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, Setfield.IdentityLens, typeof(BifurcationKit.plot_default), typeof(BifurcationKit.record_sol_default)}, Vector{Float64}, Int64}([1.0, 1.0, 1.0, 1.0000000000241298, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], ┌─ Bifurcation Problem with uType Vector{Float64}
├─ Inplace:  false
├─ Symmetric: false
└─ Parameter: p, [2.364428928249313, 4.7772870036750244e10, 1.4154924455074142e10, 4.1940516901330786e9, 1.2426819820023928e9, 3.682020684821897e8, 1.0909690892064789e8, 3.2325009791302186e7, 9.577780419644209e6, 2.837860605819689e6  …  568.5824574777324, 168.2100072939458, 49.58205748789596, 14.43612851434234, 4.032663831950709, 0.9812223952474366, 0.15307310648766137, 0.006673689932859883, 1.4736761521749742e-5, 7.238942778542423e-11], false, 25, 25)

Example

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