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Arb 2.9.0 released

December 2, 2016

I have tagged Arb 2.9.0. The last public release (Arb 2.8.1) is almost one year and 700 commits old, which is far too much! I hope to get back to more regular releases (every 1-2 months) next year. I want to thank Pascal Molin, Alex Griffing, Isuru Fernando, Tommy Hofmann, Ricky Farr and Andreas Enge who contributed code to this version. Many others have helped out by reporting bugs or providing feedback.

The list of changes from Arb 2.8 to Arb 2.9 is reproduced below. As usual, the Arb documentation (now with its own domain, arblib.org) describes the methods in more detail.

A few highlights: Arb now supports Dirichlet L-functions (implemented by Pascal Molin and myself), Alex Griffing has done a lot of nice work on matrix operations, and the code for hypergeometric functions has been improved substantially. There are lots of new utility methods, build improvements, optimizations, and at least one serious bug fix. I'm not going to write in detail about the new features here, but do see the following posts: Taking the error out of the error function, The Riemann-Siegel formula in Arb, Dirichlet L-functions in Arb.

Another significant change is that the license has been switched from GPL to LGPL. The same switch was also made in FLINT, so Arb and its dependencies are now all LGPL.

In other news, I have written a preprint titled Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic. This paper describes how Arb implements floating-point numbers and interval operations at a low level and clarifies how the precision and error bounds are managed. Assuming that it gets published, this paper will supplant my extended abstract from 2013 (which is woefully out of date) as a citable reference.

