Book - Websites 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.) EXERCISE: Find the fixed points of the map h(x)=x^4+x-1 a. assuming that the domain of h is the real numbers b. assuming that the domain of h is the complex numbers restart: h:=x->x^4+x-1; solve(h(x)=x,x); with(RealDomain): solve(h(x)=x,x); 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 points x, f(x), f^2(x), f^3(x) .... is called the orbit of x under f EXERCISE: Find the fixed points of the Newton function associated to the polynomial x^3-1, What can you say about those fixed points? restart; f:=x->x^3-1; N:=x->x-f(x)/D(f)(x); fixedPointsOfN:=solve(N(x)=x,x); for i from 1 to 3 do simplify(f(fixedPointsOfN[i])) end do; The fixed points of N are the roots of f. If x0 is root of f, (f(x0)=0). Then N(x0)=x0-f(x0)/f'(x0)= x0 -0/f'(x0) = x0 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 10-th iteration of the Newton's method, with initial point a+b*I is close to r1 and 0.7 if the 10-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. p2:=x-> x^2 -3*I*x-4; exA:=proc(a,b) local p2, rootsOfp2, N, initial, i; p2:=x-> x^2 -(1+3*I)*x-4+5*I; rootsOfp2:=solve(p2(x),x); N:=x-> x- p2(x)/D(p2)(x); initial := a + b*I; for i from 1 to 10 do initial :=evalf(N(initial)) end do; if (abs(evalf (rootsOfp2[1]-initial) ) < 1 ) then return 0.4; else if(abs(evalf (rootsOfp2[2]-initial ) )< 1 ) then return 0.7; end if; end if; end proc: exA(1,0); with(plots): densityplot( exA , -3..3, -3..3, colorstyle=HUE, grid=[75,75],style=patchnogrid); evalf(abs (rootsOfp2[1]-initial) ) < 1 ) Make a more efficient program. Does any initial point converge to one of the roots by applying the Newton's method? If so, study how can a point fail to converge to a root. If the sequences converges, how long does it take to get close enough to the root.?