newton_refine.hpp File Reference

Standalone Newton refinement of a point against a System – no tracker required. More...

#include "bertini2/system/system.hpp"
#include "bertini2/common/config.hpp"
#include "bertini2/linalg/lu_solver.hpp"

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Classes
struct	bertini::algorithm::NewtonRefineResult< ComplexT >
	The outcome of a standalone Newton refinement. More...

Namespaces
namespace	bertini
namespace	bertini::algorithm

Functions
template<typename ComplexT>
NewtonRefineResult< ComplexT >	bertini::algorithm::NewtonRefine (System const &S, Vec< ComplexT > const &start, NumErrorT tolerance, unsigned max_iterations, ComplexT time=ComplexT(0))
	Newton-refine a point against a square System, without a tracker.

Detailed Description

Standalone Newton refinement of a point against a System – no tracker required.

The sharpening primitive: given a square system and an approximate solution, iterate Newton's method until consecutive approximations agree to a requested tolerance. The system may be autonomous (no path variable – the ordinary case for deflated critical point systems) or carry a path variable with a caller-supplied time.

The whole point of the standalone (versus the tracker-owned Refine) is refinement on DEFLATED systems at singular points: deflation restores quadratic convergence exactly where the tracker's own system cannot converge, and a deflated system is a plain autonomous System that no tracker is configured around. Overdetermined systems are refused with an instructive message – square them by randomization first, as the tracking layer does.

Evaluation runs at the precision of the supplied system and point; lift both before calling to sharpen beyond their current precision.