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^4-x:
r:=[fsolve(p(x),x,complex)]:
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 100 while( evalf( abs(p(x0)))>0.01 ) do
x0:=evalf(N(x0));
end do;
return (Im(x0)+Re(x0)+1.4)/3;
end proc:for i from 1 to 4 do [ r[i] ,new1(Re(r[i]), Im(r[i]))] od;with(plots):new1(0.01,0.01);display(densityplot(new1,-1..1,-1..1,colorstyle=HUE,style=patchnogrid,grid=[100,100]),complexplot(r,style=point,thickness=2,color=white));Plot some of the "other" types of orbits of the Newton function associated to x^3-1p:=x->x^3-1:
r:=[fsolve(p(x),x,complex)]:
N:=x->x-p(x)/D(p)(x):
p2:=solve((N@@2)(x)=x,x):p3:=evalf(solve((N@@3)(x)=x,x)):
display(complexplot([p2],color=blue, style=point,thickness=2),complexplot([p3],color=green, style=point,thickness=2));r1:=solve((N)(x)=0,x):r2:=solve((N@@2)(x)=0,x):r3:=evalf(solve((N@@3)(x)=0,x)):display(complexplot([r2],color=blue, style=point,thickness=2),complexplot([r3],color=green, style=point,thickness=2),complexplot([r1],color=magenta, style=point,thickness=2));EXERCISE: What do you think this program do? How do you interpret the corresponding density plot?new1(0,0);f:=x->x^3-1;
N:=x->x-f(x)/D(f)(x);
mystery:=proc(a,b)
global f,N;
local i,x0;
i:=1;
x0:=a+b*I;
for i from 1 to 20 while( evalf(abs(f(x0)))>0.1) do
x0:=evalf(N(x0));
end do;
return i/40;
end proc:densityplot(0.45,x=-2..2,y=-2..2,colorstyle=HUE);densityplot(mystery, -2..2, -2..2,colorstyle=HUE,grid=[80,80],style=patchnogrid);Definition: The Julia set of a function F (which is aNewton function) is the set of points which are not attracted to any fixed point nor any periodic orbit.Question: What is the Julia set of the Newton function associated to a quadratic polynomial?It is the line formed by the points that they are equidistant from both 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*If:=x->x^3-(0.68+1.64*I)*x-0.32-1.64*I;
N:=x->x-f(x)/D(f)(x);
mystery:=proc(a,b)
global f,N;
local i,x0;
i:=1;
x0:=a+b*I;
for i from 1 to 100 while( evalf(abs(f(x0)))>0.1) do
x0:=evalf(N(x0));
end do;
return i/40;
end proc:with(plots):
densityplot(0.5,x=-2..2,y=-2..2,colorstyle=HUE);densityplot(mystery, 1..1.1, 0.1..0.125,colorstyle=HUE,grid=[80,80],style=patchnogrid);