What Jacobians does bvp4c evaluate?

I am trying to debug my "singular Jacobian" issues with bvp4c. The question that I have is: which Jacobians are being evaluated? We have the nonlinear system of ODE's y' = f(x,y). I assume that the derivatives of f_i with respect to y_j are put into a matrix and evaluated at some values of y. Maybe this matrix should be invertible for the bvp4c algorithm, although I have found the algorithm to converge when this is not the case.

What about the residuals for the boundary conditions g(ya,yb) = 0? I assume the derivative of g_i with respect to ya_j are taken. This gives a matrix. Also, the derivative of g_i with respect to yb_j are taken. This also gives a matrix. Does bvp4c also evaluate these Jacobians? If so, then they would always be singular if any of the g_i's don't contain a ya or a yb.

Can someone point me to some theory that will illuminate the singular Jacobian error? Thank you.