Create a new ArfInterval structure.

Use ArfInterval structure.

Use new ArfInterval structure.

arf_interval_init v

arf_interval_clear v

_arf_interval_vec_init n

_arf_interval_vec_clear v n

arf_interval_set v u

arf_interval_swap v u

arf_interval_get_arb x v prec

arf_interval_printd v n

Helper functions for endpoint-based intervals.

arf_interval_fprintd file v n

Helper functions for endpoint-based intervals.

arb_calc_isolate_roots found flags func param interval maxdepth maxeval maxfound prec

Rigorously isolates single roots of a real analytic function on the interior of an interval.

This routine writes an array of n interesting subintervals of interval to found and corresponding flags to flags, returning the integer n. The output has the following properties:

• The function has no roots on interval outside of the output subintervals.
• Subintervals are sorted in increasing order (with no overlap except possibly starting and ending with the same point).
• Subintervals with a flag of 1 contain exactly one (single) root.
• Subintervals with any other flag may or may not contain roots.

If no flags other than 1 occur, all roots of the function on interval have been isolated. If there are output subintervals on which the existence or nonexistence of roots could not be determined, the user may attempt further searches on those subintervals (possibly with increased precision and/or increased bounds for the breaking criteria). Note that roots of multiplicity higher than one and roots located exactly at endpoints cannot be isolated by the algorithm.

The following breaking criteria are implemented:

• At most maxdepth recursive subdivisions are attempted. The smallest details that can be distinguished are therefore about \(2^{-\text{maxdepth}}\) times the width of interval. A typical, reasonable value might be between 20 and 50.
• If the total number of tested subintervals exceeds maxeval, the algorithm is terminated and any untested subintervals are added to the output. The total number of calls to func is thereby restricted to a small multiple of maxeval (the actual count can be slightly higher depending on implementation details). A typical, reasonable value might be between 100 and 100000.
• The algorithm terminates if maxfound roots have been isolated. In particular, setting maxfound to 1 can be used to locate just one root of the function even if there are numerous roots. To try to find all roots, LONG_MAX may be passed.

The argument prec denotes the precision used to evaluate the function. It is possibly also used for some other arithmetic operations performed internally by the algorithm. Note that it probably does not make sense for maxdepth to exceed prec.

Warning: it is assumed that subdivision points of interval can be represented exactly as floating-point numbers in memory. Do not pass \(1 \pm 2^{-10^{100}}\) as input.

arb_calc_refine_root_bisect r func param start iter prec

Given an interval start known to contain a single root of func, refines it using iter bisection steps. The algorithm can return a failure code if the sign of the function at an evaluation point is ambiguous. The output r is set to a valid isolating interval (possibly just start) even if the algorithm fails.

arb_calc_newton_conv_factor conv_factor func param conv_region prec

Given an interval \(I\) specified by conv_region, evaluates a bound for \(C = \sup_{t,u \in I} \frac{1}{2} |f''(t)| / |f'(u)|\), where \(f\) is the function specified by func and param. The bound is obtained by evaluating \(f'(I)\) and \(f''(I)\) directly. If \(f\) is ill-conditioned, \(I\) may need to be extremely precise in order to get an effective, finite bound for C.

arb_calc_newton_step xnew func param x conv_region conv_factor prec

Performs a single step with an interval version of Newton's method. The input consists of the function \(f\) specified by func and param, a ball \(x = [m-r, m+r]\) known to contain a single root of \(f\), a ball \(I\) (conv_region) containing \(x\) with an associated bound (conv_factor) for \(C = \sup_{t,u \in I} \frac{1}{2} |f''(t)| / |f'(u)|\), and a working precision prec.

The Newton update consists of setting \(x' = [m'-r', m'+r']\) where \(m' = m - f(m) / f'(m)\) and \(r' = C r^2\). The expression \(m - f(m) / f'(m)\) is evaluated using ball arithmetic at a working precision of prec bits, and the rounding error during this evaluation is accounted for in the output. We now check that \(x' \in I\) and \(r' < r\). If both conditions are satisfied, we set xnew to \(x'\) and return ARB_CALC_SUCCESS. If either condition fails, we set xnew to \(x\) and return ARB_CALC_NO_CONVERGENCE, indicating that no progress is made.

arb_calc_refine_root_newton r func param start conv_region conv_factor eval_extra_prec prec

Refines a precise estimate of a single root of a function to high precision by performing several Newton steps, using nearly optimally chosen doubling precision steps.

The inputs are defined as for arb_calc_newton_step, except for the precision parameters: prec is the target accuracy and eval_extra_prec is the estimated number of guard bits that need to be added to evaluate the function accurately close to the root (for example, if the function is a polynomial with large coefficients of alternating signs and Horner's rule is used to evaluate it, the extra precision should typically be approximately the bit size of the coefficients).

This function returns ARB_CALC_SUCCESS if all attempted Newton steps are successful (note that this does not guarantee that the computed root is accurate to prec bits, which has to be verified by the user), only that it is more accurate than the starting ball.

On failure, ARB_CALC_IMPRECISE_INPUT or ARB_CALC_NO_CONVERGENCE may be returned. In this case, r is set to a ball for the root which is valid but likely does have full accuracy (it can possibly just be equal to the starting ball).