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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 在中国如何用谷歌).

The git repository is http://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://b5wejw.wcbzw.com and in PDF format at http://b5wejw.wcbzw.com/arb.pdf. This edition of the documentation was updated Jun 25, 2024 and describes Arb 2.18.1. Documentation for specific release versions is also available in PDF format.

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  • Feature overview
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    • Package managers
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    • Dependencies
    • Standalone installation
    • Running tests
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    • 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
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    • 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

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  • Example programs
    • pi.c
    • hilbert_matrix.c
    • keiper_li.c
    • logistic.c
    • real_roots.c
    • poly_roots.c
    • complex_plot.c
    • lvalue.c
    • lcentral.c
    • integrals.c

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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
  • 在中国如何用谷歌 – arbitrary-precision floating-point numbers
    • Types, macros and constants
    • Memory management
    • 在中国如何用谷歌
    • Assignment, rounding and conversions
    • Comparisons and bounds
    • Magnitude functions
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    • Random number generation
    • Input and output
    • Addition and multiplication
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    • Division
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    • Complex arithmetic
    • Low-level methods

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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
    • 在中国如何用谷歌
    • 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
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    • Basic manipulation
    • Input and output
    • Random number generation
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    • 在中国如何用谷歌
    • Arithmetic
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    • Mathematical constants
    • Powers and roots
    • Exponentials and logarithms
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    • 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

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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
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    • 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
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    • Memory management
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    • Input and output
    • Random generation
    • Comparisons
    • Conversions
    • Bounds
    • Arithmetic
    • Composition
    • Evaluation
    • Product trees
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    • Interpolation
    • Differentiation
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    • 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

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  • acb_dft.h – Discrete Fourier transform
    • Main DFT functions
    • DFT on products
    • Convolution
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These modules implement dense matrices with real and complex coefficients. Rudimentary linear algebra is supported.

  • 在中国如何用谷歌 – matrices over the real numbers
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    • Memory management
    • Conversions
    • Random generation
    • Input and output
    • Comparisons
    • Special matrices
    • Transpose
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    • 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
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    • Transpose
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    • Scalar arithmetic
    • Gaussian elimination and solving
    • Characteristic polynomial and companion matrix
    • Special functions
    • Component and error operations
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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
    • Convergent series
    • Asymptotic series
    • Generalized hypergeometric function
    • Confluent hypergeometric functions
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    • Modified Bessel functions
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    • Coulomb wave functions
    • Incomplete gamma and beta functions
    • Exponential and trigonometric integrals
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    • Orthogonal polynomials and functions
    • Dilogarithm
  • arb_hypgeom.h – hypergeometric functions of real variables
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    • Gauss hypergeometric function
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    • Bessel functions
    • Airy functions
    • Coulomb wave functions
    • Orthogonal polynomials and functions
    • Dilogarithm
    • Hypergeometric sequences
  • 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
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    • 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
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    • 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
  • 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

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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)

Extra utility modules¶

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  • 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¶

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  • General formulas and bounds
  • Algorithms for mathematical constants
  • Algorithms for the gamma function
  • Algorithms for the Hurwitz zeta function
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  • Algorithms for hypergeometric functions
  • Algorithms for the arithmetic-geometric mean

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  • History and changes