A degree-increasing [N to N+1] homotopy for Chebyshev and Fourier spectral methods

Unfortunately, u→N is all too often outside the domain of convergence of Newton's iteration when N is increased to (N+1). We show that a good option is the artificial parameter homotopy H→(u→;τ)≡R→N+1(u→)−(1−τ)R→N+1(u→N), τ∈[0,1]. Marching in small steps in τ, we proceed smoothly from the N-term to the N+1-term approximations.