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1 ## Copyright (C) 2000 Paul Kienzle |
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2 ## |
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3 ## This program is free software; you can redistribute it and/or modify |
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4 ## it under the terms of the GNU General Public License as published by |
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5 ## the Free Software Foundation; either version 2 of the License, or |
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6 ## (at your option) any later version. |
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7 ## |
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8 ## This program is distributed in the hope that it will be useful, |
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9 ## but WITHOUT ANY WARRANTY; without even the implied warranty of |
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10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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11 ## GNU General Public License for more details. |
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12 ## |
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13 ## You should have received a copy of the GNU General Public License |
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14 ## along with this program; if not, write to the Free Software |
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15 ## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA |
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16 |
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17 ## -*- texinfo -*- |
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18 ## @deftypefn {Function File} {} primes (@var{n}) |
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19 ## Return all primes up to @var{n}. |
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20 ## |
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21 ## Note that if you need a specific number of primes, you can use the |
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22 ## fact the distance from one prime to the next is on average |
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23 ## proportional to the logarithm of the prime. Integrating, you find |
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24 ## that there are about @math{k} primes less than @math{k \log ( 5 k )}. |
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25 ## |
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26 ## The algorithm used is called the Sieve of Erastothenes. |
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27 ## @end deftypefn |
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28 |
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29 ## Author: Paul Kienzle, Francesco Potort� and Dirk Laurie |
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30 |
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31 function x=primes(p) |
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32 if nargin != 1 |
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33 usage("p = primes(n)"); |
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34 endif |
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35 if (p > 100000) |
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36 ## optimization: 1/6 less memory, and much faster (asymptotically) |
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37 ## 100000 happens to be the cross-over point for Paul's machine; |
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38 ## below this the more direct code below is faster. At the limit |
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39 ## of memory in Paul's machine, this saves .7 seconds out of 7 for |
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40 ## p=3e6. Hardly worthwhile, but Dirk reports better numbers. |
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41 lenm = floor((p+1)/6); # length of the 6n-1 sieve |
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42 lenp = floor((p-1)/6); # length of the 6n+1 sieve |
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43 sievem = ones (1, lenm); # assume every number of form 6n-1 is prime |
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44 sievep = ones (1, lenp); # assume every number of form 6n+1 is prime |
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45 for i=1:(sqrt(p)+1)/6 # check up to sqrt(p) |
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46 if (sievem(i)) # if i is prime, eliminate multiples of i |
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47 sievem(7*i-1:6*i-1:lenm) = 0; |
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48 sievep(5*i-1:6*i-1:lenp) = 0; |
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49 endif # if i is prime, eliminate multiples of i |
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50 if (sievep(i)) |
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51 sievep(7*i+1:6*i+1:lenp) = 0; |
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52 sievem(5*i+1:6*i+1:lenm) = 0; |
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53 endif |
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54 endfor |
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55 x = sort([2, 3, 6*find(sievem)-1, 6*find(sievep)+1]); |
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56 elseif (p > 352) # nothing magical about 352; just has to be greater than 2 |
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57 len = floor((p-1)/2); # length of the sieve |
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58 sieve = ones (1, len); # assume every odd number is prime |
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59 for i=1:(sqrt(p)-1)/2 # check up to sqrt(p) |
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60 if (sieve(i)) # if i is prime, eliminate multiples of i |
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61 sieve(3*i+1:2*i+1:len) = 0; # do it |
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62 endif |
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63 endfor |
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64 x = [2, 1+2*find(sieve)]; # primes remaining after sieve |
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65 else |
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66 a=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, \ |
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67 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, \ |
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68 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193,\ |
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69 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269,\ |
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70 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349]; |
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71 x = x (x<=p); |
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72 endif |
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73 endfunction |
