# History and changes¶

For more details, view the commit log in the git repository https://github.com/fredrik-johansson/arb

Old releases of the code can be accessed from https://github.com/fredrik-johansson/arb/releases

## Old versions of the documentation¶

## 2015-12-31 - version 2.8.1¶

- Fixed 32-bit test failure for the Laguerre function.
- Made the Laguerre function indeterminate at negative integer orders, to be consistent with the test code.

## 2015-12-29 - version 2.8.0¶

- Compatibility and build system
- Windows64 support (contributed by Bill Hart).
- Fixed a bug that broke basic arithmetic on targets where FLINT uses fallback code instead of assembly code, such as PPC64 (contributed by Jeroen Demeyer).
- Fixed configure to use EXTRA_SHARED_FLAGS/LDFLAGS, and other build system fixes (contributed by Tommy Hofmann, Bill Hart).
- Added soname versioning (contributed by Julien Puydt).
- Fixed test code on MinGW (contributed by Hrvoje Abraham).
- Miscellaneous fixes to simplify interfacing Arb from Julia.

- Arithmetic and elementary functions
- Fixed arf_get_d to handle underflow/overflow correctly and to support round-to-nearest.
- Added more complex inverse hyperbolic functions (acb_asin, acb_acos, acb_asinh, acb_acosh, acb_atanh).
- Added arb_contains_int and acb_contains_int for testing whether an interval contains any integer.
- Added acb_quadratic_roots_fmpz.
- Improved arb_sinh to use a more accurate formula for x < 0.
- Added sinc function (arb_sinc) (contributed by Alex Griffing).
- Fixed bug in arb_exp affecting convergence for huge input.
- Faster implementation of arb_div_2expm1_ui.
- Added mag_root, mag_geom_series.
- Improved and added test code for arb_add_error functions.
- Changed arb_pow and acb_pow to make pow(0,positive) = 0 instead of nan.
- Improved acb_sqrt to return finite output for finite input straddling the branch cut.
- Improved arb_set_interval_arf so that [inf,inf] = inf instead of an infinite interval.
- Added computation of Bell numbers (arb_bell_fmpz).
- Added arb_power_sum_vec for computing power sums using Bernoulli numbers.
- Added computation of the Fujiwara root bound for acb_poly.
- Added code to identify all the real roots of a real polynomial (acb_poly_validate_real_roots).
- Added several convenient assignment functions, including arb_set_d, acb_set_d, acb_set_d_d, acb_set_fmpz_fmpz (contributed by Ricky Farr).
- Added many accessor functions (_arb/acb_vec_entry_ptr, arb_get_mid/rad_arb, acb_real/imag_ptr, arb_mid/rad_ptr, acb_get_real/imag).
- Added missing functions acb_add_si, acb_sub_si.
- Renamed arb_root to arb_root_ui (keeping alias) and added acb_root_ui.

- Special functions
- Implemented the Gauss hypergeometric function 2F1 and its regularized version.
- Fixed two bugs in acb_hypgeom_pfq_series_direct discovered while implementing 2F1. In rare cases, these could lead to incorrect values for functions depending on parameter derivatives of hypergeometric series.
- The first bug involved incorrect handling of negative integer parameters. The bug only affected 2F1 and higher functions; it did not affect correctness of any previously implemented functions that relied on acb_hypgeom_pfq_series_direct (such as Bessel Y and K functions of integer order).
- The second bug involved a too small bound being computed for the sum of a geometric series. The geometric series bound is nearly tight for 2F1, and the incorrect version caused immediate test failures for that function. Theoretically, this bug affected correctness of some previously-implemented functions that relied on acb_hypgeom_pfq_series_direct (such as Bessel Y and K functions of integer order), but since the geometric bound is not as tight in those cases, those functions were still reliable in practice (no failing test case has been found).

- Implemented Airy functions and their derivatives (acb_hypgeom_airy).
- Implemented the confluent hypergeometric function 0F1 (acb_hypgeom_0f1).
- Implemented associated Legendre functions P and Q.
- Implemented Chebyshev, Jacobi, Gegenbauer, Laguerre, Hermite functions.
- Implemented spherical harmonics.
- Added function for computing Bessel J and Y functions simultaneously.
- Added rising factorials for non-integer n (arb_rising, acb_rising).
- Made rising factorials use gamma function for large integer n.
- Faster algorithm for theta constants and Dedekind eta function at very high precision.
- Fixed erf to give finite values instead of +/-inf for big imaginary input.
- Improved acb_zeta (and arb_zeta) to automatically use fast code for integer zeta values.
- Added double factorial (arb_doublefac_ui).
- Added code for generating Hilbert class polynomials (acb_modular_hilbert_class_poly).

