Newton's method

Newton's method

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

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.

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.
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.

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

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

NewtonCache(F::AbstractSystem, x)

Cache for the newton function.