Calcium

Calcium (pronounced “kalkium”) is a C library for exact computation with real and complex numbers. It is capable of rigorously deciding the truth of any constant relation involving algebraic numbers and many relations involving transcendental numbers, for example

\[\frac{\log(\sqrt{2}+\sqrt{3})}{\log(5+2\sqrt{6})} = \frac{1}{2}, \quad i^{\,i} = \exp\left(\frac{\pi}{ \left({\left(\sqrt{-2}\right)}^{\sqrt{2}}\right)^{\sqrt{2}}}\right), \quad 10^{-30} < \frac{640320^3 + 744}{e^{\pi \sqrt{163}}} - 1 < 10^{-29}.\]

Calcium is free software (LGPL). It depends on GMP, MPFR, Flint, Antic and Arb.

• Source code: https://github.com/fredrik-johansson/calcium

• Bug reports and feature requests: https://github.com/fredrik-johansson/calcium/issues

• Mailing list: flint-devel

This documentation is available in HTML format at http://fredrikj.net/calcium/ and in PDF format at http://fredrikj.net/calcium/calcium.pdf. This edition of the documentation was updated May 28, 2021 and describes Calcium 0.4.0.

Tutorial

For new users, this introductory Jupyter notebook is a good place to start.

General information

• Introduction
  • Exact numbers in Calcium
  • FAQ
• Setup
  • Installation
  • Running tests
  • Running code
• Example programs
  • elementary.c
  • binet.c
  • machin.c
  • swinnerton_dyer_poly.c
  • huge_expr.c
  • hilbert_matrix.c
  • dft.c

General modules

• calcium.h – global definitions
  • Version
  • Test code
  • Triple-valued logic
  • Flint, Arb and Antic types
  • Flint, Arb and Antic extras
  • Input and output

Numbers

• ca.h – exact real and complex numbers
  • Introduction: numbers
  • Introduction: special values
  • Number objects
  • Context objects
  • Memory management for numbers
  • Symbolic expressions
  • Printing
  • Special values
  • Assignment and conversion
  • Conversion of algebraic numbers
  • Random generation
  • Representation properties
  • Value predicates
  • Comparisons
  • Field structure operations
  • Arithmetic
  • Powers and roots
  • Complex parts
  • Exponentials and logarithms
  • Trigonometric functions
  • Special functions
  • Numerical evaluation
  • Rewriting and simplification
  • Factorization
  • Context options
  • Internal representation
• ca_vec.h – vectors of real and complex numbers
  • Types, macros and constants
  • Memory management
  • Length
  • Assignment
  • Special vectors
  • Input and output
  • List operations
  • Arithmetic
  • Comparisons and properties
  • Internal representation

Matrices and polynomials

• ca_poly.h – dense univariate polynomials over the real and complex numbers
  • Types, macros and constants
  • Memory management
  • Assignment and simple values
  • Random generation
  • Input and output
  • Degree and leading coefficient
  • Comparisons
  • Arithmetic
  • Evaluation and composition
  • Derivative and integral
  • Power series division
  • Elementary functions
  • Greatest common divisor
  • Roots and factorization
  • Vectors of polynomials
• ca_mat.h – matrices over the real and complex numbers
  • Types, macros and constants
  • Memory management
  • Assignment and conversions
  • Random generation
  • Input and output
  • Special matrices
  • Comparisons and properties
  • Conjugate and transpose
  • Arithmetic
  • Powers
  • Polynomial evaluation
  • Gaussian elimination and LU decomposition
  • Solving and inverse
  • Rank and echelon form
  • Determinant and trace
  • Characteristic polynomial
  • Eigenvalues and eigenvectors
  • Jordan canonical form
  • Matrix functions

Field and extension number constructions

These modules are used internally by the ca_t type to construct towers of algebraic and transcendental number fields. The user does not normally need to use these modules directly outside of advanced applications requiring inspection of the symbolic representations of numbers.

• ca_ext.h – real and complex extension numbers
  • Type and macros
  • Memory management
  • Structure
  • Input and output
  • Numerical evaluation
  • Cache
• ca_field.h – extension fields
  • Type and macros
  • Memory management
  • Input and output
  • Ideal
  • Structure operations
  • Cache

Symbolic expressions

• fexpr.h – flat-packed symbolic expressions
  • Introduction
  • Types and macros
  • Memory management
  • Size information
  • Comparisons
  • Atoms
  • Input and output
  • LaTeX output
  • Function call structure
  • Composition
  • Subexpressions and replacement
  • Arithmetic expressions
  • Vectors
• fexpr_builtin.h – builtin symbols
  • C helper functions
  • Variables and iteration
  • Booleans and logic
  • Tuples, lists and sets
  • Numbers and arithmetic
  • Operators and calculus
  • Matrices and linear algebra
  • Polynomials, series and rings
  • Special functions

Basic algebraic structures

The following modules implement useful exact structures independently of the ca_t type. They are used internally in Calcium, but the interfaces are stable and use in appropriate external applications is encouraged.

• fmpz_mpoly_q.h – multivariate rational functions over Q
  • Types and macros
  • Memory management
  • Assignment
  • Canonicalisation
  • Properties
  • Special values
  • Input and output
  • Random generation
  • Comparisons
  • Arithmetic
  • Content
• qqbar.h – algebraic numbers represented by minimal polynomials
  • Types and macros
  • Memory management
  • Assignment
  • Properties
  • Conversions
  • Special values
  • Input and output
  • Random generation
  • Comparisons
  • Complex parts
  • Integer parts
  • Arithmetic
  • Powers and roots
  • Numerical enclosures
  • Numerator and denominator
  • Conjugates
  • Polynomial evaluation
  • Polynomial roots
  • Roots of unity and trigonometric functions
  • Guessing and simplification
  • Symbolic expressions and conversion to radicals
  • Internal functions
• utils_flint.h – extra methods for Flint types
  • General methods for multivariate polynomials
  • Ideals and Gröbner bases
  • Index pairs

Python interface

• Python interface (pycalcium / pyca)

Credits and references

• Bibliography

Version history

• History and changes

Indices and tables

• Index