######################################################################## ## ## Copyright (C) 2000-2021 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or . ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## . ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {@var{pf} =} factor (@var{q}) ## @deftypefnx {} {[@var{pf}, @var{n}] =} factor (@var{q}) ## Return the prime factorization of @var{q}. ## ## The prime factorization is defined as @code{prod (@var{pf}) == @var{q}} ## where every element of @var{pf} is a prime number. If @code{@var{q} == 1}, ## return 1. The output @var{pf} is of the same numeric class as the input. ## ## With two output arguments, return the unique prime factors @var{pf} and ## their multiplicities. That is, ## @code{prod (@var{pf} .^ @var{n}) == @var{q}}. ## ## Implementation Note: The input @var{q} must be less than @code{flintmax} ## when the input is a floating point class (double or single). ## @seealso{gcd, lcm, isprime, primes} ## @end deftypefn function [pf, n] = factor (q) if (nargin < 1) print_usage (); endif if (! isscalar (q) || ! isreal (q) || q < 0 || q != fix (q)) error ("factor: Q must be a real non-negative integer"); endif ## Special case of no primes less than sqrt (q). if (q < 4) pf = q; n = 1; return; endif cls = class (q); # store class if (isfloat (q) && q > flintmax (q)) error ("factor: Q too large to factor (> flintmax)"); endif # The basic idea is to divide by the prime numbers from 1 to sqrt(q). # But primes(sqrt(q)) can be very time-consuming to compute for q > 1e16, # so we divide by smaller primes first. # This won't make a difference for prime q, but it makes a big (100x) difference for large composite q. # Since there are many more composites than primes, this leads overall to a speedup. ## There is at most one prime greater than sqrt(q), and if it exists, ## it has multiplicity 1, so no need to consider any factors greater ## than sqrt(q) directly. [If there were two factors p1, p2 > sqrt(q), ## then q >= p1*p2 > sqrt(q)*sqrt(q) == q. Contradiction.] # The following calculation of transition and number of divisors to use was detemined empirically. # As of 2021 it gives the best overall performance over the range of 1 <= q <= intmax ("uint64"). # For future programmers, check periodically for performance improvements and tune this transition as required. if (q < 10e9) # sufficiently small q smallprimes = feval (cls, [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97]); # going up to primes(100) is 3% faster than going up to primes(30) even for the very small q < 1e6. # increases to 7% faster for 1e9 < q < 10e9. # smallprimes(smallprimes > q) = []; # only relevant for really small q, unnecessary and slows down for large q. else # sufficiently large q smallprimes = primes (feval (cls, q^0.2)); # go up to the 5th root of q for now. end # Trials that didn't yield success in 2021: # persistent smallprimes = primes (FOO) for various fixed FOO in the range 10 <= FOO <= 10e6. (FOO is independent of q) # The thought had been that making it persistent would cache it so it didn't need to be recomputed for subsequent calls, # but it slowed it down overall. It seems calling primes twice with smaller q is still faster than one persistent call for a large q. # # Also tried variants with: # smallprimes = primes (q ^ FOO) for various values of FOO. # For FOO >= 0.25 or FOO <= 0.16, the performance is very poor. # FOO needs to be in the 0.17 to 0.24 range, somewhat. # Benchmark experiments indicate it should increase gently from 0.18 to 0.21 as q goes from 10^11 to 10^18. # But putting in such an expression would require calculating the log of q, which defeats any performance improvement. # Or a stepwise approximation like: # foo = 0.18 + 0.01 * (q > 1e12) + 0.01 * (q > 1e14) + 0.01 * (q > 1e16); # smallprimes = primes (feval (cls, q^foo)); # where the RHS of foo would go from 0.18 to 0.21 over several orders of magnitude without calling the log. # Obviously that is overly empirical, so putting in q^0.2 seems to be the most robust overall for 64-bit q. pf = feval (cls, []); # pf will be the list of prime factors returned. We cast it to be the same class as the input. [pf, q] = reducefactors (q, pf, smallprimes); # pf will now contain all prime factors of q within smallprimes, including repetitions, in ascending order. # q itself will be divided by those prime factors to become smaller, unless q was prime to begin with. # Now we go all the way to sqrt(q), where q is smaller than the original q in most cases. largeprimes = primes(sqrt(q)); # do not try to weed out the smallprimes inside largeprimes, whether using length(smallprimes) or max(smallprimes) -- it slows it down! [pf, q] = reducefactors (q, pf, largeprimes); # at this point, all prime factors <= the sqrt of the original q have been pulled out in ascending order. # if q = 1, then no further primes are left. # if q > 1, then q itself must be prime, and it must be the single prime factor that was larger than the sqrt of the original q. if (q > 1) pf(end+1) = q; endif # at this point, all prime factors have been pulled out of q in ascending order, # so no need to sort(pf). ## Determine multiplicity. if (nargout > 1) idx = find ([0, pf] != [pf, 0]); pf = pf(idx(1:length (idx)-1)); n = diff (idx); endif endfunction function [pf, q] = reducefactors (qin, pfin, divisors) pf = pfin; q = qin; divisors = divisors (mod (q, divisors) == 0); # a few milliseconds faster than "divisors(mod(q,divisors) ~= 0) = []" for pp = divisors # for each factor in turn, while (mod (q, pp) == 0) # keep extracting all occurrences of that factor pf(end+1) = pp; # before going to larger factors q /= pp; # mod() was marginally faster than rem() when assessed over 10e6 trials of the whole factor() function. endwhile endfor endfunction ## Test special case input %!assert (factor (1), 1) %!assert (factor (2), 2) %!assert (factor (3), 3) %!test %! for i = 2:20 %! pf = factor (i); %! assert (prod (pf), i); %! assert (all (isprime (pf))); %! [pf, n] = factor (i); %! assert (prod (pf.^n), i); %! assert (all ([0,pf] != [pf,0])); %! endfor %!assert (factor (uint8 (8)), uint8 ([2 2 2])) %!assert (factor (single (8)), single ([2 2 2])) %!test %! [pf, n] = factor (int16 (8)); %! assert (pf, int16 (2)); %! assert (n, double (3)); ## Test input validation %!error factor () %!error factor ([1,2]) %!error factor (6i) %!error factor (-20) %!error factor (1.5) %!error factor (flintmax ("single") + 2) %!error factor (flintmax ("double") + 2)