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 Dec 02, 2023 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
  • 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
• 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