How to solve f(x)=0 numerically? A globally convergent modification of the Newton-Raphson method

Statistical Physics and Complexity Group meeting

How to solve f(x)=0 numerically? A globally convergent modification of the Newton-Raphson method

Event details

The usual way of solving numerically a nonlinear system of equations f(x)=0 is the Newton-Raphson method (NRM). Unfortunately it only converges to a solution if the initial guess is very close to the actual solution. Typically, the larger dimensionality of x is, the smaller is the radius of convergence of the NRM. Coming up with an accurate initial guess is thus crucial to finding a solution. Failure to do so will typically result in a blow-up of the iteration scheme. It appears that there exist several versions of the globally convergent modification of the NRM. These methods are very robust (no blow-ups) and converge from an arbitrary initial condition! I will discuss one realisation of these methods -- the so-called hook-step method. I will also show how to apply this method to very large systems of algebraic equations dim(x)~O(100).
References
1. J.E.Dennis, Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996
2. L.N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia,1997
3. D. Viswanath, The critical layer in pipe flow at high Reynolds number, Phil. Trans. Royal Soc. A, 367 561 (2009)