Writing functions f(x,y)

How do I correctly write f(x,y)=4(x-1)^2+3(y-2)^2+2(x-2)^2(y-3)^2 in MatLab code for a fiminsearch? I have been trying for days and I still cant get it right. Please help!!!

Answer 1

[Torsten]

f = @(x,y)4*(x-1)^2+3*(y-2)^2+2*(x-2)^2*(y-3)^2;
z0 = [1 1]

z0 = 1×2

1 1

sol = fminsearch(@(z)f(z(1),z(2)),z0)

sol = 1×2

1.1963 2.3010

f(sol(1),sol(2))

ans = 1.0571

[Kaleina]

Thanks so much.

Answer 2

[Stephen23]

fh1 = @(x,y) 4*(x-1).^2 + 3*(y-2).^2 + 2*(x-2).^2.*(y-3).^2;
fh2 = @(v) fh1(v(1),v(2)); % function with one input
sol = fminsearch(fh2,[1,1])

sol = 1×2

1.1963 2.3010

val = fh2(sol)

val = 1.0571

An important part of any calculation is checking the result. Lets do that now:

fsurf(fh1)
hold on
plot3(sol(1),sol(2),val,'*r')
view(54,31)

[Kaleina]

Thank you for the insight.

Answer 3

[Sam Chak]

Hi @Kaleina

For a static function, especially a function of two variables, this is how I systematically find the minimum point.

% STEP 1: Find the approximate location of the minimum point via contour plot
[X, Y]   = meshgrid(-1:0.1:5);          % adjust these until you see the basin
Z        = staticfun(X, Y);             % scroll to the bottom of the script
figure(1)
contour(X, Y, Z, 30), xline(1.5, '--'), yline(2.5, '--')
xlabel('x'), ylabel('y'), colorbar

% STEP 2: Find minimum of unconstrained function f(x, y)
funhd    = @(v) staticfun(v(1), v(2));  % create a function handle to pass it to fminsearch
v0       = [1.5, 2.5];                  % initial guess values based on the contour plot
solution = fminsearch(funhd, v0)        % applying fminsearch

solution = 1×2

1.1963 2.3010

% STEP 3a: Plot the surface and the minimum point
figure(2)
surf(X, Y, Z), hold on
minValue = funhd(solution)  % insert xsol & ysol into static funtion to find the min value

minValue = 1.0571

plot3(solution(1), solution(2), minValue, 'r.', 'MarkerSize', 25)   % plot the red dot

% STEP 3b: rotate for better viewing
[az, el] = view;
az       = az + 180;
view(az, el)
xlabel('x'), ylabel('y'), zlabel('f(x,y)')

% Part of STEP 1: to create a function for the surface f(x, y).
function fun = staticfun(x, y)
    fun = 4*(x - 1).^2 + 3*(y - 2).^2 + 2*(x - 2).^2.*(y - 3).^2;
end

[Kaleina]

Thats very detailed thank you.

Answer 4

[Alexander]

The solutions above are very good and should be used to avoid loops. But anyway I post my little script here, because I think it is good for the understanding how a multible dimension function is built up.

X = 0:0.1:4; % Taylor it to your needs (precision, boundary)
Y = X; % Y = -1:0.1:3;
for(ix = 1:length(X));
    for(iy = 1:length(Y))
        Z(ix,iy) = 4*(X(ix) - 1).^2 + 3*(Y(iy) - 2).^2 + 2*(X(ix) - 2).^2.*(Y(iy) - 3).^2;
    end
end
figure(1); contour(X,Y,Z);xlabel('X');ylabel('Y');
figure(2); mesh(X,Y,Z);xlabel('X');ylabel('Y');ylabel('Z');
[C, ind] = min(Z); [zMin, xInd] = min(C);
[C, ind] = min(Z'); [zMin, yInd] = min(C);
fprintf('Minimum of Z: %3.4f at pos. [X,Y] = [%i, %i], value = [%3.4f, %3.4f]\n',zMin,xInd,yInd,X(xInd),Y(yInd))

[Stephen23]

Also without nested FOR loops:

[X,Y] = meshgrid(-1:0.1:5);
Z = X;
Z(:) = 4*(X(:) - 1).^2 + 3*(Y(:) - 2).^2 + 2*(X(:) - 2).^2.*(Y(:) - 3).^2;
contour(X,Y,Z, 30)

[Alexander]

Great!