Newton's method

Sometimes it is necessary to refine obtained solutions. For this we provide an interface to Newton's method.

HomotopyContinuation.newton — Function.

newton(F::AbstractSystem, x₀, norm=euclidean_norm, cache=NewtonCache(F, x₀); tol=1e-6, miniters=1, maxiters=3, simplified_last_step=true)

An ordinary Newton's method. If simplified_last_step is true, then for the last iteration the previously Jacobian will be used. This uses an LU-factorization for square systems and a QR-factorization for overdetermined.

HomotopyContinuation.NewtonResult — Type.

NewtonResult{T}

Structure holding information about the outcome of the newton function. The fields are.

return_code::NewtonReturnCode.codes: The return code of computation. NewtonReturnCode.converged means that accuracy ≤ tol.
accuracy::T: |xᵢ-xᵢ₋₁| for i = iters and x₀,x₁,…,xᵢ₋₁,xᵢ are the Newton iterates.
iters::Int: The number of iterations used.
digits_lost::Float64 Estimate of the (relative) lost digits in the linear algebra.

HomotopyContinuation.NewtonReturnCode.codes — Type.

NewtonReturnCode.codes

The possible return codes of Newton's method

NewtonReturnCode.converged
NewtonReturnCode.terminated
NewtonReturnCode.terminated_no_approximate
NewtonReturnCode.maximal_iterations

source

For high performance applications we also provide an in-place version of Newton's method which avoids any temporary allocations.

HomotopyContinuation.newton! — Function.

newton!(out, F::AbstractSystem, x₀, norm, cache::NewtonCache; tol=1e-6, miniters=1, maxiters=3, simplified_last_step=true)

In-place version of newton. Needs a NewtonCache and norm as input.

HomotopyContinuation.NewtonCache — Type.

NewtonCache(F::AbstractSystem, x)

Cache for the newton function.