ζ(s) Arb Arb Documentation Release 2.11.0 Fredrik Johansson Jul 09, 2017 CONTENTS 1 Introduction 1 2 General information 3 2.1 Feature overview.........................................3 2.2 Setup...............................................4 2.2.1 Package managers...................................4 2.2.2 Download........................................4 2.2.3 Dependencies......................................4 2.2.4 Standalone installation.................................5 2.2.5 Running tests......................................5 2.2.6 Installation as part of FLINT (deprecated)......................5 2.2.7 Building with MSVC..................................6 2.2.8 Running code......................................6 2.3 Using ball arithmetic......................................7 2.3.1 Ball semantics......................................7 2.3.2 Binary and decimal...................................7 2.3.3 Quality of enclosures..................................8 2.3.4 Predicates........................................9 2.3.5 A worked example: the sine function......................... 10 2.3.6 More on precision and accuracy............................ 12 2.3.7 Polynomial time guarantee............................... 13 2.4 Technical conventions and potential issues.......................... 13 2.4.1 Integer types...................................... 13 2.4.2 Integer overflow..................................... 14 2.4.3 Aliasing......................................... 15 2.4.4 Thread safety and caches............................... 16 2.4.5 Use of hardware floating-point arithmetic...................... 16 2.4.6 Interface changes.................................... 16 2.4.7 General note on correctness.............................. 17 2.5 Example programs........................................ 17 2.5.1 pi.c............................................ 17 2.5.2 hilbert_matrix.c.................................... 17 2.5.3 keiper_li.c........................................ 18 2.5.4 logistic.c......................................... 19 2.5.5 real_roots.c....................................... 19 2.5.6 poly_roots.c....................................... 22 2.5.7 complex_plot.c..................................... 24 2.5.8 lvalue.c.......................................... 25 3 Floating-point numbers 27 3.1 mag.h – fixed-precision unsigned floating-point numbers for bounds............ 27 3.1.1 Types, macros and constants............................. 27 3.1.2 Memory management.................................. 27 3.1.3 Special values...................................... 28 i 3.1.4 Comparisons...................................... 28 3.1.5 Input and output.................................... 28 3.1.6 Random generation................................... 28 3.1.7 Conversions....................................... 29 3.1.8 Arithmetic........................................ 29 3.1.9 Fast, unsafe arithmetic................................. 30 3.1.10 Powers and logarithms................................. 30 3.1.11 Special functions.................................... 31 3.2 arf.h – arbitrary-precision floating-point numbers...................... 32 3.2.1 Types, macros and constants............................. 32 3.2.2 Memory management.................................. 33 3.2.3 Special values...................................... 33 3.2.4 Assignment, rounding and conversions........................ 34 3.2.5 Comparisons and bounds................................ 36 3.2.6 Magnitude functions.................................. 36 3.2.7 Shallow assignment................................... 37 3.2.8 Random number generation.............................. 37 3.2.9 Input and output.................................... 37 3.2.10 Addition and multiplication.............................. 38 3.2.11 Summation....................................... 39 3.2.12 Division......................................... 39 3.2.13 Square roots....................................... 39 3.2.14 Complex arithmetic................................... 40 3.2.15 Low-level methods................................... 40 4 Real and complex numbers 41 4.1 arb.h – real numbers...................................... 41 4.1.1 Types, macros and constants............................. 42 4.1.2 Memory management.................................. 42 4.1.3 Assignment and rounding............................... 43 4.1.4 Assignment of special values.............................. 43 4.1.5 Input and output.................................... 44 4.1.6 Random number generation.............................. 44 4.1.7 Radius and interval operations............................ 45 4.1.8 Comparisons...................................... 47 4.1.9 Arithmetic........................................ 48 4.1.10 Powers and roots.................................... 50 4.1.11 Exponentials and logarithms.............................. 51 4.1.12 Trigonometric functions................................ 52 4.1.13 Inverse trigonometric functions............................ 53 4.1.14 Hyperbolic functions.................................. 53 4.1.15 Inverse hyperbolic functions.............................. 53 4.1.16 Constants........................................ 53 4.1.17 Lambert W function.................................. 54 4.1.18 Gamma function and factorials............................ 54 4.1.19 Zeta function...................................... 55 4.1.20 Bernoulli numbers and polynomials.......................... 56 4.1.21 Polylogarithms..................................... 57 4.1.22 Other special functions................................. 57 4.1.23 Internals for computing elementary functions.................... 58 4.1.24 Vector functions..................................... 59 4.2 acb.h – complex numbers................................... 60 4.2.1 Types, macros and constants............................. 61 4.2.2 Memory management.................................. 61 4.2.3 Basic manipulation................................... 61 4.2.4 Input and output.................................... 62 4.2.5 Random number generation.............................. 63 4.2.6 Precision and comparisons............................... 63 ii 4.2.7 Complex parts..................................... 64 4.2.8 Arithmetic........................................ 65 4.2.9 Mathematical constants................................ 66 4.2.10 Powers and roots.................................... 66 4.2.11 Exponentials and logarithms.............................. 67 4.2.12 Trigonometric functions................................ 67 4.2.13 Inverse trigonometric functions............................ 68 4.2.14 Hyperbolic functions.................................. 68 4.2.15 Inverse hyperbolic functions.............................. 68 4.2.16 Lambert W function.................................. 68 4.2.17 Rising factorials..................................... 69 4.2.18 Gamma function.................................... 70 4.2.19 Zeta function...................................... 71 4.2.20 Polylogarithms..................................... 71 4.2.21 Arithmetic-geometric mean.............................. 71 4.2.22 Other special functions................................. 71 4.2.23 Vector functions..................................... 72 5 Polynomials and power series 75 5.1 arb_poly.h – polynomials over the real numbers...................... 75 5.1.1 Types, macros and constants............................. 75 5.1.2 Memory management.................................. 75 5.1.3 Basic manipulation................................... 76 5.1.4 Conversions....................................... 77 5.1.5 Input and output.................................... 77 5.1.6 Random generation................................... 77 5.1.7 Comparisons...................................... 77 5.1.8 Bounds.......................................... 77 5.1.9 Arithmetic........................................ 78 5.1.10 Composition....................................... 80 5.1.11 Evaluation........................................ 81 5.1.12 Product trees...................................... 82 5.1.13 Multipoint evaluation.................................. 83 5.1.14 Interpolation...................................... 83 5.1.15 Differentiation...................................... 83 5.1.16 Transforms....................................... 84 5.1.17 Powers and elementary functions........................... 84 5.1.18 Lambert W function.................................. 88 5.1.19 Gamma function and factorials............................ 88 5.1.20 Zeta function...................................... 88 5.1.21 Root-finding....................................... 89 5.1.22 Other special polynomials............................... 90 5.2 acb_poly.h – polynomials over the complex numbers.................... 90 5.2.1 Types, macros and constants............................. 90 5.2.2 Memory management.................................. 91 5.2.3 Basic properties and manipulation.......................... 91 5.2.4 Input and output.................................... 92 5.2.5 Random generation................................... 92 5.2.6 Comparisons...................................... 92 5.2.7 Conversions....................................... 93 5.2.8 Bounds.......................................... 93 5.2.9 Arithmetic........................................ 93 5.2.10 Composition....................................... 95 5.2.11 Evaluation........................................ 96