Definition A fixed point of a function G is an element u in the domain of G, such that G(u)=u. Example : 1/2 is a fixed point of the function G(x)=-x+1.(Note that a function can have fixed points if and only if the range is included in the domain.)Definition: The set of points that converge to a root r under the iteration of the Newton's method is called the basin of attraction of the root r.Definition If f is a function with range included in the domain, then the sequence of pointsx, f(x), f^2(x), f^3(x) .... is called the orbit of x under fWe will be interesting in describing the orbits of certain maps from a qualitative point of view.Definition: If n is a positive integer then a point period n of a function G is an element u in the domain of G, such that G^n(u)=u but u is not a periodic point of period stritly less than n.Note: All the above definitions are for any function not necessarily for the associated Newton function.EXERCISE: Find the fixed points of the Newton function associated to the polynomial x^3-1, What can you say about those fixed points? Fixed points of N = Roots of fThe derivative of N evaluated at a root of F is 0Let's put together two facts we have observed up to now:The derivative of N evaluate at the fixed points of N is 0.The fixed points of N "attract" points.EXERCISE: a. Choose a polynomial of degree 2. Denote one of the roots by r1 and the other by r2. Create a maple procedure (analogous to the one we did the previous class) that takes as input a pair of numbers (a,b), and gives as a ouput 0.4 if the 20-th iteration of the Newton's method, with initial point a+b*I is close to r1 and 0.7 if the 20-th iterate is close to r2. b. Repeat the above with a polynmial of degree 4 (now you need to study the 4 roots).c. Make the density plots of both procedures.p:=x->x^2-x+I;
r:=solve(p(x),x);
N:=x->x-p(x)/D(p)(x);
new1:=proc(a,b)
local x0, i;
global N,p,r;
x0:=a+b*I;
for i from 1 to 20 do
x0:=evalf(N(x0));
if (abs(evalf(x0-r[1]))<0.1) then return 0.4;
else if (abs(evalf(x0-r[2]))<0.1) then return 0.7; end if; end if;
end do;
end proc:with(plots):densityplot(new1, -3..3, -3..3, colorstyle=HUE, grid=[76,76],style=patchnogrid);new1:=proc(a,b)
local p, r,N,x0, i;
p:=x->3*x^2-1;
r:=solve(p(x),x);
N:=x->x-p(x)/D(p)(x);
x0:=a+b*I;
for i from 1 to 20 do
x0:=evalf(N(x0));
if (abs(evalf(x0-r[1]))<0.1) then return 0.4;
else if (abs(evalf(x0-r[2]))<0.1) then return 0.7; end if; end if;
end do;
end proc:
densityplot(new1, -3..3, -3..3, colorstyle=HUE, grid=[45,45]);new1:=proc(a,b)
local p, r,N,x0, i;
p:=x->3*x^2-1;
r:=solve(p(x),x);
N:=x->x-p(x)/D(p)(x);
x0:=a+b*I;
for i from 1 to 20 do
x0:=evalf(N(x0));
end do;
if (abs(evalf(x0-r[1]))<0.1) then return 0.4;
else if (abs(evalf(x0-r[2]))<0.1) then return 0.7; end if; end if;
end proc:
densityplot(new1, -3..3, -3..3, colorstyle=HUE, grid=[45,45]);p:=x->3*x^2-1;
r:=solve(p(x),x);
N:=x->x-p(x)/D(p)(x);N(u);simplify(N(I*u));p:=x->x^2-x+I;
r:=evalf(solve(p(x),x));
N:=x->x-p(x)/D(p)(x);
new2:=proc(a,b)
local x0, i;
global N,p,r;
x0:=a+b*I;
for i from 1 to 20 while(abs(evalf(p(x0)))>0.1) do
x0:=evalf(N(x0));
end do;
return (Im(x0)+0.7)/2;
end proc:evalf(subs(x0=r[1],(Im(x0)+0.7)/2));evalf(subs(x0=r[2],(Im(x0)+0.7)/2));new2(1,1);densityplot(new2, -3..3, -3..3, colorstyle=HUE, grid=[45,45]);N(u);N(1/2) is not defined.Make a more efficient program.QUESTION-EXERCISE Consider your favorite polynomial with complex coefficients and degree 2.
Does any initial point converge to one of the roots by applying the Newton's method?If not, study how can a point fail to converge to a root.Give a qualitative description of the orbits of the Newton function associated to your favorite polynomial.Definition: The Julia set of a function is the set of points The Julia set of your favorite polynomial is...QUESTION -EXERCISE Consider the polynomial x^3-1.
Does any initial point converge to one of the roots by applying the Newton's method?
Give a qualitative description of the orbits of the Newton function associated to your favorite polynomial.EXERCISE Consider a function f and the associated Newton function N. Use a maple procedure to study "how long" (how many iterates) does it take to get close to roots.EXERCISE: Let us study the orbits of the functiosn N1, N2, F1 and F2 whereN1 and N2 will be the Newton function associated to two different polynomials: One will be the polynomial f(x)=x^3-1and the other, f1(x)=x^3-(0.68+1.64*I)*x-0.32-1.64*I
F1 will be defined as F1(x)=x^2+0.27+0.53*I
F2 will be defined as F2(x)=x^2+ 0.75*I