ln :

y α β γ Inverse y t α β γ F ≡ t solve maple : α β y exp γ * + /

x α β γ myInv α β x exp γ * + / :

α 1 :

β 1 :

γ 1 :

0 x α β γ myInv 2 1 sys 0 x α β γ F 2 1 sys

x Pnorm 12 π * sqrt / x 2^- 2 / exp * :

x cnorm x CDF.Normal :

x Pnorm x cnorm 2 1 sys x ICDF.Normal

x Cn 11 0.07056 x 3^* 1.5976 x * + (- exp + / :

init 0 :

x q xx Cn x - 0 ≡ xx init roots :

x Pnorm x cnorm x q 31 sys

x Pnorm x Cn x q 3 1 sys

======= copy/paste in new sheet =======

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

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

Shadow in the context

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 2 1 sys

=====================================

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 * + / + :

maple("symbolicExpression") - evaluates symbolic expression.

# # 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 : 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 num

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

μ 11000 0.76 1.35 5.037 41 mat :

colony 500 :

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

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

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

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 interpolate 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 + / :

L H N xd U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx i * dx - (+ : for U 41 line :

L 0 : H 10 : N 200 : 13 mat

U L H N xd :

Ant i 1 U rows range AntCol i el U i el AntColony : for Ant U AntCol augment : 21 line :

AntColony symbolic model

Ant colony 2 1 sys