' - -  File: III_Zero3.txt

' Find the zeros of a function, that has a plotting range beyond the program's limits.
' We want to find the zeroes of 10^x - x^10.  Follow the steps below.

' STEP 1: ***** Lets compress the function to Plot to help locate the zeros. *****
'         The following procedure is used:
'                          If Abs[F{x}] > 1 --> Sign[F{x}] * Ln[Abs[F{x}]] + Sign[F{x}  else
'                         -1 < F{x} < 1      -->  Sign[F{x} * (Exp[Abs[F{x}]] - 1) / (Exp[1] - 1)

F{x} = 10^x - x^10     ' Let Co{x} be our "compressed" function:
Sa{x} =  Unit[Abs[F{x}] - 1]              '  *** Unit[...] = 1  if    | F{x} | > 1   ***
Sb{x}  = Unit[1 - Abs[F{x}]]              '  *** Unit[...] = 1  if    | F{x} | <  1   ***

Co{x} = Sa{x} * (Sign[F{x}] *Ln[Abs[F{x}]] + Sign[F{x}]) + Sb{x} *Sign[F{x}] * (Exp[Abs[F{x}]] -1) / (Exp[1] -1)
' -   -   -   -   -   -   -   -   -   -   this part -   -   -   -   -   -   -   -  OR  -   -   -   -   -   this part. -   -   -   -   -   -   -   -   -   -

H<x> = <x, Co, 0>                          ' Plot to estimate the zeros of F{x}
Plot[H, {x, -3,11,200}]                    ' Read these estimate zeros:

'     Estimated visual brackets from Plots:  (-1, 0); (1, 2);  &  (9.5, 10.5)
   
' ***********************************************************************************************************************
'STEP 2:  Insert the above estimates into the Zero command below.
to = 0.00001                           ' Tolerance to try.

Zero[F, {-1,0, z1}to]
'@Zero at @z1 !

Zero[F,{1,2, z2} to]       
'@Zero at @z2!

Zero[F,{9.5,10.5, z3} to]
'@Zero at @z3
<t = 1>                                    ' Use any value to display the zeroes.




