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[CreaPhase.git] / octave_packages / m / specfun / primes.m

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+## Copyright (C) 2000-2012 Paul Kienzle
+##
+## 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
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {} primes (@var{n})
+##
+## Return all primes up to @var{n}.
+##
+## The algorithm used is the Sieve of Eratosthenes.
+##
+## Note that if you need a specific number of primes you can use the
+## fact that the distance from one prime to the next is, on average,
+## proportional to the logarithm of the prime.  Integrating, one finds
+## that there are about @math{k} primes less than
+## @tex
+## $k \log (5 k)$.
+## @end tex
+## @ifnottex
+## k*log(5*k).
+## @end ifnottex
+## @seealso{list_primes, isprime}
+## @end deftypefn
+
+## Author: Paul Kienzle
+## Author: Francesco Potortì
+## Author: Dirk Laurie
+
+function x = primes (n)
+
+  if (nargin != 1)
+    print_usage ();
+  endif
+
+  if (! isscalar (n))
+    error ("primes: N must be a scalar");
+  endif
+
+  if (n > 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
+    ## n = 3e6.  Hardly worthwhile, but Dirk reports better numbers.
+    lenm = floor ((n+1)/6);       # length of the 6n-1 sieve
+    lenp = floor ((n-1)/6);       # length of the 6n+1 sieve
+    sievem = true (1, lenm);      # assume every number of form 6n-1 is prime
+    sievep = true (1, lenp);      # assume every number of form 6n+1 is prime
+
+    for i = 1:(sqrt(n)+1)/6       # check up to sqrt(n)
+      if (sievem(i))              # if i is prime, eliminate multiples of i
+        sievem(7*i-1:6*i-1:lenm) = false;
+        sievep(5*i-1:6*i-1:lenp) = false;
+      endif                       # if i is prime, eliminate multiples of i
+      if (sievep(i))
+        sievep(7*i+1:6*i+1:lenp) = false;
+        sievem(5*i+1:6*i+1:lenm) = false;
+      endif
+    endfor
+    x = sort([2, 3, 6*find(sievem)-1, 6*find(sievep)+1]);
+  elseif (n > 352)                # nothing magical about 352; must be >2
+    len = floor ((n-1)/2);        # length of the sieve
+    sieve = true (1, len);        # assume every odd number is prime
+    for i = 1:(sqrt(n)-1)/2       # check up to sqrt(n)
+      if (sieve(i))               # if i is prime, eliminate multiples of i
+        sieve(3*i+1:2*i+1:len) = false; # 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 = a(a <= n);
+  endif
+
+endfunction
+
+%!error primes ();
+%!error primes (1, 2);
+
+%!assert (size (primes (350)), [1, 70]);
+%!assert (size (primes (350)), [1, 70]);
+
+%!assert (primes (357)(end), 353);