Changes from Arb 2.8 to 2.9

• License
  • Changed license from GPL to LGPL.
• Build system and compatibility
  • Fixed FLINT includes to use flint/foo.h instead of foo.h, simplifying compilation on many systems.
  • Added another alias for the dynamic library to fix make check on certain systems (contributed by Andreas Enge).
  • Travis CI support (contributed by Isuru Fernando).
  • Added support for ARB_TEST_MULTIPLIER environment variable to control the number of test iterations.
  • Support building with CMake (contributed by Isuru Fernando).
  • Support building with MSVC on Windows (contributed by Isuru Fernando).
  • Fixed unsafe use of FLINT_ABS for slong -> ulong conversion in arf.h, which caused failures on MIPS and ARM systems.
• Basic arithmetic and methods
  • Fixed mag_addmul(x,x,x) with x having a mantissa of all ones. This could produce a non-normalized mag_t value, potentially leading to incorrect results in arb and acb level arithmetic. This bug was caught by new test code, and fortunately would have been hard to trigger accidentally.
  • Added fasth paths for error bound calculations in arb_sqrt and arb_div, speeding up these operations significantly at low precision
  • Added support for round-to-nearest in all arf methods.
  • Added fprint methods (contributed by Alex Griffing).
  • Added acb_printn and acb_fprintn methods to match arb_printn.
  • Added arb_equal_si and acb_equal_si.
  • Added arb_can_round_mpfr.
  • Added arb_get_ubound_arf, arb_get_lbound_arf (contributed by Tommy Hofmann).
  • Added sign function (arb_sgn).
  • Added complex sign functions (acb_sgn, acb_csgn).
  • Rewrote arb_contains_fmpq to make the test exact.
  • Optimized mag_get_fmpq.
  • Optimized arf_get_fmpz and added more robust test code.
  • Rewrote arb_get_unique_fmpz and arb_get_interval_fmpz_2exp, reducing overhead, making them more robust with huge exponents, and documenting their behavior more carefully.
  • Optimized arb_union.
  • Optimized arf_is_int, arf_is_int_2exp_si and changed these from inline to normal functions.
  • Added mag_const_pi, mag_sub, mag_expinv.
  • Optimized binary-to-decimal conversion for huge exponents by using exponential function instead of binary powering.
  • Added arb_intersection (contributed by Alex Griffing).
  • Added arb_min, arb_max (contributed by Alex Griffing).
  • Fixed a bug in arb_log and in test code on 64-bit Windows due to unsafe use of MPFR which only uses 32-bit exponents on Win64.
  • Improved some test functions to reduce the chance of reporting spurious failures.
  • Added squaring functions (arb_sqr, acb_sqr) (contributed by Ricky Farr).
  • Added arf_frexp.
  • Added arf_cmp_si, arf_cmp_ui, arf_cmp_d.
  • Added methods to count allocated bytes (arb_allocated_bytes, _arb_vec_allocated_bytes, etc.).
  • Added methods to predict memory usage for large vectors (_arb/_acb_vec_estimate_allocated_bytes).
  • Changed clear() methods from inline to normal functions, giving 8% faster compilation and 25% smaller libarb.so.
  • Added acb_unit_root and _acb_vec_unit_roots (contributed by Pascal Molin).
• Polynomials
  • Added sinh and cosh functions of power series (arb/acb_poly_sinh/cosh_series and sinh_cosh_series).
  • Use basecase series inversion algorithm to improve speed and error bounds in arb/acb_poly_inv_series.
  • Added functions for fast polynomial Taylor shift (arb_poly_taylor_shift, acb_poly_taylor_shift and variants).
  • Fast handling of special cases in polynomial composition.
  • Added acb_poly scalar mul and div convenience methods (contributed by Alex Griffing).
  • Added set_trunc, set_trunc_round convenience methods.
  • Added add_series, sub_series methods for truncating addition.
  • Added polynomial is_zero, is_one, is_x, valuation convenience methods.
  • Added hack to arb_poly_mullow and acb_poly_mullow to avoid overhead when doing an in-place multiplication with length at most 2.
  • Added binomial and Borel transform methods for acb_poly.
• Matrices
  • Added Cholesky decomposition plus solving and inverse for positive definite matrices (arb_mat_cho, arb_mat_spd_solve, arb_mat_spd_inv and related methods) (contributed by Alex Griffing).
  • Added LDL decomposition and inverse and solving based on LDL decomposition for real matrices (arb_mat_ldl, arb_mat_solve_ldl_precomp, arb_mat_inv_ldl_precomp) (contributed by Alex Griffing).
  • Improved the entrywise error bounds in matrix exponential computation to preserve sparsity and give exact entries where possible in many cases (contributed by Alex Griffing).
  • Added public functions for computing the truncated matrix exponential Taylor series (arb_mat_exp_taylor_sum, acb_mat_exp_taylor_sum).
  • Added functions related to sparsity structure (arb_mat_entrywise_is_zero, arb_mat_count_is_zero, etc.) (contributed by Alex Griffing).
  • Entrywise multiplication (arb_mat_mul_entrywise, acb_mat_mul_entrywise) (contributed by Alex Griffing).
  • Added is_empty and is_square convenience methods (contributed by Alex Griffing).
  • Added the bool_mat helper module for matrices over the boolean semiring (contributed by Alex Griffing).
  • Added Frobenius norm computation (contributed by Alex Griffing).
• Miscellaneous special functions
  • Added evaluation of Bernoulli polynomials (arb_bernoulli_poly_ui, acb_bernoulli_poly_ui).
  • Added convenience function for evaluation of huge Bernoulli numbers (arb_bernoulli_fmpz).
  • Added Euler numbers (arb_euler_number_ui, arb_euler_number_fmpz).
  • Added fast approximate partition function (arb_partitions_fmpz/ui).
  • Optimized partition function for n < 1000 by using recurrence for the low 64 bits.
  • Improved the worst-case error bound in arb_atan.
  • Added arb_log_base_ui.
  • Added complex sinc function (acb_sinc).
