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, 1.0, 1.0, 1.0, 1.0, 15.163854490967951, 1.0, 1.0], ┌─ Bifurcation Problem with uType Vector{Float64}
├─ Inplace:  false
├─ Symmetric: false
└─ Parameter: p, [2.350046036011986, 1.66310933847056e16, 4.927731373246102e15, 1.4600685550358828e15, 4.3261290519581706e14, 1.2818160153950108e14, 3.797973378948153e13, 1.1253254456142416e13, 3.334297616634531e12, 9.879400345581201e11  …  1.9806792134957582e8, 5.868679125172619e7, 1.7388678630141098e7, 5.1522008163381135e6, 1.5265777603965215e6, 452319.0771546703, 134020.20804631952, 39709.4320153824, 11765.498380524208, 3485.814353856415], false, 25, 25)

Example

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