# Lack of quadratic convergence in Newton's method

It is well-known that Newton's method can converge quadratically, if initial guess is close enough and if the arising linear systems are solved accurately.

I am applying Newton's method to highly ill-conditioned systems (with condition number around $$10^{14}$$). I am sure that my Hessian is computed correctly, however I never obtain quadratic convergence.

I assume that the problem comes from the inability to solve linear systems with sufficient accuracy (I am using LU decomposition with iterative refinement).

Does anybody know about any reference, which discusses how speed of convergence of Newton's method depends on accuracy of the obtained correction?

• Can you write down the system that you are optimizing? – nicoguaro Jan 28 '19 at 16:54
• I am minimizing total potential energy of nonlinear/non-convex orthotropic-elasticity model of shell discretized by subdivision finite elements – computational_scientist Jan 28 '19 at 18:55
• I would say that rather than an insufficient precision of the LSE solution your problem are the possible other local optima of your problem. Especially if you are using LU with iterative refinement. – Nox Jan 28 '19 at 19:22
• BTW, how did you compute your condition number? Sometimes you might get a number spat out of a condition number estimation that just means "reallll big!" 10^14 sounds like it's one of those numbers. – user14717 Jan 28 '19 at 21:26
• The asymptotic quadratic convergence of Newton's method also requires that the Jacobian be non-singular at the root. Since your matrix is very badly conditioned, this condition isn't being met. – Brian Borchers Jan 28 '19 at 23:25