《Fractal Basins of Attraction paper》.doc

Fractal Basins of Attraction paper Liudong 101237 Considering using the Newton downhill method to iterate the polynomial, we can choose the points that the Newton iteration converges and at the same time we plot the historical points of the successful iteration points with a small point shown on the picture. As for the different color of each basin we can determine the final iteration point on which basin and for each basin we have different colors on it; and the program code is shown as follows: function basinsz3 t0=clock; n=100; for j=1:n for k=1:n a=-1+2*(j-1)/(n-1); b=1-2*(k-1)/(n-1); z(j,k)=a+b*i; [z,hist,flag]=newton(z(j,k),f,g); if flag==1 hold on if real(z)0.8 plot(hist,.b) elseif imag(z)0.3 plot(hist,.r) else plot(hist,.g) end title(Basins of attraction); axis ([-1,1,-1,1]); hold off end end end t=etime(clock,t0) end function [z,hist,flag]=newton(z0,f,g) maxit=100; tol=1e-6; flag=1; z=z0; hist=z; fz=feval(f,z); for it=1:maxit, if norm(fz)=tol, return; end d=feval(g,z)\fz; lambda=1.0; zn=z-d; fn=feval(f,zn); while norm(fn)=norm(fz), lambda=lambda*0.5; if lambdatol, flag=0; break; end zn=z-lambda*d; fn=feval(f,zn); end z=zn; fz=fn; hist=[hist z]; end end function v=f(z) v=z^3-1; end function v=g(z) v=3*z^2; end For this kind of program will debugs for a lot of time, here we just choose 100*100 pixels and we get the picture as shown below: For p(z)=z^3-1: t