Mercurial > octave
view scripts/specfun/factor.m @ 31548:c8ad083a5802 stable
maint: Clean up m-files before Octave 8.1 release.
* external.txi, oop.txi, Table.h, documentation.cc, gui-preferences-ed.h,
lo-specfun.cc, range.tst : Eliminate triple newlines.
* Map.m, MemoizedFunction.m, delaunayn.m, inputParser.m,
__publish_latex_output__.m, publish.m, unpack.m, fminbnd.m,
__add_default_menu__.m, gammainc.m, gallery.m, hadamard.m, weboptions.m:
Add newline after keyword "function" or before keyword "endfunction" for
readability.
* getaudiodata.m, pkg.m : Add semicolon to end of line for error() statement.
* movegui.m: Combine mutliple calls to set() into one for performance.
* __unimplemented__.m (missing_functions): Remove missing functions that have
been implemented.
* __vectorize__.m, check_default_input.m, betaincinv.m, gammaincinv.m:
Remove semicolon at end of line with "function" declaration.
* weboptions.m: Remove semicolon at end of line with "if" keyword.
* integrate_adaptive.m, factor.m: Use keyword "endif" rather than bare "end".
author | Rik <rik@octave.org> |
---|---|
date | Fri, 25 Nov 2022 21:23:54 -0800 |
parents | 658cce403bc7 |
children | 597f3ee61a48 |
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######################################################################## ## ## Copyright (C) 2000-2022 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## 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 ## <https://www.gnu.org/licenses/>. ## ######################################################################## ## -*- 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: If the input @var{q} is @code{single} or @code{double}, ## then it must not exceed the corresponding @code{flintmax}. For larger ## inputs, cast them to @code{uint64} if they're less than 2^64: ## ## @example ## @group ## factor (uint64 (18446744073709011493)) ## @result{} 571111 761213 42431951 ## @end group ## @end example ## ## For even larger inputs, use @code{sym} if you have the Symbolic package ## installed and loaded: ## ## @example ## @group ## factor (sym ('9444733049654361449941')) ## @result{} (sym) ## 1 1 ## 1099511627689 ⋅8589934669 ## @end group ## @end example ## @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 if q is prime, because isprime() is now much faster than ## factor(). This also absorbs the case of q < 4, where there are no primes ## less than sqrt(q). if (q < 4 || isprime (q)) pf = q; n = 1; return; endif ## If we are here, then q is composite. cls = class (q); # store class if (isfloat (q) && q > flintmax (q)) error ("factor: Q too large to factor (> flintmax)"); endif ## The overall flow is this: ## 1. Divide by small primes smaller than q^0.2, if any. ## 2. Use Pollard Rho to reduce the value below 1e10 if possible. ## 3. Divide by primes smaller than sqrt (q), if any. ## 4. At all stages, stop if the remaining value is prime. ## First divide by primes (q ^ 0.2). ## For q < 1e10, we can hard-code the primes. if (q < 1e10) 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]); else smallprimes = primes (feval (cls, q ^ 0.2)); endif ## pf is the list of prime factors returned with type of input class. pf = feval (cls, []); [pf, q] = reducefactors (q, pf, smallprimes); ## pf now contains all prime factors of q within smallprimes, including ## repetitions, in ascending order. ## ## q itself has been divided by those prime factors to become smaller, ## unless q was prime to begin with. sortflag = false; if (isprime (q)) pf(end+1) = q; elseif (q > 1) ## Use Pollard Rho technique to pull factors one at a time. while (q > 1e10 && ! isprime (q)) pr = feval (cls, __pollardrho__ (q)); # pr is a factor of q. ## There is a small chance (13 in 1e5) that pr is not actually prime. ## To guard against that, factorize pr, which will force smaller factors ## to be found. The use of isprime above guards against infinite ## recursion. if (! isprime (pr)) pr = factor (pr); endif [pf, q] = reducefactors (q, pf, pr); ## q is now divided by all occurrences of factor(s) pr. sortflag = true; endwhile if (isprime (q)) pf(end+1) = q; elseif (q > 1) ## If we are here, then q is composite but less than 1e10, ## and that is fast enough to test by division. largeprimes = primes (feval (cls, sqrt (q))); [pf, q] = reducefactors (q, pf, largeprimes); ## If q is still not 1, then it must be a prime of power 1. if (q > 1) pf(end+1) = q; endif endif endif ## The Pollard Rho technique can give factors in arbitrary order, ## so we need to sort pf if that was used. if (sortflag) pf = sort (pf); endif ## 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); for pp = divisors # for each factor in turn ## Keep extracting all occurrences of that factor before going to larger ## factors. while (mod (q, pp) == 0) pf(end+1) = pp; q /= pp; 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 ## Make sure that all factors returned are indeed prime, even when ## __pollardrho__ returns a composite factor. %!assert (all (isprime (factor (uint64 (18446744073707633197))))) %!assert (all (isprime (factor (uint64 (18446744073707551733))))) %!assert (all (isprime (factor (uint64 (18446744073709427857))))) %!assert (all (isprime (factor (uint64 (18446744073709396891))))) %!assert (all (isprime (factor (uint64 (18446744073708666563))))) %!assert (all (isprime (factor (uint64 (18446744073708532009))))) %!assert (all (isprime (factor (uint64 (18446744073708054211))))) %!assert (all (isprime (factor (uint64 (18446744073707834741))))) %!assert (all (isprime (factor (uint64 (18446744073707298053))))) %!assert (all (isprime (factor (uint64 (18446744073709407383))))) %!assert (all (isprime (factor (uint64 (18446744073708730121))))) %!assert (all (isprime (factor (uint64 (18446744073708104447))))) %!assert (all (isprime (factor (uint64 (18446744073709011493))))) %!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 <Invalid call> factor () %!error <Q must be a real non-negative integer> factor ([1,2]) %!error <Q must be a real non-negative integer> factor (6i) %!error <Q must be a real non-negative integer> factor (-20) %!error <Q must be a real non-negative integer> factor (1.5) %!error <Q too large to factor> factor (flintmax ("single") + 2) %!error <Q too large to factor> factor (flintmax ("double") + 2)