function x = full_calibration_min(M)

m11 = M(1,1); m13 = M(1,3); m14 = M(1,4); m15 = M(1,5);
m22 = M(2,2); m25 = M(2,5);
m34 = M(3,4); m35 = M(3,5);
m44 = M(4,4); 
m55 = M(5,5);

a = m11*m22^2 - m22*m13^2;

b = 2*m11*m22^2*m44 - m22^2*m14^2 - 2*m22*m13^2*m44 ...
     - 2*m11*m22*m34^2 - 2*m11*m22*m35^2 - m22^2*m15^2 ...
     + 2*m13*m22*m34*m14 + m13^2*m34^2 + 2*m13*m22*m35*m15 + m13^2*m35^2;

c = - 2*m13*m35^3*m15 - m22*m13^2*m44^2 + m11*m22^2*m44^2 ...
     + m13^2*m35^2*m44 + 2*m13*m22*m34*m14*m44 + m13^2*m34^2*m44 ...
     - 2*m11*m22*m34^2*m44 - 2*m13*m34^3*m14 - 2*m11*m22*m35^2*m44 ...
     + 2*m11*m35^2*m34^2 + m22*m14^2*m35^2 - 2*m13*m35^2*m34*m14 ...
     - 2*m13*m34^2*m35*m15 + m11*m34^4 + m22*m15^2*m34^2 ...
     + m22*m35^2*m15^2 + m11*m35^4 - m22^2*m14^2*m44  ...
     + 2*m13*m22*m35*m15*m44 + m22*m34^2*m14^2 - m22^2*m15^2*m44;

r = roots([a b c]);

W = [zeros(3,3) zeros(3,2); zeros(2,3) eye(2)];

if isreal(r(1)) | isreal(r(2))
	
	x1 =  x_given_lambda(M, r(1), W);
	e1 = x1' * M * x1;
	x2 =  x_given_lambda(M, r(2), W);
	e2 = x2' * M * x2;
	
	if e1 < e2
		x = x1;
	else
		x = x2;
	end
else
 	sprintf('bad thing: imaginary solution\n');
end


if false
	f1 = @(l) det(M  +  l * W);
	f2 = @(l) polyval([c3 c2 c1], l);

	figure; hold on;
	fplot(f1, [ - 10 10], 'k - ');
	fplot(f2, [ - 10 10], 'r - ');
end

function x = x_given_lambda(M, lambda, W)
	Z = (M  +  lambda * W);
	[a, b] = eig(Z * Z');
	v0 = a(:,1);
	norm(v0(4:5))
	x = v0 * (sign(v0(1)) / norm(v0(4:5)) );