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73 lines
3.3 KiB
ReStructuredText
73 lines
3.3 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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Other implementations of ball arithmetic include
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`iRRAM <http://irram.uni-trier.de/>`_ and
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`Mathemagix <http://www.mathemagix.org/www/mmdoc/doc/html/main/index.en.html>`_.
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In contrast to those systems, Arb is more focused on low-level arithmetic
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and computation of transcendental functions needed for
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number theory. Arb also differs in some technical aspects of
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the implementation.
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Arb 2.x contains:
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* A module (:ref:`arf <arf>`) for correctly rounded arbitrary-precision
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floating-point arithmetic. Arb's floating-point numbers have a few special
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features, such as arbitrary-size exponents (useful for combinatorics and
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asymptotics) and dynamic allocation (facilitating implementation of hybrid
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integer/floating-point and mixed-precision algorithms).
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* A module (:ref:`mag <mag>`) for representing magnitudes (error bounds)
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more efficiently than with an arbitrary-precision floating-point type.
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* A module (:ref:`arb <arb>`) for real ball arithmetic, where a ball is
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implemented as an *arf* midpoint and a *mag* radius.
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* A module (:ref:`acb <acb>`) for complex numbers in rectangular form,
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represented as pairs real balls.
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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:`arb_poly <arb-poly>`, :ref:`acb_poly <acb-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:`arb_mat <arb-mat>`, :ref:`acb_mat <acb-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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Arb 1.x used a different set of numerical base types (*fmpr*, *fmprb*
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and *fmpcb*). These types had a slightly simpler internal representation,
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but generally had worse performance. Almost all methods for the Arb 1.x types
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have now been ported to faster equivalents for the Arb 2.x types.
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The last version to include both the Arb 1.x and Arb 2.x types and methods
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was Arb 2.2. As of Arb 2.3, only a small set of *fmpr* and *fmprb*
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methods are left for fallback and testing purposes.
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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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