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

NonLinearSolution{Vector{Float64}, BifurcationProblem{BifFunction{typeof(Main.F), BifurcationKit.var"#115#117"{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, [2.3984454057219446, 3.7181357083321125e6, 1.1016695802366924e6, 326420.35709843866, 96716.88357845171, 28656.595132570434, 8490.583749602316, 2515.4692852610356, 745.065061723623, 220.50105695480048, 65.07539125656805, 19.025709794891615, 5.389177288601556, 1.373222340575611, 0.2478347354342172, 0.01604555305157951, 8.4316433557996e-5, 2.3695319040228924e-9, 0.0], true, 18, 18)

Example

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