New set of problems is posted in the course schedule page.We continue studying Newton's methodNow, let us consider the function f(x)=x^3-1 with domain the complex numbers.EXERCISE: Define a procedure to help you find the roots of f by the Netwon's method.f:=x->x^3-1;
N:=x->x-f(x)/D(f)(x);
We use a for loop to find a root of fx0:=0.9:
for i from 1 to 10 do x0:=N(x0) end do; x0:=90;
for i from 1 to 20 do x0:=evalf(N(x0)) end do; x0:=-0.5;
for i from 1 to 10 do x0:=N(x0) end do; N(-0.5);N(u);x0:=I:
for i from 1 to 10 do x0:=evalf(N(x0)) end do; 1 is one of the roots of f. Since we are working in the complex domain, there are two more roots.Let's try to find themQUESTION: Is it possible to find all the roots of f by the Newton's method. Why?newton:=proc(x0)
local x, i, n;
global N;
n:=50;
x:=x0;
for i from 1 to n do
x:=evalf(N(x));
od;
return(x);
end:
newton(2);newton(-1+0.2*I);newton(-1+0.02*I);newton1:=proc(x0, n)
local x, i;
global N;
x:=x0;
for i from 1 to n do
x:=evalf(N(x));
od;
return(x);
end:newton1(3+I,10);We are going to modify the above procedure so it takes as parameters the real and imaginary parts of a complex number newton2:=proc(x0,y0)
local x, i;
global N;
x:=x0+y0*I;
for i from 1 to 10 do
x:=evalf(N(x));
od;
return(x);
end:newton2(0,3);jWHO GOES WHERE??? WHICH INITIAL POINTS CONVERGE TO WHICH ROOTS???The map f we are working with has three roots. We saw that if e take an initial pointst "close" to one of the roots, and apply Newton's method, then the sequence of iterates of the associated Newton's function converges to the close root. Now we'd like to study what happens with the rest of points, whether they converge to a root or not when we apply Newton method, and if they converge, to which root.NOTE: Now, we are changing the point of view. We are not longer worried about "finding roots", but we are studing how Newton's iterates behave.EXERCISE: Find all the roots of the above map f. Denote one of them as r1, another as r2 and the third one as r3 in any order you want. Define a procedure that takes as an input a complex number z, computes 50 iterates of z by Newton's method, and returns 0.1 if the result is close to r1, 0.2 if it is close to r2 and 0.3 if it is close to r3.sol:=fsolve(f(x),x,complex);r1:=sol[1];
r2:=sol[2];
r3:=sol[3];
newton3:=proc(x0,y0)
local x, i;
global N, r1,r2,r3;
x:=x0+y0*I;
for i from 1 to 10 do
x:=evalf(N(x));
od;
if( abs(x- r1)< 0.1) then return 0.1 ;
else if( abs(x- r2)< 0.1) then return 0.2;
else if( abs(x- r3)< 0.1) then return 0.3 end if; end if; end if;
end:newton3(-4,-3);newton3(-0.5,1);plot3d(newton3, -2..2,-2..2,axes=boxed,style=patchcontour);with(plots):densityplot(newton3, -2..2,-2..2, grid=[100,100],colorstyle=HUE,style=patchnogrid);color1:=proc(x,y)
return x;
end proc:
densityplot(color1,-2..2,-2..2,colorstyle=HUE);F:=(x,y)->y;color2:=proc(x,y)
return y;
end proc:
densityplot(color2,-2..2,-2..2,colorstyle=HUE);densityplot(F(x,y),x=-2..2,y=-2..2,colorstyle=HUE);color2:=proc(x,y)
return y;
end proc:
densityplot(color2,-3..10,-20..23,colorstyle=HUE);densityplot(color2,-3..10,0..1,colorstyle=HUE);densityplot(color2,-3..10,0.2..0.6,colorstyle=HUE);How do you think Maple does the coloring?Here is the idea: Remeber that we have a procedure "color" that takes an ordered pair of real numbers (x,y) and return another real number r.If the all the values that output of the procedure are between 0 and 1, then Maple paint in read all points (x,y) such that color(x,y) is equal (or very close ) to zero, in orange all the points such that color(x,y) is 0.1, and so on.If the values of the output of the procedure are between the real numbers a and b, (and a and b are reached), then first the interval [a, b] is scaled to [0,1] then the points are painted as before. Hence, for instance, the points such that n(x,y)=a, are going to be painted in red (as if we had n(x,y)=0) Definition A fixed point of a function G is an element u in the domain of G, such that N(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.)EXERCISE: Find the fixed points of the map h(x)=x^4+x-1a. assuming that the domain of h is the real numbersb. assuming that the domain of h is the complex numbersEXERCISE: Find the fixed points of the Newton function associated to the polynomial x^3-1, What can you say about those fixed points?