- Matrices
- Added faster matrix squaring (arb/acb_mat_sqr) (contributed by Alex Griffing).
- Added matrix trace (arb/acb_mat_trace) (contributed by Alex Griffing).
- Added arb/acb_mat_set_round_fmpz_mat, acb_mat_set(_round)_arb_mat (contributed by Tommy Hofmann).
- Added arb/acb_mat_transpose (contributed by Tommy Hofmann).
- Added comparison methods arb/acb_mat_eq/ne (contributed by Tommy Hofmann).

- Other
- Added complex_plot example program.
- Added Airy functions to real_roots example program.
- Other minor patches were contributed by Alexander Kobel, Marc Mezzarobba, Julien Puydt.
- Removed obsolete file config.h.

## 2015-07-14 - version 2.7.0¶

- Hypergeometric functions
- Implemented Bessel I and Y functions (acb_hypgeom_bessel_i, acb_hypgeom_bessel_y).
- Fixed bug in Bessel K function giving the wrong branch for negative real arguments.
- Added code for evaluating complex hypergeometric series binary splitting.
- Added code for evaluating complex hypergeometric series using fast multipoint evaluation.

- Gamma related functions
- Implemented the Barnes G-function and its continuous logarithm (acb_barnes_g, acb_log_barnes_g).
- Implemented the generalized polygamma function (acb_polygamma).
- Implemented the reflection formula for the logarithmic gamma function (acb_lgamma, acb_poly_lgamma_series).
- Implemented the digamma function of power series (arb_poly_digamma_series, acb_poly_digamma_series).
- Improved acb_poly_zeta_series to produce exact zero imaginary parts in most cases when the result should be real-valued.
- Made the real logarithmic gamma function (arb_lgamma, arb_poly_lgamma_series) abort more quickly for negative input.

- Elementary functions
- Added arb_exp_expinv and acb_exp_expinv functions for simultaneously computing exp(x), exp(-x).
- Improved acb_tan, acb_tan_pi, acb_cot and acb_cot_pi for input with large imaginary parts.
- Added complex hyperbolic functions (acb_sinh, acb_cosh, acb_sinh_cosh, acb_tanh, acb_coth).
- Added acb_log_sin_pi for computing the logarithmic sine function without branch cuts away from the real line.
- Added arb_poly_cot_pi_series, acb_poly_cot_pi_series.
- Added arf_root and improved speed of arb_root.
- Tuned algorithm selection in arb_pow_fmpq.
- Other
- Added documentation for arb and acb vector functions.

## 2015-04-19 - version 2.6.0¶

- Special functions
- Added the Bessel K function.
- Added the confluent hypergeometric functions M and U.
- Added exponential, trigonometric and logarithmic integrals ei, si, shi, ci, chi, li.
- Added the complete elliptic integral of the second kind E.
- Added support for computing hypergeometric functions with power series as parameters.
- Fixed special cases in Bessel J function returning useless output.
- Fixed precision of zeta function accidentally being capped at 7000 digits (bug in 2.5).
- Special-cased real input in the gamma functions for complex types.
- Fixed exp of huge numbers outputting unnecessarily useless intervals.
- Fixed broken code in erf that sometimes gave useless output.
- Made selection of number of terms in hypergeometric series more robust.

- Polynomials and power series.
- Added sin_pi, cos_pi and sin_cos_pi for real and complex power series.
- Speeded up series reciprocal and division for length = 2.
- Added add_si methods for polynomials.
- Made inv_series and div_series with zero input produce indeterminates instead of aborting.
- Added arb_poly_majorant, acb_poly_majorant.

- Basic functions
- Added comparison methods arb_eq, arb_ne, arb_lt, arb_le, arb_gt, arb_ge, acb_eq, acb_ne.
- Added acb_rel_accuracy_bits and improved the real version.
- Fixed precision of constants like pi behaving more nondeterministically than necessary.
- Fixed arf_get_mag_lower(nan) to output 0 instead of inf.

- Other
- Removed call to fmpq_dedekind_sum which only exists in the git version of flint.
- Fixed a test code bug that could cause crashes on some systems.
- Added fix for static build on OS X (thanks Marcello Seri).
- Miscellaneous corrections to the documentation.

