Feature: user defined damping in Newton type methods

[jarlebring]

This PR adds the possibility to do damping in the newton-type methods. This is useful when you want further control of the convergence, e.g., if the eigenvalue jumps around too much in the beginning of the iteration, one can freeze it for a couple of iterations. This happens eg if your eigval approximation is much better than your eigvec approximation.

julia> dampfun(j,λ,v,Δλ,Δv,err) = (j<5 ? 0 : Δλ, Δv); # Damp eigval update to zero for the first five its
julia> nep=nep_gallery("dep0");
julia> quasinewton(nep,v=ones(size(nep,1)), λ=-0.1595, damping=dampfun,logger=1)
Precomputing linsolver
iter 1 err:0.40385648969108706 λ=-0.1595 + 0.0im
iter 2 err:4.9268122755813825e-5 λ=-0.1595 + 0.0im
iter 3 err:1.693098602693684e-5 λ=-0.1595 + 0.0im
iter 4 err:1.6930148875939743e-5 λ=-0.1595 + 0.0im
iter 5 err:1.6930148908063535e-5 λ=-0.1595 + 0.0im
iter 6 err:3.534207222540244e-10 λ=-0.15955391952211395 + 0.0im
iter 7 err:1.3189598399245775e-10 λ=-0.15955391865305488 + 0.0im
iter 8 err:6.342596145681362e-15 λ=-0.15955391823297246 + 0.0im
(-0.15955391823297246 + 0.0im, Complex{Float64}[6874.829341069828 + 0.0im, 46596.517747917926 + 0.0im, -5998.0275339435675 + 0.0im, 1523.6237897132014 + 0.0im, 5978.311709367396 + 0.0im])

Compare without damping:

julia> quasinewton(nep,v=ones(size(nep,1)), λ=-0.1595, logger=1)
Precomputing linsolver
iter 1 err:0.40385648969108706 λ=-0.1595 + 0.0im
iter 2 err:0.4016720362965377 λ=-3.3722880129157558 + 0.0im
iter 3 err:0.34209754911455964 λ=-2.353155480506111 + 0.0im
iter 4 err:0.24201665248389861 λ=-1.3736098413798286 + 0.0im
iter 5 err:0.11032443885756192 λ=-0.5854977681397857 + 0.0im
iter 6 err:0.016780803794660546 λ=-0.2144694794858884 + 0.0im
iter 7 err:0.00027978305577110147 λ=-0.1604064664410386 + 0.0im
iter 8 err:2.117065649874461e-5 λ=-0.1595002456505578 + 0.0im
iter 9 err:4.241786204533617e-5 λ=-0.15950000000003325 + 0.0im
iter 10 err:2.2749491979568696e-5 λ=-0.15962637415973216 + 0.0im
iter 11 err:5.422980704846321e-10 λ=-0.15955391728395485 + 0.0im
iter 12 err:2.3829635069539644e-10 λ=-0.1595539189919185 + 0.0im
iter 13 err:1.1465551278495235e-14 λ=-0.15955391823295587 + 0.0im
(-0.15955391823295587 + 0.0im, Complex{Float64}[-1.2308999948560478e7 + 0.0im, -8.34284759820621e7 + 0.0im, 1.0739135030701244e7 + 0.0im, -2.7279637382659465e6 + 0.0im, -1.070383494226937e7 + 0.0im])

Not yet ready to be merged. At the moment only implemented for quasinewton.