.. _arb-calc:
**arb_calc.h** -- calculus with real-valued functions
===============================================================================
This module provides functions for operations of calculus
over the real numbers (intended to include root-finding,
optimization, integration, and so on). It is planned that the module
will include two types of algorithms:
* Interval algorithms that give provably correct results. An example
would be numerical integration on an interval by dividing the
interval into small balls and evaluating the function
on each ball, giving rigorous upper and lower bounds.
* Conventional numerical algorithms that use heuristics
to estimate the accuracy of a result, without guaranteeing
that it is correct. An example would be numerical integration
based on pointwise evaluation, where the error is estimated
by comparing the results with two different sets of evaluation
points. Ball arithmetic then still tracks the accuracy
of the function evaluations.
Any algorithms of the second kind will be clearly
marked as such.
Types, macros and constants
-------------------------------------------------------------------------------
.. type:: arb_calc_func_t
Typedef for a pointer to a function with signature::
int func(arb_ptr out, const arb_t inp, void * param, slong order, slong prec)
implementing a univariate real function `f(x)`.
When called, *func* should write to *out* the first *order*
coefficients in the Taylor series expansion of `f(x)` at the point *inp*,
evaluated at a precision of *prec* bits.
The *param* argument may be used to pass through
additional parameters to the function.
The return value is reserved for future use as an
error code. It can be assumed that *out* and *inp* are not
aliased and that *order* is positive.
.. macro:: ARB_CALC_SUCCESS
Return value indicating that an operation is successful.
.. macro:: ARB_CALC_IMPRECISE_INPUT
Return value indicating that the input to a function probably needs
to be computed more accurately.
.. macro:: ARB_CALC_NO_CONVERGENCE
Return value indicating that an algorithm has failed to convergence,
possibly due to the problem not having a solution, the algorithm
not being applicable, or the precision being insufficient
Debugging
-------------------------------------------------------------------------------
.. var:: int arb_calc_verbose
If set, enables printing information about the calculation
to standard output.
Subdivision-based root finding
-------------------------------------------------------------------------------
.. type:: arf_interval_struct
.. type:: arf_interval_t
An :type:`arf_interval_struct` consists of a pair of :type:`arf_struct`,
representing an interval used for subdivision-based root-finding.
An :type:`arf_interval_t` is defined as an array of length one of type
:type:`arf_interval_struct`, permitting an :type:`arf_interval_t`
to be passed by reference.
.. type:: arf_interval_ptr
Alias for ``arf_interval_struct *``, used for vectors of intervals.
.. type:: arf_interval_srcptr
Alias for ``const arf_interval_struct *``, used for vectors of intervals.
.. function:: void arf_interval_init(arf_interval_t v)
.. function:: void arf_interval_clear(arf_interval_t v)
.. function:: arf_interval_ptr _arf_interval_vec_init(slong n)
.. function:: void _arf_interval_vec_clear(arf_interval_ptr v, slong n)
.. function:: void arf_interval_set(arf_interval_t v, const arf_interval_t u)
.. function:: void arf_interval_swap(arf_interval_t v, arf_interval_t u)
.. function:: void arf_interval_get_arb(arb_t x, const arf_interval_t v, slong prec)
.. function:: void arf_interval_printd(const arf_interval_t v, slong n)
Helper functions for endpoint-based intervals.
.. function:: void arf_interval_fprintd(FILE * file, const arf_interval_t v, slong n)
Helper functions for endpoint-based intervals.
.. function:: slong arb_calc_isolate_roots(arf_interval_ptr * found, int ** flags, arb_calc_func_t func, void * param, const arf_interval_t interval, slong maxdepth, slong maxeval, slong maxfound, slong 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.
.. function:: int arb_calc_refine_root_bisect(arf_interval_t r, arb_calc_func_t func, void * param, const arf_interval_t start, slong iter, slong 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.
Newton-based root finding
-------------------------------------------------------------------------------
.. function:: void arb_calc_newton_conv_factor(arf_t conv_factor, arb_calc_func_t func, void * param, const arb_t conv_region, slong 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*.
.. function:: int arb_calc_newton_step(arb_t xnew, arb_calc_func_t func, void * param, const arb_t x, const arb_t conv_region, const arf_t conv_factor, slong 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.
.. function:: int arb_calc_refine_root_newton(arb_t r, arb_calc_func_t func, void * param, const arb_t start, const arb_t conv_region, const arf_t conv_factor, slong eval_extra_prec, slong 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).