The Sparse Newton's method uses at each step a sparse linear iterative solver to find the next approximation of the solutions. The user can choose from five solvers: BiConjugate Gradient (see http://mathworld.wolfram.com/BiconjugateGradientMethod.html), BiConjugate Gradient Stabilized (see http://mathworld.wolfram.com/BiconjugateGradientStabilizedMethod.html), Iterative Refinement (preconditioned Richardson method)(see http://math.nist.gov/iml++/ir.h.txt or ), Conjugate Gradients Squared
(see http://mathworld.wolfram.com/ConjugateGradientSquaredMethod.html), Quasi Minimal Residual
According to the performance testing the Iterative Refinement(preconditioned Richardson method) is the fastest among them in most of the occurring cases. However if the system of equations hast the determinant of the Jacobian matrix close or equal to zero at the point were it has the LFP solution, this method is very slow. In such a case user should choose Quasi Minimal Residual method that doesn't have this problem and is just a little bit slower on the average.