## Abbildungen - Kapitel 12

 ``` h=plot([-pi,0,pi],0,'k.'); set(h,'markersize',40); hold on; pdl = inline('[x(2);-9.81*sin(x(1))]','t','x'); opt=odeset('RelTol',1e-5); for x1=[-9 -8 -7 1 2 3 4 5 6 7 8 9], [t,x]=ode45(pdl,[0 5],[0;x1],opt); h=plot(x(:,1),x(:,2),'k-'); set(h,'linewidth',2); [t,x]=ode45(pdl,[0 -5],[0;x1],opt); h=plot(x(:,1),x(:,2),'k-'); set(h,'linewidth',2); end axis([-3.2 3.2 -8 8]) [t,x]=ode45(pdl,[0 -4],[3.14;0],opt); h=plot(x(:,1),x(:,2),'-','Color',[0.6 0.6 0.6]); set(h,'linewidth',4); [t,x]=ode45(pdl,[0 4],[3.14;0],opt); h=plot(x(:,1),x(:,2),'-','Color',[0.6 0.6 0.6]); set(h,'linewidth',4); ``` ``` n = 30; tau = 0.1; x = zeros(2,n); x(:,1) = [0;4]; for i=1:n, x(:,i+1) = x(:,i) + tau*feval(pdl,0,x(:,i)); end h=plot(x(1,:),x(2,:),'k.-'); set(h,'linewidth',2,'markersize',20); ``` ``` x0 = [0;4]; n = 30; tau = 0.1; x = zeros(2,n); x(:,1) = x0; TOL = 1e-6; N_max = 20; d = size(x0,1); x(:,1) = x0; for k = 1:n x(:,k+1) = x(:,k); for j = 1:N_max % Newton step F = x(:,k+1) - (x(:,k) + tau * feval(pdl,0,x(:,k+1))); DF = eye(d) - tau * feval(dpdl,0,x(:,k+1)); dx = DF\F; x(:,k+1) = x(:,k+1) - dx; if (max(abs(dx)) < TOL) break; end; end; end; h=plot(x(1,:),x(2,:),'k.-'); set(h,'linewidth',2,'markersize',20); ``` ``` x0 = [0;4]; n = 30; tau = 0.1; x = zeros(2,n); x(:,1) = x0; TOL = 1e-6; N_max = 20; d = size(x0,1); for k = 1:n x(:,k+1) = x(:,k); for j = 1:N_max % Newton step F = x(:,k+1) - (x(:,k) + tau * feval(pdl,0,(x(:,k)+x(:,k+1))/2)); DF = eye(d) - tau * feval(dpdl,0,x(:,k+1)); dx = DF\F; x(:,k+1) = x(:,k+1) - dx; if (max(abs(dx)) < TOL) break; end; end; end; h=plot(x(1,:),x(2,:),'k.-'); set(h,'linewidth',2,'markersize',20); ```

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