Krylov-Newton algorithm

• Krylov-Newton algorithm
• • Space of solutions
  • Different Jacobians
  • Simple example
  • Example

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 solve.

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 = ODEBifProblem(F, x0, nothing)
sol = BifurcationKit.solve(prob, Newton(), NewtonPar(verbose = true))

NonLinearSolution{Vector{Float64}, ODEBifProblem{BifFunction{typeof(Main.F), BifurcationKit.var"#122#124"{typeof(Main.F)}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Float64, Nothing}, Vector{Float64}, Nothing, typeof(identity), typeof(BifurcationKit.plot_default), typeof(BifurcationKit.record_sol_default), typeof(BifurcationKit.save_solution_default), typeof(BifurcationKit.update_default)}, Vector{Float64}, Int64}([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], ┌─ Bifurcation problem with uType Vector{Float64}
├─ Inplace: false
├─ Dimension: 10
├─ Symmetric: false
└─ Parameter: p, [2.1258225863459166, 43304.57117258433, 12830.621697048233, 3801.3033958597316, 1125.9499962888833, 333.25316120371616, 98.38124303848589, 28.792885477997615, 8.184716788447812, 2.1117397568303806, 0.39935914823034313, 0.03083169307185825, 0.0002725300441754056, 2.4277330624439216e-8, 0.0], true, 14, 14)

Example

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