Minimize a function using Newton's Method

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Shikhar Singh
Shikhar Singh 2022-4-13
评论： Matt J 2022-4-13
Ran in:
I am trying to minimise the function stated below using Newton's method, however I am not able to display a plot which illustrates points as they iterate down towards the minimum:
% Minimise z =(3-X).^2 + 30*((Y-(X.^2)).^2) with a starting point of x=y=0
% My implementation:
X = -4:0.01:4; % Declare range of x values
[X,Y] = meshgrid(X); % return 2-D grid coordinates
f =(3-X).^2 + 30*((Y-(X.^2)).^2); % Initiallize function for plotting
surf(X,Y,f,'FaceColor','y','FaceAlpha',0.3,'EdgeColor','none'); % Plot function
syms X Y;
f =(3-X).^2 + 30*((Y-(X.^2)).^2);
x(1) = 0; % Initial value of x
y(1) = 0;% Initial value of y
df_dx = diff(f, X);
df_dy = diff(f, Y);
J = [subs(df_dx,[X,Y], [x(1),y(1)]) subs(df_dy, [X,Y], [x(1),y(1)])]; % Gradient
ddf_ddx = diff(df_dx,X);
ddf_ddy = diff(df_dy,Y);
ddf_dxdy = diff(df_dx,Y);
ddf_ddx_1 = subs(ddf_ddx, [X,Y], [x(1),y(1)]);
ddf_ddy_1 = subs(ddf_ddy, [X,Y], [x(1),y(1)]);
ddf_dxdy_1 = subs(ddf_dxdy, [X,Y], [x(1),y(1)]);
H = [ddf_ddx_1, ddf_dxdy_1; ddf_dxdy_1, ddf_ddy_1]; % Hessian
B = inv(H);
for i = 1:60
A = [x(i),y(i)]';
x(i+1) = A(1)-B(1,:)*J';
y(i+1) = A(2)-B(2,:)*J';
i = i+1;
J = [subs(df_dx,[X,Y], [x(i),y(i)]) subs(df_dy, [X,Y], [x(i),y(i)])]; % Update Jacobian
ddf_ddx_1 = subs(ddf_ddx, [X,Y], [x(i),y(i)]);
ddf_ddy_1 = subs(ddf_ddy, [X,Y], [x(i),y(i)]);
ddf_dxdy_1 = subs(ddf_dxdy, [X,Y], [x(i),y(i)]);
H = [ddf_ddx_1, ddf_dxdy_1; ddf_dxdy_1, ddf_ddy_1]; % Update Hessian
B = inv(H); % New Search Direction
end
% Plot:
hold on;
plot3(x,y,f,'o','Markersize',3,'Color','red'); % Illustrate descent
Error using plot3
Data must be numeric, datetime, duration or an array convertible to double.
xlabel x; ylabel y; zlabel z;
title('Newton Method');
hold off
grid on;

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回答（1 个）

Matt J
Matt J 2022-4-13
  6 个评论
显示 4更早的评论
Shikhar Singh
Shikhar Singh 2022-4-13
I have updated the code above with plot3 implemented.
Matt J
Matt J 2022-4-13
You cannot pass symbolic variables (like f) to plot3. In your previous post, you computed a sequence of z values that you fed to plot3,
z(i+1) = ((3-x(i+1)).^2) + (30.*(y(i+1)-(x(i+1).^2)).^2);
You have not done a similar thing here.

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