## 2015-01-28 - version 2.5.0¶

- String conversion
- Added arb_set_str.
- Added arb_get_str and arb_printn for pretty-printed rigorous decimal output.
- Added helper functions for binary to decimal conversion.

- Core arithmetic
- Improved speed of division when using GMP instead of MPIR.
- Improved complex division with a small denominator.
- Removed a little bit of overhead for complex squaring.

- Special functions
- Faster code for atan at very high precision, used instead of mpfr_atan.
- Optimized elementary functions slightly for small input.
- Added modified error functions erfc and erfi.
- Added the generalized exponential integral.
- Added the upper incomplete gamma function.
- Implemented the complete elliptic integral of the first kind.
- Implemented the arithmetic-geometric mean of complex numbers.
- Optimized arb_digamma for small integers.
- Made mag_log_ui, mag_binpow_uiui and mag_polylog_tail proper functions.
- Added pow, agm, erf, elliptic_k, elliptic_p as functions of complex power series.
- Added incomplete gamma function of complex power series.
- Improved code for bounding complex rising factorials (the old code could potentially have given wrong results in degenerate cases).
- Added arb_sqrt1pm1, arb_atanh, arb_asinh, arb_atanh.
- Added arb_log1p, acb_log1p, acb_atan.
- Added arb_hurwitz_zeta.
- Improved parameter selection in the Hurwitz zeta function to try to avoid stalling when given enormous input.
- Optimized sqrt and rsqrt of power series when given a binomial as input.
- Made arb_bernoulli_ui(2^64-2) not crash.
- Fixed rgamma of negative integers returning indeterminate.

- Polynomials and matrices
- Added characteristic polynomial computation for real and complex matrices.
- Added polynomial set_round methods.
- Added is_real methods for more types.
- Added more get_unique_fmpz methods.
- Added code for generating Swinnerton-Dyer polynomials.
- Improved error bounding in det() and exp() of complex matrices to recognize when the result is real-valued.
- Changed polynomial divrem to return success/fail instead of aborting on divide by zero.

- Miscellaneous
- Added logo to documentation.
- Made inlined functions build as part of the library.
- Silenced a clang warning.
- Made _acb_vec_sort_pretty a library function.

## 2014-11-15 - version 2.4.0¶

- Arithmetic and core functions
- Made evaluation of sin, cos and exp at medium precision faster using the sqrt trick.
- Optimized arb_sinh and arb_sinh_cosh.
- Optimized complex division with a small denominator.
- Optimized cubing of complex numbers.
- Added floor and ceil functions for the arf and arb types.
- Added acb_poly powering functions.
- Added acb_exp_pi_i.
- Added functions for evaluation of Chebyshev polynomials.
- Fixed arb_div to output nan for input containing nan.

- Added a module acb_hypgeom for hypergeometric functions
- Evaluation of the generalized hypergeometric function in convergent cases.
- Evaluation of confluent hypergeometric functions using asymptotic expansions.
- The Bessel function of the first kind for complex input.
- The error function for complex input.

- Added a module acb_modular for modular forms and elliptic functions
- Support for working with modular transformations.
- Mapping a point to the fundamental domain.
- Evaluation of Jacobi theta functions and their series expansions.
- The Dedekind eta function.
- The j-invariant and the modular lambda and delta function.
- Eisenstein series.
- The Weierstrass elliptic function and its series expansion.

- Miscellaneous
- Fixed mag_print printing a too large exponent.
- Fixed printd methods to use a fallback instead of aborting when printing numbers too large for MPFR.
- Added version number string (arb_version).
- Various additions to the documentation.

## 2014-09-25 - version 2.3.0¶

- Removed most of the legacy (Arb 1.x) modules.
- Updated build scripts, hopefully fixing various issues.
- New implementations of arb_sin, arb_cos, arb_sin_cos, arb_atan, arb_log, arb_exp, arb_expm1, much faster up to a few thousand bits.
- Ported the bit-burst code for high-precision exponentials to the arb type.
- Speeded up arb_log_ui_from_prev.
- Added mag_exp, mag_expm1, mag_exp_tail, mag_pow_fmpz.
- Improved various mag functions.
- Added arb_get/set_interval_mpfr, arb_get_interval_arf, and improved arb_set_interval_arf.
- Improved arf_get_fmpz.
- Prettier printing of complex numbers with negative imaginary part.
- Changed some frequently-used functions from inline to non-inline to reduce code size.

