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Arb 0.5

March 28, 2013

After a busy February and March, I’ve finally tagged Arb version 0.5. A summary of the changes:

• 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

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