(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)




(* Lazy implementation of Newton's method *)

f[x_] := x^3 - 2x - 2
nif[x_] := x - f[x] / f'[x]

(* This computes the Newton iterating function nif[x] for the given function f[x].  Changing the definition of f[x] above and rerunning both lines of code will automatically change nif[x] *)

NestList[nif, 1.5`20, 10]

(* This computes the first ten iterates of the Newton iterating function applied to 1.5, where 1.5 starts with 20 decimal places of precision. *)

(* Note that the quote mark is the angle quote `, which on most keyboards is to the left of the 1 key, above the Tab key.  The notation a`b tells Mathematica that the number a is given to b digits of precision. *)

(* Mathematica's internal system will keep track of precision losses during computations and display fewer digits accordingly -- exceedingly useful in numerical computation. *)


(* For problem 4 *)

Clear[a]

(* This clears the definition of the variable a, in case it was previously defined to be something else.  If you want to define the variable a again later, simply assign it a new value. *)

p[x_] := x^3 - a x

(* This defines the function p[x].  Note that you must include underscores after each variable in the function definition to tell Mathematica that x is one of the variables that p depends on. *)

Simplify[ (p[p[x]] - x) / (p[x] - x)]

(* This will simplify the quotient (p[p[x]] - x) / (p[x] - x), useful for finding 2-cycles *)




(* For problem 7 *)

p[x_]:= 1 + x - 3 x^2 - 15/4 x^3 + 3/2 x^4 + 9/4 x^5

(* This is the function in part b *)




(* For problem 8 *)

s1[x_]:= -x + x^2 - x^3 + 3/2 x^4 - 5/2 x^5 + 19/8 x^6 - 139/80 x^8 + 653/1000 x^10
s2[x_]:= -x + x^2 - x^3 + 3/2 x^4 - 5/2 x^5 + 19/8 x^6 - 139/80 x^8 + 652/1000 x^10

(* These are the two functions in the problem *)

NestList[s1, 0.1`50, 10]

(* This will compute the first 10 iterates of 0.1 under s1, where 0.1 starts with 50 decimal places of precision. *)

Expand[s1[s1[x]]]

(* This will expand out the expression s1[s1[x]] *)

Series[s1[s1[x]], {x,0,14}]

(* This will compute the Taylor series expansion for s1[s1[x]] in the variable x, centered at x=0, to degree 14 in x *)




(* For problem 9 *)

f[x_] := x + x Sin[1/x]

NestList[f, 0.1`20, 10]

(* This will compute the first 10 iterates of 0.1 under the function f, where 0.1 starts with 20 decimal places of precision. *)

NestList[f, 0.00015494157427205179, 10]

(* This will compute the first 10 iterates of 0.00015494157427205179 under the function f *)