Mercurial > forge
diff main/specfun/primes.m @ 0:6b33357c7561 octave-forge
Initial revision
author | pkienzle |
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date | Wed, 10 Oct 2001 19:54:49 +0000 |
parents | |
children | 1f628db16a0d |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/main/specfun/primes.m Wed Oct 10 19:54:49 2001 +0000 @@ -0,0 +1,73 @@ +## Copyright (C) 2000 Paul Kienzle +## +## This program 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 2 of the License, or +## (at your option) any later version. +## +## This program 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 this program; if not, write to the Free Software +## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + +## -*- texinfo -*- +## @deftypefn {Function File} {} primes (@var{n}) +## Return all primes up to @var{n}. +## +## Note that if you need a specific number of primes, you can use the +## fact the distance from one prime to the next is on average +## proportional to the logarithm of the prime. Integrating, you find +## that there are about @math{k} primes less than @math{k \log ( 5 k )}. +## +## The algorithm used is called the Sieve of Erastothenes. +## @end deftypefn + +## Author: Paul Kienzle, Francesco Potort́ and Dirk Laurie + +function x=primes(p) + if nargin != 1 + usage("p = primes(n)"); + endif + if (p > 100000) + ## optimization: 1/6 less memory, and much faster (asymptotically) + ## 100000 happens to be the cross-over point for Paul's machine; + ## below this the more direct code below is faster. At the limit + ## of memory in Paul's machine, this saves .7 seconds out of 7 for + ## p=3e6. Hardly worthwhile, but Dirk reports better numbers. + lenm = floor((p+1)/6); # length of the 6n-1 sieve + lenp = floor((p-1)/6); # length of the 6n+1 sieve + sievem = ones (1, lenm); # assume every number of form 6n-1 is prime + sievep = ones (1, lenp); # assume every number of form 6n+1 is prime + for i=1:(sqrt(p)+1)/6 # check up to sqrt(p) + if (sievem(i)) # if i is prime, eliminate multiples of i + sievem(7*i-1:6*i-1:lenm) = 0; + sievep(5*i-1:6*i-1:lenp) = 0; + endif # if i is prime, eliminate multiples of i + if (sievep(i)) + sievep(7*i+1:6*i+1:lenp) = 0; + sievem(5*i+1:6*i+1:lenm) = 0; + endif + endfor + x = sort([2, 3, 6*find(sievem)-1, 6*find(sievep)+1]); + elseif (p > 352) # nothing magical about 352; just has to be greater than 2 + len = floor((p-1)/2); # length of the sieve + sieve = ones (1, len); # assume every odd number is prime + for i=1:(sqrt(p)-1)/2 # check up to sqrt(p) + if (sieve(i)) # if i is prime, eliminate multiples of i + sieve(3*i+1:2*i+1:len) = 0; # do it + endif + endfor + x = [2, 1+2*find(sieve)]; # primes remaining after sieve + else + a=[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, 101, 103, 107, 109, 113, 127, \ + 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193,\ + 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269,\ + 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349]; + x = x (x<=p); + endif +endfunction