From Thomas Henlich on 20 February 2020: Implement the cotangent function. From Joseph Myers 12 Apr 2015: https://sympa.inria.fr/sympa/arc/mpc-discuss/2015-04/msg00009.html Try implementing tan z = (sin 2x + i sinh 2y) / (cos 2x + cosh 2y) or (sin(x)*cos(x) + i*sinh(y)*cosh(y))/(cos(x)^2 + sinh(y)^2) as in glibc. From Karim Belabas 9 Jan 2014: Implement Hurwitz(s,x) -> gives Zeta for x=1. Cf http://arxiv.org/abs/1309.2877 From Andreas Enge 27 August 2012: Implement im(atan(x+i*y)) as 1/4 * [log1p (4y / (x^2 +(1-y)^2))] (see https://sympa.inria.fr/sympa/arc/mpc-discuss/2012-08/msg00002.html) From Andreas Enge 23 July 2012: go through tests and move them to the data files if possible (see, for instance, tcos.c) From Andreas Enge 31 August 2011: implement mul_karatsuba with three multiplications at precision around p, instead of two at precision 2*p and one at precision p requires analysis of error propagation From Andreas Enge 1 December 2022: think about, implement and document the possibility of having signed zeros as real and imaginary parts of results of multiplication. We might follow IEEE 754-2019, 3rd paragraph from section 6.3: "When the sum of two operands with opposite signs (or the difference of two operands with like signs) is exactly zero, the sign of that sum (or difference) shall be +0 under all rounding-direction attributes except roundTowardNegative; under that attribute, the sign of an exact zero sum (or difference) shall be -0. However, under all rounding-direction attributes, when x is zero, x + x and x - (-x) have the sign of x." From Andreas Enge and Paul Zimmermann 6 July 2012: Improve speed of Im (atan) for x+i*y with small y, for instance by using the Taylor series directly. See also the discussion https://sympa.inria.fr/sympa/arc/mpc-discuss/2012-08/msg00002.html and the timing program on https://sympa.inria.fr/sympa/arc/mpc-discuss/2013-08/msg00005.html For example with Sage 5.11: sage: %timeit atan(MPComplexField()(1,1)) 10000 loops, best of 3: 42.2 us per loop sage: %timeit atan(MPComplexField()(1,1e-1000)) 100 loops, best of 3: 5.29 ms per loop Same for asin: sage: %timeit asin(MPComplexField()(1,1)) 10000 loops, best of 3: 83.7 us per loop sage: %timeit asin(MPComplexField()(1,1e-1000)) 100 loops, best of 3: 17 ms per loop -> should be much faster with revision 1402 (check) Same for acos: sage: %timeit acos(MPComplexField()(1,1)) 10000 loops, best of 3: 90.8 us per loop sage: %timeit acos(MPComplexField()(1,1e-1000)) 1 loops, best of 3: 2.29 s per loop Same for asinh: sage: %timeit asinh(MPComplexField()(1,1)) 10000 loops, best of 3: 84 us per loop sage: %timeit asinh(MPComplexField()(1,1e-1000)) 100 loops, best of 3: 2.1 ms per loop sage: %timeit acosh(MPComplexField()(1,1)) 10000 loops, best of 3: 92 us per loop sage: %timeit acosh(MPComplexField()(1,1e-1000)) 1 loops, best of 3: 2.28 s per loop Bench: - from Andreas Enge 9 June 2009: Scripts and web page comparing timings with different systems, as done for mpfr at http://www.mpfr.org/mpfr-2.4.0/timings.html New functions to implement: - from Joseph S. Myers <joseph at codesourcery dot com> 19 Mar 2012: mpc_erf, mpc_erfc, mpc_exp2, mpc_expm1, mpc_log1p, mpc_log2, mpc_lgamma, mpc_tgamma https://sympa.inria.fr/sympa/arc/mpc-discuss/2012-03/msg00009.html See the article by Pascal Molin (hal.archives-ouvertes.fr/hal-00580855). - implement a root-finding algorithm using the Durand-Kerner method (cf http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method). See also the CEVAL algorithm from Yap and Sagraloff: http://www.mpi-inf.mpg.de/~msagralo/ceval.pdf A good starting point for the Durand-Kerner and Aberth methods is the paper by Dario Bini "Numerical computation of polynomial zeros by means of Aberth's method", Numerical Algorithms 13 (1996), 179-200. New tests to add: - from Andreas Enge and Philippe Théveny 9 April 2008 correct handling of Nan and infinities in the case of intermediate overflows while the result may fit (we need special code)
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