## 2014-08-01 - version 2.2.0¶

- Added functions for computing polylogarithms and order expansions of polylogarithms, with support for real and complex s, z.
- Added a missing cast affecting C++ compatibility.
- Generalized powsum functions to allow a geometric factor.
- Improved powsum functions slightly when the exponent is an integer.
- Faster arb_log_ui_from_prev.
- Added mag_sqrt and mag_rsqrt functions.
- Fixed various minor bugs and added missing tests and documentation entries.

## 2014-06-20 - version 2.1.0¶

- Ported most of the remaining functions to the new arb/acb types,
including:
- Elementary functions (log, atan, etc.).
- Hypergeometric series summation.
- The gamma function.
- The Riemann zeta function and related functions.
- Bernoulli numbers.
- The partition function.
- The calculus modules (rigorous real root isolation, rigorous numerical integration of complex-valued functions).
- Example programs.

- Added several missing utility functions to the arf and mag modules.

## 2014-05-27 - version 2.0.0¶

- New modules mag, arf, arb, arb_poly, arb_mat, acb, acb_poly, acb_mat for higher-performance ball arithmetic.
- Poly_roots2 and hilbert_matrix2 example programs.
- Vector dot product and norm functions (contributed by Abhinav Baid).

## 2014-05-03 - version 1.1.0¶

- Faster and more accurate error bounds for polynomial multiplication (error bounds are now always as good as with classical multiplication, and multiplying high-degree polynomials with approximately equal coefficients now has proper quasilinear complexity).
- Faster and much less memory-hungry exponentials at very high precision.
- Improved the partition function to support n bigger than a single word, and enabled the possibility to use two threads for the computation.
- Fixed a bug in floating-point arithmetic that caused a too small bound for the rounding error to be reported when the result of an inexact operation was rounded up to a power of two (this bug did not affect the correctness of ball arithmetic, because operations on ball midpoints always round down).
- Minor optimizations to floating-point arithmetic.
- Improved argument reduction of the digamma function and short series expansions of the rising factorial.
- Removed the holonomic module for now, as it did not really do anything very useful.

## 2013-12-21 - version 1.0.0¶

- New example programs directory
- poly_roots example program.
- real_roots example program.
- pi_digits example program.
- hilbert_matrix example program.
- keiper_li example program.

- New fmprb_calc module for calculus with real functions
- Bisection-based root isolation.
- Asymptotically fast Newton root refinement.

- New fmpcb_calc module for calculus with complex functions
- Numerical integration using Taylor series.

- Scalar functions
- Simplified fmprb_const_euler using published error bound.
- Added fmprb_inv.
- Added fmprb_trim, fmpcb_trim.
- Added fmpcb_rsqrt (complex reciprocal square root).
- Fixed bug in fmprb_sqrtpos with nonfinite input.
- Slightly improved fmprb powering code.
- Added various functions for bounding fmprs by powers of two.
- Added fmpr_is_int.

- Polynomials and power series
- Implemented scaling to speed up blockwise multiplication.
- Slightly faster basecase power series exponentials.
- Improved sin/cos/tan/exp for short power series.
- Added complex sqrt_series, rsqrt_series.
- Implemented the Riemann-Siegel Z and theta functions for real power series.
- Added fmprb_poly_pow_series, fmprb_poly_pow_ui and related methods.
- Added fmprb/fmpcb_poly_contains_fmpz_poly.
- Faster composition by monomials.
- Implemented Borel transform and binomial transform for real power series.

- Matrices
- Implemented matrix exponentials.
- Multithreaded fmprb_mat_mul.
- Added matrix infinity norm functions.
- Added some more matrix-scalar functions.
- Added matrix contains and overlaps methods.

- Zeta function evaluation
- Multithreaded power sum evaluation.
- Faster parameter selection when computing many derivatives.
- Implemented binary splitting to speed up computing many derivatives.

- Miscellaneous
- Corrections for C++ compatibility (contributed by Jonathan Bober).
- Several minor bugfixes and test code enhancements.