  • Special handling of z = 1 when computing polylogarithms.
  • Fixed agm(-1,-1) to output 0 instead of indeterminate.
  • Made working precision in arb_gamma and acb_gamma more sensitive to the input accuracy.
• Hypergeometric functions
  • Compute erf and erfc without cancellation problems for large or complex z.
  • Avoid re-computing the square root of pi in several places.
  • Added generalized hypergeometric function (acb_hypgeom_pfq).
  • Implement binary splitting and rectangular splitting for evaluation of hypergeometric series with a power series parameter, greatly speeding up Y_n, K_n and other functions at high precision, as well as speeding up high-order parameter derivatives.
  • Use binary splitting more aggressively in acb_hypgeom_pfq_sum to reduce error bound inflation.
  • Asymptotic expansions of hypergeometric functions: more accurate parameter selection, and better handling of terminating cases.
  • Tweaked algorithm selection and working precision in acb_hypgeom_m.
  • Avoid dividing by the denominator of the next term in acb_hypgeom_sum, which would lead to a division by zero when evaluating hypergeometric polynomials.
  • Fixed a bug in hypergeometric series evaluation resulting in near-integers not being skipped in some cases, leading to unnecessary loss of precision.
  • Added series expansions of Airy functions (acb_hypgeom_airy_series, acb_hypgeom_airy_jet).
  • Fixed a case where Airy functions accidentally chose the worst algorithm instead of the best one.
  • Added functions for computing erf, erfc, erfi of power series in the acb_hypgeom module.
  • Added series expansion of the logarithmic integral (acb_hypgeom_li_series).
  • Added Fresnel integrals (acb_hypgeom_fresnel, acb_hypgeom_fresnel_series).
  • Added the lower incomplete gamma function (acb_hypgeom_gamma_lower) (contributed by Alex Griffing).
  • Added series expansion of the lower incomplete gamma function (acb_hypgeom_gamma_lower_series) (contributed by Alex Griffing).
  • Added support for computing the regularized incomplete gamma functions.
  • Use slightly sharper error bound for analytic continuation of 2F1.
  • Added support for computing finite limits of 2F1 with inexact parameters differing by integers.
  • Added the incomplete beta function (acb_hypgeom_beta_lower, acb_hypgeom_beta_lower_series)
  • Improved acb_hypgeom_u to use a division-avoiding algorithm for small polynomial cases.
  • Added arb_hypgeom module, wrapping the complex hypergeometric functions for more convenient use with the arb_t type.
• Dirichlet L-functions and Riemann zeta function
  • New module dirichlet for working algebraically with Dirichlet groups and characters (contributed by Pascal Molin).
  • New module acb_dirichlet for numerical evaluation of Dirichlet characters and L-functions (contributed by Pascal Molin).
  • Efficient representation and manipulation of Dirichlet characters using the Conrey representation (contributed by Pascal Molin).
  • New module dlog for word-size discrete logarithm evaluation, used to support algorithms on Dirichlet characters (contributed by Pascal Molin).
  • Methods for properties, evaluation, iteration, pairing, lift, lowering etc. of Dirichlet characters (contributed by Pascal Molin).
  • Added acb_dirichlet_roots methods for fast evaluation of many roots of unity (contributed by Pascal Molin).
  • Added acb_dirichlet_hurwitz_precomp methods for fast multi-evaluation of the Hurwitz zeta function for many parameter values.
  • Added methods for computing Gauss, Jacobi and theta sums over Dirichlet characters (contributed by Pascal Molin).
  • Added methods (acb_dirichlet_l, acb_dirichlet_l_jet, acb_dirichlet_l_series) for evaluation of Dirichlet L-functions and their derivatives.
  • Implemented multiple algorithms for evaluation of Dirichlet L-functions depending on the argument (Hurwitz zeta function decomposition, Euler product, functional equation).
  • Added methods (acb_dirichlet_hardy_z, acb_dirichlet_hardy_z_series, etc.) for computing the Hardy Z-function corresponding to a Dirichlet L-function.
  • Added fast bound for Hurwitz zeta function (mag_hurwitz_zeta_uiui).
  • Improved parameter selection in Hurwitz zeta function to target relative instead of absolute error for large positive s.
  • Improved parameter selection in Hurwitz zeta function to avoid computing unnecessary Bernoulli numbers for large imaginary s.
  • Added Dirichlet eta function (acb_dirichlet_eta).
  • Implemented the Riemann-Siegel formula for faster evaluation of the Riemann zeta function at large height.
  • Added smooth-index algorithm for the main sum when evaluating the Riemann zeta function, avoiding the high memory usage of the full sieving algorithm when the number of terms gets huge.
  • Improved tuning for using the Euler product when computing the Riemann zeta function.
• Example programs
  • Added logistic map example program.
  • Added lvalue example program.
  • Improved poly_roots in several ways: identify roots that are exactly real, automatically perform squarefree factorization, use power hack, and allow specifying a product of polynomials as input on the command line.
• Housekeeping
  • New section in the documentation giving an introduction to ball arithmetic and using the library.
  • Tidied, documented and added test code for the fmpz_extras module.
  • Added proper documentation and test code for many helper methods.
  • Removed the obsolete fmprb module entirely.
  • Documented more algorithms and formulas.
  • Clarified integer overflow issues and use of ARF_PREC_EXACT in the documentation.
  • Added .gitignore file.
  • Miscellaneous improvements to the documentation.

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