% illustration of numerical integration % compare the Forward Euler method, which is globally O(h)  % with Midpoint method, which is globally O(h^2) % and the exact solution  function main()     f = inline ('2-y', 't', 'y');    % will solve y' = f(t, y)     a=0; b=1; % endpoints of the interval where we will solve the ODE    A = -0.5*b; B = 1.5*b; % a bit of an expanded interval    N = 2; T = linspace(a, b, N); h = T(2)-T(1); % the grid    y0 = 1; % initial condition  %   % One step of the midpoint method    Y = solve_ODE (N, f, y0,  h, T, 2); % midpoint method     % exact solution to the right      hh=0.05; TT = a:hh:B; NN = length(TT);    YY = solve_ODE (NN, f, y0,  hh, TT, 2); % midpoint method     % exact solution to the left     TTl = a:hh:(-A); NN = length(TTl);    ZZ = solve_ODE (NN, f, y0,  -hh, TTl, 2); % midpoint method  %  the tangent line at the midpoint    tmid = (a+b)/2;    I = find(TT >= tmid); m = I(1);    tmid = TT(m); ymid = YY(m); slope = f(tmid, ymid);    Tan_l = 0.5*b; Tant = (tmid-Tan_l):hh:(tmid+Tan_l); Tany = slope*(Tant-tmid)+ymid;   %  prepare the plotting window    lw = 3; % curves linewidth    lw_thin  = 2; % thinner curves    fs = 30; % font size    figure(1); clf; hold on; axis equal; axis off;  % colors    red=[0.867 0.06 0.14];    blue = [0, 129, 205]/256;    green = [0, 200,  70]/256;    black = [0, 0, 0];  % coordinate axes    shifty=0.2;    arrowsize=0.1; arrow_type=1; angle=20; % in degrees    arrow([A, shifty], [B, shifty], lw_thin, arrowsize, angle, arrow_type, black)  % plot auxiliary lines    I = find(TT >= a); m = I(1);  ya = YY(m);    plot([a, a], [0+shifty, ya], 'linewidth', lw_thin, 'linestyle', '--', 'color', black)     I = find(TT >= tmid); m = I(1);  ymid = YY(m);    plot([tmid, tmid], [0+shifty, ymid], 'linewidth', lw_thin, 'linestyle', '--', 'color', black)     I = find(TT >= b); m = I(1);  yb = YY(m);    plot([b, b], [0+shifty, yb], 'linewidth', lw_thin, 'linestyle', '--', 'color', black)  % plot the solutions    plot(TT, YY, 'color', blue,   'linewidth', lw);    plot(-TTl, ZZ, 'color', blue,   'linewidth', lw)    plot(T, Y, 'color', red, 'linewidth', lw)     % plot the tangent line    plot(Tant, Tany+0.003*lw, 'color', green, 'linewidth', lw)     smallrad = 0.02;    ball (T(1), Y(1), smallrad, red)    ball (T(length(T)), Y(length(Y)), smallrad, red)     % text    small = 0.15;     text(a, shifty-small, '\it{t_n}', 'fontsize', fs)    text(tmid, shifty-small, '\it{t_n+h/2}', 'fontsize', fs)    text(b, shifty-small, '\it{t_{n+1}}', 'fontsize', fs)    text(T(1)-1.5*small, Y(1), '\it{y_n}', 'fontsize', fs, 'color', red)    text(T(length(T))+0.6*small, Y(length(Y)), '\it{y_{n+1}}', 'fontsize', fs, 'color', red)    text(-TTl(length(TTl))+0.1*small, ZZ(length(ZZ))+3*small, '\it{y(t)}', 'fontsize', fs, 'color', blue)            % axes aspect ratio %   pbaspect([1 1.5 1]);  %% save to disk    saveas(gcf, sprintf('Midpoint_method_illustration.eps', h), 'psc2');     function Y = solve_ODE (N, f, y0,  h, T, method)     Y = 0*T;        Y(1)=y0;    for i=1:(N-1) 	  t = T(i); y = Y(i);  	  if method == 1 % forward Euler method 		  		 Y(i+1) = y + h*f(t, y); 		  	  elseif method == 2 % explicit one step midpoint method 		  		 K = y + 0.5*h*f(t, y); 		 Y(i+1) =  y + h*f(t+h/2, K); 		  	  else 		 disp ('Don`t know this type of method'); 		 return; 		  	  end    end      function arrow(start, stop, thickness, arrow_size, sharpness, arrow_type, color)  % Function arguments: % start, stop:  start and end coordinates of arrow, vectors of size 2 % thickness:    thickness of arrow stick % arrow_size:   the size of the two sides of the angle in this picture -> % sharpness:    angle between the arrow stick and arrow side, in degrees % arrow_type:   1 for filled arrow, otherwise the arrow will be just two segments % color:        arrow color, a vector of length three with values in [0, 1]  % convert to complex numbers    i=sqrt(-1);    start=start(1)+i*start(2); stop=stop(1)+i*stop(2);    rotate_angle=exp(i*pi*sharpness/180);  % points making up the arrow tip (besides the "stop" point)    point1 = stop - (arrow_size*rotate_angle)*(stop-start)/abs(stop-start);    point2 = stop - (arrow_size/rotate_angle)*(stop-start)/abs(stop-start);     if arrow_type==1 % filled arrow        % plot the stick, but not till the end, looks bad       t=0.5*arrow_size*cos(pi*sharpness/180)/abs(stop-start); stop1=t*start+(1-t)*stop;       plot(real([start, stop1]), imag([start, stop1]), 'LineWidth', thickness, 'Color', color);        % fill the arrow       H=fill(real([stop, point1, point2]), imag([stop, point1, point2]), color);       set(H, 'EdgeColor', 'none')     else % two-segment arrow       plot(real([start, stop]), imag([start, stop]),   'LineWidth', thickness, 'Color', color);       plot(real([stop, point1]), imag([stop, point1]), 'LineWidth', thickness, 'Color', color);       plot(real([stop, point2]), imag([stop, point2]), 'LineWidth', thickness, 'Color', color);    end  function ball(x, y, r, color)    Theta=0:0.1:2*pi;    X=r*cos(Theta)+x;    Y=r*sin(Theta)+y;    H=fill(X, Y, color);    set(H, 'EdgeColor', 'none');