Krylov-Newton algorithm

BifurcationKit is built upon the newton algorithm for solving (large-dimensional) nonlinear equations

\[F(x)=0\in\mathbb R^n,\quad x\in\mathbb R^n.\]

Writing $J(x)\in\mathcal L(\mathbb R^n)$ the jacobian, the algorithm reads

\[x_{n+1} = x_n - J(x_n)^{-1}F(x_n)\]

with initial guess $x_0$.

The crux of the algorithm is to solve the linear system in $y$:

\[J(x_n)\cdot y = F(x_n).\]

To this end, we never form $J^{-1}$ like with pinv(J) but solve the linear system directly.

Space of solutions

For the algorithm to be defined, a certain number of operations on x need to be available. If you pass x::AbstractArray, you should not have any problem. Otherwise, your x must comply with the requirements listed in Requested methods for Custom State.

Different Jacobians

There are basically two ways to specify the jacobian:

1. Matrix based
2. Matrix-free.

In case you pass a matrix (in effect an AbstractMatrix like a sparse one,...), you can use the default linear solver from LinearAlgebra termed the backslash operator \. This is a direct method. This is the case 1 above.

Another possibility is to pass a function J(dx) and to use iterative linear solvers. In this case, this is termed a Krylov-Newton method. This is the case 2 above. In comparison to the Matrix-based case, there is no restriction to the number of unknowns $n$.

The available linear solvers are explained in the section Linear solvers (LS).

One can find a full description of the Krylov-Newton method in the API.

Simple example

Here is a quick example to show how the basics work. In particular, the problem generates a matrix based jacobian using automatic differentiation.

using BifurcationKit
F(x, p) = x.^3 .- 1
x0 = rand(10)
prob = BifurcationProblem(F, x0, nothing)
sol = newton(prob, NewtonPar(verbose = true))

NonLinearSolution{Vector{Float64}, BifurcationProblem{BifFunction{typeof(Main.F), BifurcationKit.var"#8#24", Nothing, BifurcationKit.var"#6#22", Nothing, BifurcationKit.var"#11#28"{BifurcationKit.var"#d1Fad#26"}, BifurcationKit.var"#13#30", BifurcationKit.var"#15#32", BifurcationKit.var"#17#34", Bool, Float64}, Vector{Float64}, Nothing, typeof(identity), typeof(BifurcationKit.plot_default), typeof(BifurcationKit.record_sol_default), typeof(BifurcationKit.save_solution_default)}, Vector{Float64}, Int64}([1.0, 1.0, 1.0, 2.2997249592167996, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], ┌─ Bifurcation Problem with uType Vector{Float64}
├─ Inplace:  false
├─ Symmetric: false
└─ Parameter: p, [2.4859929889241528, 5.4980215237763984e13, 1.6290434144522408e13, 4.826795302080457e12, 1.4301615709865425e12, 4.23751576588346e11, 1.2555602269258398e11, 3.720178450124712e10, 1.1022750963073227e10, 3.266000285095771e9  …  654786.2241290528, 194010.47381635505, 57484.32557636174, 17032.133507972918, 5046.2988302237545, 1494.9404381697766, 442.6862043207388, 130.90752454506045, 38.529842231297394, 11.162635624717087], false, 25, 25)

Example

The (basic) tutorial Temperature model presents all cases (direct and iterative ones).