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Arb - a C library for arbitrary-precision ball arithmetic — Arb 2.23.0 documentationArb - a C library for arbitrary-precision ball arithmetic — Arb 2.23.0 documentation Navigation index next | Arb 2.23.0 documentation » Arb - a C library for arbitrary-precision ball arithmetic Note Arb was merged into FLINT in 2023. The documentation on arblib.org will no longer be updated. See the FLINT documentation instead. Arb - a C library for arbitrary-precision ball arithmetic¶ Welcome to Arb’s documentation! Arb is a C library for rigorous real and complex arithmetic with arbitrary precision. Arb tracks numerical errors automatically using ball arithmetic, a form of interval arithmetic based on a midpoint-radius representation. On top of this, Arb provides a wide range of mathematical functionality, including polynomials, power series, matrices, integration, root-finding, and many transcendental functions. Arb is designed with efficiency as a primary goal, and is usually competitive with or faster than other arbitrary-precision packages. The code is thread-safe, portable, and extensively tested. Arb is free software distributed under the GNU Lesser General Public License (LGPL), version 2.1 or later (see License). The git repository is https://github.com/fredrik-johansson/arb/ Arb is developed by Fredrik Johansson (fredrik.johansson@gmail.com), with help from many contributors (see Credits and references). Questions and discussion about Arb are welcome on the flint-devel mailing list. There is also an issue tracker for bug reports and feature requests. Development progress is sometimes covered on Fredrik’s blog. This documentation is available in HTML format at http://arblib.org and in PDF format at http://arblib.org/arb.pdf. This edition of the documentation was updated Mar 16, 2024 and describes Arb 2.23.0. Documentation for specific release versions is also available in PDF format. General information¶ Feature overview Setup Package managers Download Dependencies Standalone installation Running tests Building with MSVC Running code Computer algebra systems and wrappers Using ball arithmetic Ball semantics Binary and decimal Quality of enclosures Predicates A worked example: the sine function More on precision and accuracy Polynomial time guarantee Technical conventions and potential issues Integer types Integer overflow Aliasing Thread safety and caches Use of hardware floating-point arithmetic Interface changes General note on correctness Contributing to Arb Code conventions Test code Credits and references License Authors Funding Software Citing Arb Bibliography Example programs¶ Example programs pi.c zeta_zeros.c bernoulli.c class_poly.c hilbert_matrix.c keiper_li.c logistic.c real_roots.c poly_roots.c zeta_zeros.c complex_plot.c lvalue.c lcentral.c integrals.c fpwrap.c functions_benchmark.c Floating-point numbers¶ Arb uses two custom floating-point types in its implementation of ball arithmetic. The radius of a ball is represented using the type mag_t which is unsigned and has a fixed precision. The midpoint is represented using the type arf_t which has arbitrary precision. mag.h – fixed-precision unsigned floating-point numbers for bounds Types, macros and constants Memory management Special values Assignment and conversions Comparisons Input and output Random generation Arithmetic Fast, unsafe arithmetic Powers and logarithms Special functions arf.h – arbitrary-precision floating-point numbers Types, macros and constants Memory management Special values Assignment, rounding and conversions Comparisons and bounds Magnitude functions Shallow assignment Random number generation Input and output Addition and multiplication Summation Dot products Division Square roots Complex arithmetic Low-level methods Real and complex numbers¶ Real numbers (arb_t) are represented as midpoint-radius intervals, also known as balls. Complex numbers (acb_t) are represented in rectangular form, with arb_t balls for the real and imaginary parts. arb.h – real numbers Types, macros and constants Memory management Assignment and rounding Assignment of special values Input and output Random number generation Radius and interval operations Comparisons Arithmetic Dot product Powers and roots Exponentials and logarithms Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Constants Lambert W function Gamma function and factorials Zeta function Bernoulli numbers and polynomials Polylogarithms Other special functions Internals for computing elementary functions Vector functions acb.h – complex numbers Types, macros and constants Memory management Basic manipulation Input and output Random number generation Precision and comparisons Complex parts Arithmetic Dot product Mathematical constants Powers and roots Exponentials and logarithms Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Lambert W function Rising factorials Gamma function Zeta function Polylogarithms Arithmetic-geometric mean Other special functions Piecewise real functions Vector functions Polynomials and power series¶ These modules implement dense univariate polynomials with real and complex coefficients. Truncated power series are supported via methods acting on polynomials, without introducing a separate power series type. arb_poly.h – polynomials over the real numbers Types, macros and constants Memory management Basic manipulation Conversions Input and output Random generation Comparisons Bounds Arithmetic Composition Evaluation Product trees Multipoint evaluation Interpolation Differentiation Transforms Powers and elementary functions Lambert W function Gamma function and factorials Zeta function Root-finding Other special polynomials acb_poly.h – polynomials over the complex numbers Types, macros and constants Memory management Basic properties and manipulation Input and output Random generation Comparisons Conversions Bounds Arithmetic Composition Evaluation Product trees Multipoint