with(plots):EXERCISE: What do you think this program do? How do you interpret the corresponding density plot?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 200 while( evalf(abs(f(x0)))>0.1) do
x0:=evalf(N(x0));
end do;
return i/400;
end proc:densityplot(mystery, -2..2, -2..2,colorstyle=HUE,grid=[10,10],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.In the plot colored by the "mystery" procedure, the set of points colored in teal correspond to an approximating set of the Newton function associated to the polynomial x^3-1.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*IThe aspect of the orbits of F1 and F2 we want to study is whether they go to infinity or not.c:=0.31+1.64*I;
f:=x->x^3-(c-1)*x-c*I:
N:=x->x-f(x)/D(f)(x):
new3:=proc(a,b)
global f,N;
local i,x0;
i:=1;
x0:=a+b*I;
for i from 1 to 150 while( evalf(abs(f(x0)))>0.1) do
x0:=evalf(N(x0));
end do;
return i/300;
end proc:with(plots):
densityplot(new3, -2..2 ,-2..2,colorstyle=HUE,grid=[140,140],style=patchnogrid);F1:=x->x^2+0.27+0.53*I;
orbits:=proc(a,b)
global F1;
local x0, i, C;
x0:=a + b*I;
C:=evalf(abs(0.27+0.53*I));
for i from 1 to 50 while (evalf(abs(x0)) <=2 or evalf(abs(x0))<=C )
do x0:=F1(x0) end do;
return i/100;
end proc:orbits(0,0);densityplot(orbits, -2..2 , -2..2 , style= patchnogrid, grid=[200,200],colorstyle=HUE);F1:=x->x^2+0.27+0.53*I;
orbits:=proc(a,b)
global F1;
local x0, i, C;
x0:=a + b*I;
C:=evalf(abs(0.27+0.53*I));
for i from 0 to 80 while (evalf(abs(x0)) <=2 or evalf(abs(x0))<=C )
do x0:=F1(x0) end do;
return modp(i,10)/10;
end proc:orbits(0,0);densityplot(orbits, -2..2 , -2..2 , style= patchnogrid, grid=[200,200],colorstyle=HUE);F1:=x->x^2+ 0.75*I;
orbits:=proc(a,b)
global F1;
local x0, i, C;
x0:=a + b*I;
C:=evalf(abs(0.27+0.53*I));
for i from 1 to 50 while (evalf(abs(x0)) <=2 or evalf(abs(x0))<=C )
do x0:=F1(x0) end do;
return i/100;
end proc:orbits(0,0);densityplot(orbits, -2..2 , -2..2 , style= patchnogrid, grid=[200,200],colorstyle=HUE);F1:=x->x^2+ 1.5;
orbits:=proc(a,b)
global F1;
local x0, i, C;
x0:=a + b*I;
C:=evalf(1.5);
for i from 1 to 50 while (evalf(abs(x0)) <=2 or evalf(abs(x0))<=C )
do x0:=F1(x0) end do;
return i/100;
end proc:orbits(0,0);densityplot(orbits, -2..2 , -2..2 , style= patchnogrid, grid=[200,200],colorstyle=HUE);EXERCISE: Plot the Mandelbrot set.