I'm currently working with Galerkin's method to solve differential equations and I have to retrieve unknown coefficients for the truncate expansion. This is just to set the background for why I need to solve a system of possible non-linear equations.
Several of my problems turn out to have a singular Jacobian which prevents me from using Newton's method. The following figure is a "spy" from my Jacobian:
In the figure, a blue dot means a non-zero entry in the Jacobian and a red circle indicates a "missing" (zeroed) entry that could be helpful if I want to turn my Jacobian into a non-singular one. I'm outrageous cause the Jacobian J was built as a 64x64 matrix and just 3 elements "made" it singular. (this is a gag only, I know those missing elements have physical meaning :-D )
The Newton's method I am referring to is the multivariate version:
$$X^{(k+1)} = X^{(k)} - J^{-1} \cdot [f_1(X^{(k)}),\,f_2(X^{(k)}),\ldots f_N(X^{(k)})]^T$$
So, my questions are:
(1) What is the meaning for this sparse Jacobian to be singular?
(2) Between 64 variables -- i.e. 4,096 derivatives from which 652 are non-zeros --, 3 strategically placed derivatives are evaluated as zeros. This is enough to throw away all valid information?
(3) Is there any workaround to apply Newton's method with this Jacobian? For example, could I use parts of the Jacobian to update at least some of the coordinates?