evaluation Interpolation Differentiation Transforms Elementary functions Lambert W function Gamma function Power sums Zeta function Other special functions Root-finding arb_fmpz_poly.h – extra methods for integer polynomials Evaluation Utility methods Polynomial roots Special polynomials Transforms¶ acb_dft.h – Discrete Fourier transform Main DFT functions DFT on products Convolution FFT algorithms Matrices¶ These modules implement dense matrices with real and complex coefficients. Rudimentary linear algebra is supported. arb_mat.h – matrices over the real numbers Types, macros and constants Memory management Conversions Random generation Input and output Comparisons Special matrices Transpose Norms Arithmetic Scalar arithmetic Gaussian elimination and solving Cholesky decomposition and solving Characteristic polynomial and companion matrix Special functions Sparsity structure Component and error operations Eigenvalues and eigenvectors acb_mat.h – matrices over the complex numbers Types, macros and constants Memory management Conversions Random generation Input and output Comparisons Special matrices Transpose Norms Arithmetic Scalar arithmetic Gaussian elimination and solving Characteristic polynomial and companion matrix Special functions Component and error operations Eigenvalues and eigenvectors Special functions¶ These modules implement mathematical functions with complexity that goes beyond the basics covered directly in the arb and acb modules. acb_hypgeom.h – hypergeometric functions of complex variables Rising factorials Gamma function Convergent series Asymptotic series Generalized hypergeometric function Confluent hypergeometric functions Error functions and Fresnel integrals Bessel functions Modified Bessel functions Airy functions Coulomb wave functions Incomplete gamma and beta functions Exponential and trigonometric integrals Gauss hypergeometric function Orthogonal polynomials and functions Dilogarithm arb_hypgeom.h – hypergeometric functions of real variables Rising factorials Gamma function Binomial coefficients Generalized hypergeometric function Confluent hypergeometric functions Gauss hypergeometric function Error functions and Fresnel integrals Incomplete gamma and beta functions Exponential and trigonometric integrals Bessel functions Airy functions Coulomb wave functions Orthogonal polynomials and functions Dilogarithm Hypergeometric sums acb_elliptic.h – elliptic integrals and functions of complex variables Complete elliptic integrals Legendre incomplete elliptic integrals Carlson symmetric elliptic integrals Weierstrass elliptic functions acb_modular.h – modular forms of complex variables The modular group Modular transformations Addition sequences Jacobi theta functions Dedekind eta function Modular forms Elliptic integrals and functions Class polynomials dirichlet.h – Dirichlet characters Dirichlet characters Multiplicative group modulo q Character type Character properties Character evaluation Character operations acb_dirichlet.h – Dirichlet L-functions, Riemann zeta and related functions Roots of unity Truncated L-series and power sums Riemann zeta function Riemann-Siegel formula Hurwitz zeta function Hurwitz zeta function precomputation Lerch transcendent Stieltjes constants Dirichlet character evaluation Dirichlet character Gauss, Jacobi and theta sums Discrete Fourier transforms Dirichlet L-functions Hardy Z-functions Gram points Riemann zeta function zeros Riemann zeta function zeros (Platt’s method) bernoulli.h – support for Bernoulli numbers Generation of Bernoulli numbers Caching Bounding Isolated Bernoulli numbers hypgeom.h – support for hypergeometric series Strategy for error bounding Types, macros and constants Memory management Error bounding Summation partitions.h – computation of the partition function Calculus¶ Using ball arithmetic, it is possible to do rigorous root-finding and integration (among other operations) with generic functions. This code should be considered experimental. arb_calc.h – calculus with real-valued functions Types, macros and constants Debugging Subdivision-based root finding Newton-based root finding acb_calc.h – calculus with complex-valued functions Types, macros and constants Integration Local integration algorithms Integration (old) Wrappers¶ Floating-point wrappers for Arb functions. arb_fpwrap.h – floating-point wrappers of Arb mathematical functions Option and return flags Types Functions Calling from C Interfacing from Python Interfacing from Julia Extra utility modules¶ Mainly for internal use. fmpzi.h – Gaussian integers acf.h – complex floating-point numbers double_interval.h – double-precision interval arithmetic and helpers fmpz_extras.h – extra methods for FLINT integers bool_mat.h – matrices over booleans dlog.h – discrete logarithms mod ulong primes fmpr.h – Arb 1.x floating-point numbers (deprecated) Supplementary algorithm notes¶ Here, we give extra proofs, error bounds, and formulas that would be too lengthy to reproduce in the documentation for each module. General formulas and bounds Algorithms for mathematical constants Algorithms for the gamma function Algorithms for the Hurwitz zeta function Algorithms for polylogarithms Algorithms for hypergeometric functions Algorithms for the arithmetic-geometric mean Version history¶ History and changes Table of Contents Arb - a C library for arbitrary-precision ball arithmetic General information Example programs Floating-point numbers Real and complex numbers Polynomials and power series Transforms Matrices Special functions Calculus Wrappers Extra utility modules Supplementary algorithm notes Version history Next topic Feature overview This Page Show Source Quick search « Navigation index next | Arb 2.23.0 documentation » Arb - a C library for arbitrary-precision ball arithmetic © Copyright 2012-2022, Fredrik Johansson. Created using Sphinx 7.2.6.enen1770994201https://arblib.org

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