Accompanying files of the paper On a Non-Archimedean Broyden Method
by X. Dahan and T. Vaccon. In proceedings of ISSAC 2020, pp:114-121, ACM Press

MAGMA code

• broyden.mgm Implementation of Broyden's method with power series coefficients.
  • BroydenSeries. Example of calling procedure.
    Input:
    • System \(FF\) with power series (univariate, say in varaiable \(t\)) coefficients.
      (embedded in the ring of Laurent series. The coefficients of the Laurent series are in a field denoted Re, either the rationals \(\mathbb{Q}\) or a finite field \( \mathbb{F}\))
    • Expansion value \(t_0\in\) Re.
    • Solution \( z_0 \in\) Re\({}^m\) at order 1: \(FF|_{t=t_0}(z_0) =0\). We assume moreover that the jacobian \({\rm Jac}_{FF}(x_0)|_{t=t_0}\) is invertible.
    • Number of iterations
    
    Settings:
    • The three systems of Section 7 are pre-written (follow instructions given in the preamble of the file).
    • Any system verifying the above assumptions can be input (follow instructions given in the preamble of the file)
  
  
  • "out, val, tim, gap:=BroydenSeries( FF, z0, t0, nb of iterations)" returns:
    1. out: power series approximation of the solution to system FF.
    2. val: list of the precisions of the approximation solutions after each iteration
    3. tim: list of times taken by each iteration
    4. gap: list of gaps between the true valuation and the approximated valuation used for the computations (see Eq. (25) to Eq. (26) of Section 6.2 for details).
       Remark: the \( \alpha\) is tuned after each iteration among the last five computed ratios \(\frac{v_{n}}{v_{n-1}},\ldots,\frac{v_{n-4}}{v_{n-5}}\)
  
  
• Newton.mgm For sake of comparisions same kind of implementation as Broyden's.