arb_calc_refine_root_newton

Refines the enclosure start of a simple root to a relative accuracy of about prec bits by Newton steps at working precisions that roughly double from the accuracy of start up to prec (plus a padding of about \log_2 C bits). eval_extra_prec is the number of guard bits needed to evaluate f accurately near the root (for a polynomial with large coefficients of alternating signs evaluated by Horner's rule, about the bit size of the coefficients). Returns ARB_CALC_SUCCESS if every step succeeded; this does not guarantee prec accurate bits, only an improved enclosure. On failure returns ARB_CALC_IMPRECISE_INPUT or ARB_CALC_NO_CONVERGENCE, and r is a valid but possibly less accurate enclosure (possibly start).