It is well known that a damped or underrelaxed Newton's method will sometimes solve a system of nonlinear equations when the full Newton's method cannot. This happens, for example, when only a poor ...

There are two correct ways to solve it. The first way to find the solution is to add the equation, then combine the sum with that of the previous equation.

C. C. Paige, M. A. Saunders, Solution of Sparse Indefinite Systems of Linear Equations, SIAM Journal on Numerical Analysis, Vol. 12, No. 4 (Sep., 1975), pp. 617-629 ...