## 2013-08-07 - version 0.7¶

- Floating-point and ball functions
- Documented, added test code, and fixed bugs for various operations involving a ball containing an infinity or NaN.
- Added reciprocal square root functions (fmpr_rsqrt, fmprb_rsqrt) based on mpfr_rec_sqrt.
- Faster high-precision division by not computing an explicit remainder.
- Slightly faster computation of pi by using new reciprocal square root and division code.
- Added an fmpr function for approximate division to speed up certain radius operations.
- Added fmpr_set_d for conversion from double.
- Allow use of doubles to optionally compute the partition function faster but without an error bound.
- Bypass mpfr overflow when computing the exponential function to extremely high precision (approximately 1 billion digits).
- Made fmprb_exp faster for large numbers at extremely high precision by skipping the log(2) removal.
- Made fmpcb_lgamma faster at high precision by speeding up the argument reduction branch computation.
- Added fmprb_asin, fmprb_acos.
- Added various other utility functions to the fmprb module.
- Added a function for computing the Glaisher constant.
- Optimized evaluation of the Riemann zeta function at high precision.

- Polynomials and power series
- Made squaring of polynomials faster than generic multiplication.
- Implemented power series reversion (various algorithms) for the fmprb_poly type.
- Added many fmprb_poly utility functions (shifting, truncating, setting/getting coefficients, etc.).
- Improved power series division when either operand is short
- Improved power series logarithm when the input is short.
- Improved power series exponential to use the basecase algorithm for short input regardless of the output size.
- Added power series square root and reciprocal square root.
- Added atan, tan, sin, cos, sin_cos, asin, acos fmprb_poly power series functions.
- Added Newton iteration macros to simplify various functions.
- Added gamma functions of real and complex power series ([fmprb/fmpcb]_poly_[gamma/rgamma/lgamma]_series).
- Added wrappers for computing the Hurwitz zeta function of a power series ([fmprb/fmpcb]_poly_zeta_series).
- Implemented sieving and other optimizations to improve performance for evaluating the zeta function of a short power series.
- Improved power series composition when the inner series is linear.
- Added many fmpcb_poly versions of nearly all fmprb_poly functions.
- Improved speed and stability of series composition/reversion by balancing the power table exponents.

- Other
- Added support for freeing all cached data by calling flint_cleanup().
- Introduced fmprb_ptr, fmprb_srcptr, fmpcb_ptr, fmpcb_srcptr typedefs for cleaner function signatures.
- Various bug fixes and general cleanup.

## 2013-05-31 - version 0.6¶

- Made fast polynomial multiplication over the reals numerically stable by using a blockwise algorithm.
- Disabled default use of the Gauss formula for multiplication of complex polynomials, to improve numerical stability.
- Added division and remainder for complex polynomials.
- Added fast multipoint evaluation and interpolation for complex polynomials.
- Added missing fmprb_poly_sub and fmpcb_poly_sub functions.
- Faster exponentials (fmprb_exp and dependent functions) at low precision, using precomputation.
- Rewrote fmpr_add and fmpr_sub using mpn level code, improving efficiency at low precision.
- Ported the partition function implementation from flint (using ball arithmetic in all steps of the calculation to guarantee correctness).
- Ported algorithm for computing the cosine minimal polynomial from flint (using ball arithmetic to guarantee correctness).
- Support using GMP instead of MPIR.
- Only use thread-local storage when enabled in flint.
- Slightly faster error bounding for the zeta function.
- Added some other helper functions.

## 2013-03-28 - version 0.5¶

- Arithmetic and elementary functions
- Added fmpr_get_fmpz, fmpr_get_si.
- Fixed accuracy problem with fmprb_div_2expm1.
- Special-cased squaring of complex numbers.
- Added various fmpcb convenience functions (addmul_ui, etc).
- Optimized fmpr_cmp_2exp_si and fmpr_cmpabs_2exp_si, and added test code for comparison functions.
- Added fmprb_atan2, also fixing a bug in fmpcb_arg.
- Added fmprb_sin_pi, cos_pi, sin_cos_pi, etc.
- Added fmprb_sin_pi_fmpq (etc.) using algebraic methods for fast evaluation of roots of unity.
- Faster fmprb_poly_evaluate and evaluate_fmpcb using rectangular splitting.
- Added fmprb_poly_evaluate2, evaluate2_fmpcb for simultaneously evaluating the derivative.
- Added fmprb_poly root polishing code using near-optimal Newton steps (experimental).
- Added fmpr_root, fmprb_root (currently based on MPFR).
- Added fmpr_min, fmpr_max.
- Added fmprb_set_interval_fmpr, fmprb_union.
- Added fmpr_bits, fmprb_bits, fmpcb_bits for obtaining the mantissa width.
- Added fmprb_hypot.
- Added complex square roots.
- Improved fmprb_log to slightly improve speed, and properly support huge arguments.
- Fixed exp, cosh, sinh to work with huge arguments.
- Added fmprb_expm1.
- Fixed sin, cos, atan to work with huge arguments.
- Improved fmprb_pow and fmpcb_pow, including automatic detection of small integer and half-integer exponents.
- Added many more elementary functions: fmprb_tan/cot/tanh/coth, fmpcb_tan/cot, and pi versions.
- Added fmprb const_e, const_log2, const_log10, const_catalan.
- Fixed ball containment/overlap checking to work operate efficiently and correctly with huge exponents.
- Strengthened test code for many core operations.

