optimization of quadratic form of [SO_2]^n

so2-opt.m

%% f2d.m
function f2d(A)
  x=0:0.05:3*pi;
  [X, Y]=meshgrid(x);
  F=zeros(size(X));

  min_val = inf;
  min_coord = [0 0];
  max_val = -inf;
  max_coord = [0 0];

  for i=1:size(X,1)
    for j=1:size(X,2)
      x=X(i,j);
      y=Y(i,j);
      v=[sin(x); cos(x); sin(y); cos(y)];
      F(i,j)=v'*A*v;

      if F(i,j) >= max_val
        max_val = F(i,j);
        max_coord = [x y];
      end

      if F(i,j) <= min_val
        min_val = F(i,j);
        min_coord = [x y];
      end
    end
  end
  surf(X,Y,F);

  min_val
  min_coord
  min_dir=[sin(min_coord(1)) cos(min_coord(1)) sin(min_coord(2)) cos(min_coord(2))]
  max_val
  max_coord
  max_dir=[sin(max_coord(1)) cos(max_coord(1)) sin(max_coord(2)) cos(max_coord(2))]
end

%% test.m
E = [
  1 0 0 0
  0 2 0 0
  0 0 2 0
  0 0 0 3
];

E = [
   2.92237  -0.13644  -2.15354  -3.46603
  -4.21442   3.55869  -3.36019   4.14869
  -1.45951  -2.99353   2.42377  -4.66483
  -1.69126   3.03281   1.89085  -3.13264
];

a=pi/30;

R = [
   cos(a) 0 -sin(a) 0
       0  1     0  0
   sin(a) 0 cos(a) 0
       0  0     0  1
];

H = R'*E*R;% rotate a small angle
f2d(H)