arb_calc.h

Initialise v (both endpoints zero), and free its memory.

Allocate and initialise a vector of n intervals, and clear and free it.

Set v to a copy of u, and swap v and u efficiently.

Sets x to a ball that contains the interval [a, b] (using arb_set_interval_arf); the ball is exact if the interval is a point and is otherwise rounded outward to prec bits.

Print v as [a, b] with the endpoints printed to n decimal digits, to standard output or to file.

Isolates the simple roots of a real analytic function f in the interior of interval. Allocates and writes to *found an array of n subintervals of interval and to *flags an array of n flags, and returns n. The caller must free them with _arf_interval_vec_clear and flint_free. The output satisfies:

• f has no root in interval outside the output subintervals;
• the subintervals are sorted in increasing order and do not overlap, except that consecutive ones may share an endpoint;
• a subinterval with flag 1 contains exactly one root, which is simple;
• a subinterval with another flag may or may not contain roots.

If all flags are 1, all roots in the interior of interval have been isolated. Roots of multiplicity greater than one and roots at the endpoints of the subdivision cannot be isolated; in their place, a subinterval of minimal size with flag other than 1 is returned. The other subintervals can be searched again with a higher precision or larger limits.

Splits block = [a, b] at its exact midpoint m into L = [a, m] and R = [m, b] and returns the sign of f(m): \pm 1 if it is determined, and 0 if f(m) contains zero.

Given an interval start with f of opposite nonzero signs at its endpoints, performs iter bisection steps, keeping the half with a sign change, and sets r to the result, which contains a root (the only one if start contains only one). Returns ARB_CALC_SUCCESS; ARB_CALC_IMPRECISE_INPUT (with r unchanged) if the signs at the endpoints of start are not determined or equal; or ARB_CALC_NO_CONVERGENCE if the sign at a bisection point cannot be determined, in which case r is the last valid interval.

Sets conv_factor to an upper bound for C = \sup_{t, u \in I} \tfrac12 |f''(t)| / |f'(u)| on the ball I = conv_region, obtained by evaluating the Taylor coefficients f'(I) and f''(I)/2 with one call to func and bounding the absolute value of their quotient. If f is ill-conditioned, I may need to be extremely narrow for the bound to be finite.

Performs one step of interval Newton iteration for a ball x = [m \pm r] that contains exactly one root of f, given a ball I \supseteq x (conv_region) and the bound C for \tfrac12 |f''| / |f'| on I (conv_factor). Computes m' = m - f(m)/f'(m) in ball arithmetic at prec bits and adds C r^2 to its radius. If the new ball x' lies in I and has a smaller radius than x, sets xnew to x' and returns ARB_CALC_SUCCESS; otherwise sets xnew to x and returns ARB_CALC_NO_CONVERGENCE.

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).