Abstract

CHAPTER 2. TEMPORAL INTEGRATION 32
2.2 Newton-Krylov Methods
To solve the nonlinear equations that come from the implicit integrator and in finding
the equilibrium Wigner distribution, f0, on different grids, we use Newton’s method.
Given a nonlinear equation F : R M → R M and an initial iterate z0, Newton’s method
is an iterative method that finds a root of F , that is find z ∗ ∈ R M such that F (z ∗ )=0,
by the iteration:
zm+1 = zm − F ′ (zm) −1 F (zm) =zm + sm
where F ′ (zm) is the Jacobian of F at zm. The Newton step, sm, is then the solution
of the linear equation:
F ′ (zm)sm = −F (zm)
Before we cite the convergence theorems of Newton’s method, we first want to
define two types of convergence, q-linear convergence and q-quadractic convergence.
A sequence {zm} ∈R M converges to z ∗ q-linearly if the sequence converges to z ∗ ,
and there exists a constant C>0 such that zm+1 − z ∗ ≤Czm − z ∗ . A sequence
{zm} ∈R M converges to z ∗ q-quadratically if the sequence converges to z ∗ ,andthere
exists a constant C>0 such that zm+1 − z ∗ ≤Czm − z ∗ 2 .
The standard assumptions [19] for this problem are:
• There exists a solution z ∗ ∈ R M .
• The Jacobian F ′ (z) is Lipschitz continuous