ArbExtras.jl

This package extends Arblib with some methods for enclosing roots, enclosing extrema and computing integrals.

The package started development during my PhD with the goal to give reusable implementations of algorithms that were used in my research. The type of methods that are implemented and which type of input they are optimized for has been heavily influenced by the projects that I worked on during that time.

Functionality

The main functionality provided by the package are methods for isolating roots and enclosing extrema of univariate real functions. In addition to this is provides a rudimentary integral routine and a few minor things.

Enclosing roots

The package provides three different functions for isolating and enclosing roots of univariate, real valued functions:

• isolate_roots(f, a::Arf, b::Arf) is used for isolating all roots of a function f on the interval [a, b].
• refine_root(f, root::Arb) is used for refining an initial enclosure root of a zero to f. It makes use of the interval Newton method.
• refine_root_bisection(f, a, b) is used for refining an initial enclosure [a, b] of a root of f. It requires that f(a) and f(b) have different signs. It makes use if bisection, checking the sign of the midpoint in each iteration.

See the documentation of the respective methods for more details, in particular the available keyword arguments. The first two methods require access to the derivative of f, this is computed using ArbSeries and for that reason f needs to support evaluation of both Arb and ArbSeries. The refine_root_bisection doesn't require access to the derivative and can therefore be used when this is not available.

Enclosing extrema

The package provides three families of methods:

• extrema_polynomial, minimum_polynomial and maximum_polynomial
• extrema_series, minimum_series and maximum_series
• extrema_enclosure, minimum_enclosure and maximum_enclosure

As indicated by the names they compute either the minimum, the maximum or both.

• extrema_polynomial(p::ArbPoly, a::Arf, b::Arb) is used for enclosing the extrema of a polynomial p on the interval [a, b]. This is done by enclosing the roots of the derivative of p.
• extrema_series(f, a::Arf, b::Arb; degree::Integer = 8) is used for enclosing the extrema of a function f on the interval [a, b]. This is done by computing a Taylor series of the given degree. The extrema of the series is computed using extrema_polynomial and then an enclosure of the remainder term is added.
• extrema_enclosure(f, a::Arf, b::Arb) is also used for enclosing the extrema of a function f on the interval [a, b]. In this case it is done by iteratively bisecting the interval and using extrema_series on each subinterval. The bisection is continued until the result satisfies the required tolerance.

See the documentation of the respective methods for more details, in particular the available keyword arguments. The minimum_ and maximum_ versions have the same interface, the only difference being that they return either the minimum or maximum instead of both.

For enclosure_series it is also possible to call it like enclosure_series(f, x::Arb), in which case it computes an enclosure of f(x) by computing the extrema and taking the union of them.

The package also provides bounded_by(f, a::Arf, b::Arf, C::Arf) which checks if the function f is bounded by C on the interval [a, b]. It is similar to maximum_enclosure but only bisects enough to be able to check that it is bounded by C. See also the ubound_tol and lbound_tol arguments to extrema_enclosure.

Other functionality

In addition to the above the package also provides some smaller things:

• integrate(f, a::Arb, b::Arb) can be used to compute the integral of f from a to b using a Gauss-Legendre quadrature of order 2, combined with bisection. In general the integrator from Arb, Arblib.integrate performs much better and should be preferred. This method does however have the benefit that it does not require evaluation on complex balls, instead it needs access to the fourth derivative of f, which is computed using ArbSeries. So it can be of use if there is no implementation of f(::Acb), but there is one for f(::ArbSeries).
• SpecialFunctions.besselj(ν::Arb, z::ArbSeries) and SpecialFunctions.bessely(ν::Arb, z::ArbSeries) for computing Taylor expansions of the Bessel functions.
• Utility functions which are mainly for internal use, but could be useful in other cases as well , see their documentation for details.
  • bisect_interval, bisect_interval_recursive and bisect_intervals
  • check_tolerance
  • check_interval
  • format_interval
  • taylor_remainder
  • enclosure_ubound, enclosure_lbound and enclosure_getinterval
  • derivative_function
• Some "temporary" functions that should possibly be added to Arblib.jl or even the Arb library itself in the future
  • iscpx
  • compose_zero and compose_zero!