(* Examples of solving equations/finding roots *)
(* Simply paste each one at the Mathemtica prompt *)


(* Execute the first lines first. You may not need them  if working within a
notebook *) 

<<JavaGraphics`

(*For some reason the one below might be needed for
FindRootPlot[]  *)

 << Optimization`UnconstrainedProblems`   

(* Now do a straightforward case. Newton's method. *) 

FindRoot[x^2 -1  == 0., {x,  0.2}, Method -> {"Newton", "StepControl" -> None}]

FindRoot[x^2 -1  == 0., {x,  -1.1}, Method -> {"Newton", "StepControl" -> None}]

(* The one below visualizaes the iterations. 
Just give the function, not f(x) == 0. Syntax quirk. *)

(* may no longer work, need "ResourceFunction" below *) 
FindRootPlot[x^2 -1, {x,  0.2}, Method -> {"Newton", "StepControl" -> None}]

ResourceFunction["FindRootPlot"][x^2 -1, {x,  0.2}, Method -> {"Newton", "StepControl" -> None}] 

(* "StepControl" -> None forces the text-book Newton's. We will not bother 
with this in what follows, it is still Newton's *)

(* Newton's method's pitfalls *)
FindRoot[x^2 -1  == 0., {x,  0}, Method -> Newton]

(* A fix *)
FindRoot[x^2 -1  == 0., {x, Random[]}, Method -> Newton]

(* The one below visualizaes the iterations. 
Just give the function, no f(x) == 0. Syntax quirk. *) 
FindRootPlot[x^2 -1, {x,  Random[]}, Method -> Newton]


(* Quickly finds the correct solution at x=0 IF you start close to it. *)

(* First, see what the function looks like *) 
Plot[x^2*Exp[-x^2], {x,  0, 4}]

FindRootPlot[x^2*Exp[-x^2], {x,  0.2}, Method -> Newton]


(* Run-away problem: unstable algorithm with the starting value not close 
to the root at x=0.  *) 

FindRootPlot[x^2*Exp[-x^2], {x,  1.2}, Method -> Newton]

(* The material below is for self-study *) 
 
(* If no analytic derivatives are available
for the Erf[x] function, Mathematica will use Newton's method 
where the derivative is quickly estimated numerically (via secant line), 
see C&K book.  *)
(* Will be a bit slower *) 
FindRoot[Erf[x] == x, {x, {0, 1}} ]

(* Below is an ill-conditioned problem: f(x) is too shallow around the root x=0, the
convergence is poor, you may never get to the correct answer. *)       
FindRootPlot[x^30, {x, 0.1}, Method -> Newton]

(* Two equations. Will not be on the quiz. *)
FindRoot[{Sin[x] == Cos[y], x+y == 1}, {x, 0.1}, {y, 0.1}]