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1 ## Copyright (C) 2000, 2006, 2007 Paul Kienzle |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 3 of the License, or (at |
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8 ## your option) any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {} primes (@var{n}) |
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21 ## |
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22 ## Return all primes up to @var{n}. |
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23 ## |
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24 ## Note that if you need a specific number of primes, you can use the |
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25 ## fact the distance from one prime to the next is on average |
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26 ## proportional to the logarithm of the prime. Integrating, you find |
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27 ## that there are about @math{k} primes less than @math{k \log ( 5 k )}. |
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28 ## |
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29 ## The algorithm used is called the Sieve of Erastothenes. |
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30 ## @end deftypefn |
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31 |
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32 ## Author: Paul Kienzle |
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33 ## Author: Francesco Potort� |
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34 ## Author: Dirk Laurie |
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35 |
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36 function x = primes (p) |
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37 |
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38 if (nargin != 1) |
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39 print_usage (); |
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40 endif |
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41 |
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42 if (! isscalar (p)) |
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43 error ("primes: n must be a scalar"); |
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44 endif |
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45 |
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46 if (p > 100000) |
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47 ## optimization: 1/6 less memory, and much faster (asymptotically) |
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48 ## 100000 happens to be the cross-over point for Paul's machine; |
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49 ## below this the more direct code below is faster. At the limit |
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50 ## of memory in Paul's machine, this saves .7 seconds out of 7 for |
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51 ## p=3e6. Hardly worthwhile, but Dirk reports better numbers. |
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52 lenm = floor ((p+1)/6); # length of the 6n-1 sieve |
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53 lenp = floor ((p-1)/6); # length of the 6n+1 sieve |
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54 sievem = ones (1, lenm); # assume every number of form 6n-1 is prime |
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55 sievep = ones (1, lenp); # assume every number of form 6n+1 is prime |
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56 |
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57 for i = 1:(sqrt(p)+1)/6 # check up to sqrt(p) |
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58 if (sievem(i)) # if i is prime, eliminate multiples of i |
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59 sievem(7*i-1:6*i-1:lenm) = 0; |
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60 sievep(5*i-1:6*i-1:lenp) = 0; |
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61 endif # if i is prime, eliminate multiples of i |
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62 if (sievep(i)) |
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63 sievep(7*i+1:6*i+1:lenp) = 0; |
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64 sievem(5*i+1:6*i+1:lenm) = 0; |
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65 endif |
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66 endfor |
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67 x = sort([2, 3, 6*find(sievem)-1, 6*find(sievep)+1]); |
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68 elseif (p > 352) # nothing magical about 352; just has to be greater than 2 |
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69 len = floor ((p-1)/2); # length of the sieve |
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70 sieve = ones (1, len); # assume every odd number is prime |
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71 for i = 1:(sqrt(p)-1)/2 # check up to sqrt(p) |
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72 if (sieve(i)) # if i is prime, eliminate multiples of i |
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73 sieve(3*i+1:2*i+1:len) = 0; # do it |
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74 endif |
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75 endfor |
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76 x = [2, 1+2*find(sieve)]; # primes remaining after sieve |
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77 else |
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78 a = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ... |
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79 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, ... |
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80 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, ... |
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81 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, ... |
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82 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, ... |
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83 293, 307, 311, 313, 317, 331, 337, 347, 349]; |
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84 x = a(a <= p); |
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85 endif |
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86 |
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87 endfunction |
