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The iterativee universal solver

x f 5953 / 11 x 2 / - (2^- (* :

1. solves for as many datum(s) as specified 2. is a discrete inverser as well 3. what matters for each application is the range of the function domain and the range of the iterative solve bloc [Lo,Hi]

Hi 4 :

x c 10 x ≤ (x Hi ≤ (& cases :

cumulative integration
over the plot canvas. z ∫ u f u 0 x int : x ∫ z ∫ : 21 sys

2 f u f u 04 int 4 ∫ 31 mat 198.3333 528.8889 528.8889 31 mat

x f

cumulative integration over domain c(x)

x ∫ x c *

y branches y x f ≡ x solve maple : 2595 sqrt 5953 y * - sqrt + (* 595 sqrt / 2595 sqrt 5953 y * - sqrt - (* 595 sqrt / 21 mat

x branches 1 el x branches 2 el 2 1 sys

x f 5953 / 11 x 2 / - (2^- (* :

max 2 f : 198.333

Two branches  iterative discrete solver/inverser ... explode/collect the 2 nests f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ 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 Lo Hi solve eval : for sol 61 line :

N 200 :

range wisely the domains sol f 0 max N 05 solveDiscrete :

j 1 sol rows range sol_1 j el sol j el 1 el : sol_2 j el sol j el 2 el : idx j el j : sol_12 idx sol_1 sol_2 augment : 41 line for

sol_1 sol_12 1 col sol_12 2 col augment : sol_2 sol_12 1 col sol_12 3 col augment : 21 line

Solutions are collected pairwise from the solve bloc. Tey must be exploded [Unested] and collected alternately for each branch, done via sub indexed [sol(j,1), sol(j,2)]

sol_1 sol_2 N 2 + 40 green 1 5 mat 3 1 sys x f 2 250 2 50 - 2 2 mat 2 1 sys

0 f 4 f 2 f 31 mat 00 198.333 31 mat

x f x 2 π * / 2 π * x / tanh * sqrt :

Inversing a function

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.

0.001 f 0.0126

One branch discrete solver/inverser export f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ 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 Lo Hi solve eval : for ω sol augment 61 line :

Sol f 0.05 0.99 100 0.00001 20 solveDiscrete :

Magnify the function to see

50 x f * Sol 2 1 sys

The native function

x f

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.

x f x 2 π * / 2 π * x / tanh * sqrt :

The other way around Maple "root"

w sol t f w - t root : w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / t root

Maple expression var root ≡ (is live in Smath 6179 but undocumented ≡

For solving, we equate the Maple 'root' to 0 and include in the Smath solve block c/w the 'roots' range. Indirectly, Mathcad/Mathsoft "tracking" implements a version of Maple "root" [or similar to Maple], but active by clicking in the curve for quick coordinates.

The symmetric solver within the range above

w t Sol w 2 sqrt * π sqrt * - t 2 π * t / tanh * sqrt + 2 sqrt π sqrt * / 0 ≡ t 20 - 20 solve :

w 0.75 :

w 0.95 ≤

w t Sol 3.804 - 3.804 21 mat

up dn locate line j 1 up range v j el j dn + : u j el locate j 0 * + : 21 line for u v augment 21 line :

Neg 21 - w t Sol 1 el line :

Pos 21 - w t Sol 2 el line :

Curve tracking Mathcad/Mathsoft style

x f w Neg Pos 4 1 sys

jmGiraud: 2015_08_23

w w t Sol 12 mat 34 / 3.804 - 3.804 21 mat 12 mat

x f x abs sin 0.5 * x 2^(* :

u 3 π * - x ≤ (x 3 π * ≤ (* :

maple will not solve any and all functions. y branches y x f u / ≡ x solve maple :

x f u / π 16 π 16 - 2 2 mat 2 1 sys

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

One branch discrete solver/inverser export f L H N Lo Hi solveDiscrete Lo/Hi are the range of the solver ω 0 : sol 0 : 12 mat refresh the solver before use ≡ 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 Lo Hi solve eval : for sol 61 line :

f x f u / :

N 100 :

set upper solver a bit higher than in x ≤ 3*π solveHigh f 01 π * N 03 π * 0.05 + solveDiscrete :

V solveHigh Unest : i 1 V rows 5 range : j 2 V rows 6 range : branches V i el V j el augment : idx 1 branches rows range : sols idx branches augment : sol_1 sols 1 col sols 2 col augment : sol_2 sols 1 col sols 3 col augment : 81 line

sol_1 sol_2 0 π o 7 black 1 5 mat 3 1 sys