mirror of
https://github.com/vale981/arb
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102 lines
3.9 KiB
Text
102 lines
3.9 KiB
Text
Core arithmetic
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-------------------------------------------------------------------------------
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* Consider changing the interface of functions such as X_set_Y, X_neg_Y
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to always take a precision parameter (and get rid of X_set_round_Y,
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X_neg_Y etc.). Perhaps have X_setexact_Y methods for convenience,
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or make an exception for _set_ in particular.
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* Make sure that excessive shifts in add/sub are detected
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with exact precision. Write tests for correctness of overlaps/contains
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in huge-exponent cases.
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* Double-check correctness of add/sub code with large shifts (rounding x+eps).
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* Work out semantics for comparisons/overlap/containment checks
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when NaNs are involved, and write test code.
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* Native string conversion code instead of relying on mpfr (so we can have
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big exponents, etc.).
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* Add functions for sloppy arithmetic (non-exact rounding). This could be
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used to speed up some ball operations with inexact output, where we don't
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need the best possible result, just a correct error bound.
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* Write functions that ignore the possibility that exponents might be
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large, and use where appropriate (e.g. polynomial and matrix multiplication
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where one bounds magnitudes in an initial pass).
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* Rewrite fmprb_div (similar to fmprb_mul)
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Polynomial and power series arithmetic
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-------------------------------------------------------------------------------
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* Verify that mullow and power series methods always truncate the inputs to
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length n.
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* Handle all input of special form ax^n + b quickly in composition and powering.
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* Implement the addition and convolution methods for Taylor shifts.
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* Add polynomial mulmid, and use in Newton iteration
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* Tune basecase/Newton selection for exp/sin/cos series (the basecase
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algorithms are more stable, and faster for quite large n)
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* Look at using the exponential to compute the complex sine/cosine series
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* Improve block multiplication, e.g. by discarding blocks that don't contribute
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to the result, and scaling individual blocks.
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Elementary functions
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-------------------------------------------------------------------------------
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* Add more transcendental functions.
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* Double-check error bounds used in the fixed-point exponential code
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* Faster elementary functions at low precision (especially log/arctan).
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Use Brent's algorithm (http://maths-people.anu.edu.au/~brent/pd/RNC7t4.pdf):
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atan(x) = atan(p/q) + atan((q*x-p)/(q+p*x))
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* Use the complex Newton iteration for cos(pi p/q) when appropriate.
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Double check the proof of correctness of the complex Newton iteration
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and make it work when the polynomial is not exact.
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* For small cos(pi p/q) and sin(pi p/q) use a lookup table of the
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1/q values and then do complex binary exponentiation.
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* Investigate using Chebyshev polynomials for elefun_cos_minpoly.
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This is certainly faster when n is prime, but might be faster for all n,
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at least if implemented cleverly.
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Special functions
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-------------------------------------------------------------------------------
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* Write a faster logarithmic rising factorial (with correct branch
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cuts) for reducing the complex log gamma function. Also implement
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the logarithmic reflection formula.
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* Tune zeta parameter selection. Also make it robust: for example,
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if FMPRB_RAD_PREC is changed to 2 (a testing tactic suggested
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by Paul Zimmermann), the parameter selection code currently hangs
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in an infinite loop for many inputs.
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* Extend Stirling series code to compute polygamma functions (i.e. starting
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the series from some derivative), and optimize for a small number of
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derivatives by using a direct recurrence instead of binary splitting.
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* Fall back to the real code when evaluating gamma functions (or their
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power series) at points that happen to be real
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* Implement more functions: error functions, Bessel functions,
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theta functions, etc.
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Other
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-------------------------------------------------------------------------------
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* Document fmpz_extras
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