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https://github.com/vale981/arb
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59 lines
2.7 KiB
ReStructuredText
59 lines
2.7 KiB
ReStructuredText
.. _overview:
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Feature overview
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===============================================================================
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Ball arithmetic, also known as mid-rad interval arithmetic, is an
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extension of floating-point arithmetic in which an error bound is
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attached to each variable. This allows doing rigorous computations
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over the real numbers, while avoiding the overhead of
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traditional (inf-sup) interval arithmetic at high precision,
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and eliminating much of the need for time-consuming
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and bug-prone manual error analysis associated with
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standard floating-point arithmetic. (See for example [Hoe2009]_.)
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At the moment, Arb contains:
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* A module (:ref:`fmpr <fmpr>`) for correctly rounded arbitrary-precision
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floating-point arithmetic. Arb numbers have a few special features, such
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as arbitrary-size exponents (useful for combinatorics and asymptotics) and
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dynamic allocation (facilitating implementation of hybrid
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integer/floating-point and mixed-precision algorithms).
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* A module (:ref:`fmprb <fmprb>`) for real ball arithmetic, where a ball is
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implemented as a pair of fmpr numbers, and a corresponding module
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(:ref:`fmpcb <fmpcb>`) for complex numbers in rectangular form.
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* Functions for fast high-precision evaluation of various
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mathematical constants and special functions, implemented using
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ball arithmetic with rigorous error bounds.
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* Modules (:ref:`fmprb_poly <fmprb-poly>`, :ref:`fmpcb_poly <fmpcb-poly>`)
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for polynomials or power series over the real and complex numbers,
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implemented using balls as coefficients,
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with asymptotically fast polynomial multiplication and
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many other operations.
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* Modules (:ref:`fmprb_mat <fmprb-mat>`, :ref:`fmpcb_mat <fmpcb-mat>`)
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for matrices over the real and complex numbers,
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implemented using balls as coefficients.
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At the moment, only rudimentary linear algebra operations are provided.
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Planned features include more transcendental functions and more extensive
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polynomial and matrix functionality, as well as further optimizations.
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Arb uses `GMP <http://mpir.org>`_ / `MPIR <http://mpir.org>`_ and
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`FLINT <http://flintlib.org/>`_
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for the underlying integer arithmetic and other functions.
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The code conventions borrow from FLINT, and the project might get
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merged back into FLINT when the code stabilizes in the future.
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Arb also uses `MPFR <http://mpfr.org/>`_ for testing purposes
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and for evaluation of some functions.
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The current version of Arb implements most of its floating-point arithmetic
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naively using high-level FLINT types. The speed at low precision is far from
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optimal, and the memory management can sometimes be wasteful. The internals
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will be rewritten in the future to fix the inefficiencies,
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which eventually should make Arb ball arithmetic about as fast as
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mpz or mpfr arithmetic at any precision.
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