- Special functions
- Reorganized zeta function related code.
- Faster evaluation of the Riemann zeta function via sieving.
- Documented and improved efficiency of the zeta constant binary splitting code.
- Calculate error bound in Borwein’s algorithm with fmprs instead of using doubles.
- Optimized divisions in zeta evaluation via the Euler product.
- Use functional equation for Riemann zeta function of a negative argument.
- Compute single Bernoulli numbers using ball arithmetic instead of relying on the floating-point code in flint.
- Initial code for evaluating the gamma function using its Taylor series.
- Much faster rising factorials at high precision, using difference polynomials.
- Much faster gamma function at high precision.
- Added complex gamma function, log gamma function, and other versions.
- Added fmprb_agm (real arithmetic-geometric mean).
- Added fmprb_gamma_fmpq, supporting rapid computation of gamma(p/q) for q = 1,2,3,4,6.
- Added real and complex digamma function.
- Fixed unnecessary recomputation of Bernoulli numbers.
- Optimized computation of Euler’s constant, and added proper error bounds.
- Avoid reliance on doubles in the hypergeometric series tail bound.
- Cleaned up factorials and binomials, computing factorials via gamma.

- Other
- Added an fmpz_extras module to collect various internal fmpz helper functions.
- Fixed detection of flint header files.
- Fixed various other small bugs.

## 2013-01-26 - version 0.4¶

- Much faster fmpr_mul, fmprb_mul and set_round, resulting in general speed improvements.
- Code for computing the complex Hurwitz zeta function with derivatives.
- Fixed and documented error bounds for hypergeometric series.
- Better algorithm for series evaluation of the gamma function at a rational point.
- Much faster generation of Bernoulli numbers.
- Complex log, exp, pow, trigonometric functions (currently based on MPFR).
- Complex nth roots via Newton iteration.
- Added code for arithmetic on fmpcb_polys.
- Code for computing Khinchin’s constant.
- Code for rising factorials of polynomials or power series
- Faster sin_cos.
- Better div_2expm1.
- Many other new helper functions.
- Improved thread safety.
- More test code for core operations.

## 2012-11-07 - version 0.3¶

- Converted documentation to Sphinx.
- New module fmpcb for ball interval arithmetic over the complex numbers
- Conversions, utility functions and arithmetic operations.

- New module fmpcb_mat for matrices over the complex numbers
- Conversions, utility functions and arithmetic operations.
- Multiplication, LU decomposition, solving, inverse and determinant.

- New module fmpcb_poly for polynomials over the complex numbers
- Root isolation for complex polynomials.

- New module fmpz_holonomic for functions/sequences
defined by linear differential/difference equations
with polynomial coefficients
- Functions for creating various special sequences and functions.
- Some closure properties for sequences.
- Taylor series expansion for differential equations.
- Computing the nth entry of a sequence using binary splitting.
- Computing the nth entry mod p using fast multipoint evaluation.

- Generic binary splitting code with automatic error bounding is now used for evaluating hypergeometric series.
- Matrix powering.
- Various other helper functions.

## 2012-09-29 - version 0.2¶

- Code for computing the gamma function (Karatsuba, Stirling’s series).
- Rising factorials.
- Fast exp_series using Newton iteration.
- Improved multiplication of small polynomials by using classical multiplication.
- Implemented error propagation for square roots.
- Polynomial division (Newton-based).
- Polynomial evaluation (Horner) and composition (divide-and-conquer).
- Product trees, fast multipoint evaluation and interpolation (various algorithms).
- Power series composition (Horner, Brent-Kung).
- Added the fmprb_mat module for matrices of balls of real numbers.
- Matrix multiplication.
- Interval-aware LU decomposition, solving, inverse and determinant.
- Many helper functions and small bugfixes.

## 2012-09-14 - version 0.1¶

- 2012-08-05 - Began simplified rewrite.
- 2012-04-05 - Experimental ball and polynomial code (first commit).