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annotate scripts/specfun/primes.m @ 10549:95c3e38098bf
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| author | Rik <code@nomad.inbox5.com> |
|---|---|
| date | Fri, 23 Apr 2010 11:28:50 -0700 |
| parents | 8c71a86c4bf4 |
| children | c6833d31f34e |
| rev | line source |
|---|---|
| 8920 | 1 ## Copyright (C) 2000, 2006, 2007, 2009 Paul Kienzle |
| 5827 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 5 ## Octave is free software; you can redistribute it and/or modify it | |
| 6 ## under the terms of the GNU General Public License as published by | |
| 7016 | 7 ## the Free Software Foundation; either version 3 of the License, or (at |
| 8 ## your option) any later version. | |
| 5827 | 9 ## |
| 10 ## Octave is distributed in the hope that it will be useful, but | |
| 11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
| 12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
| 13 ## General Public License for more details. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 7016 | 16 ## along with Octave; see the file COPYING. If not, see |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 5827 | 18 |
| 19 ## -*- texinfo -*- | |
| 20 ## @deftypefn {Function File} {} primes (@var{n}) | |
| 21 ## | |
| 22 ## Return all primes up to @var{n}. | |
| 23 ## | |
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24 ## The algorithm used is the Sieve of Erastothenes. |
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25 ## |
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26 ## Note that if you need a specific number of primes you can use the |
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27 ## fact the distance from one prime to the next is, on average, |
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28 ## proportional to the logarithm of the prime. Integrating, one finds |
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29 ## that there are about @math{k} primes less than |
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30 ## @tex |
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31 ## $k \log (5 k)$. |
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32 ## @end tex |
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33 ## @ifnottex |
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34 ## k*log(5*k). |
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35 ## @end ifnottex |
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36 ## @seealso{list_primes, isprime} |
| 5827 | 37 ## @end deftypefn |
| 38 | |
| 39 ## Author: Paul Kienzle | |
| 40 ## Author: Francesco Potort� | |
| 41 ## Author: Dirk Laurie | |
| 42 | |
| 43 function x = primes (p) | |
| 44 | |
| 45 if (nargin != 1) | |
| 46 print_usage (); | |
| 47 endif | |
| 48 | |
| 49 if (! isscalar (p)) | |
| 50 error ("primes: n must be a scalar"); | |
| 51 endif | |
| 52 | |
| 53 if (p > 100000) | |
| 8506 | 54 ## Optimization: 1/6 less memory, and much faster (asymptotically) |
| 5827 | 55 ## 100000 happens to be the cross-over point for Paul's machine; |
| 56 ## below this the more direct code below is faster. At the limit | |
| 57 ## of memory in Paul's machine, this saves .7 seconds out of 7 for | |
| 8507 | 58 ## p = 3e6. Hardly worthwhile, but Dirk reports better numbers. |
| 5827 | 59 lenm = floor ((p+1)/6); # length of the 6n-1 sieve |
| 60 lenp = floor ((p-1)/6); # length of the 6n+1 sieve | |
| 61 sievem = ones (1, lenm); # assume every number of form 6n-1 is prime | |
| 62 sievep = ones (1, lenp); # assume every number of form 6n+1 is prime | |
| 63 | |
| 8506 | 64 for i = 1:(sqrt(p)+1)/6 # check up to sqrt(p) |
| 5827 | 65 if (sievem(i)) # if i is prime, eliminate multiples of i |
| 66 sievem(7*i-1:6*i-1:lenm) = 0; | |
| 67 sievep(5*i-1:6*i-1:lenp) = 0; | |
| 68 endif # if i is prime, eliminate multiples of i | |
| 69 if (sievep(i)) | |
| 70 sievep(7*i+1:6*i+1:lenp) = 0; | |
| 71 sievem(5*i+1:6*i+1:lenm) = 0; | |
| 72 endif | |
| 73 endfor | |
| 74 x = sort([2, 3, 6*find(sievem)-1, 6*find(sievep)+1]); | |
| 8506 | 75 elseif (p > 352) # nothing magical about 352; must be >2 |
| 5827 | 76 len = floor ((p-1)/2); # length of the sieve |
| 77 sieve = ones (1, len); # assume every odd number is prime | |
| 78 for i = 1:(sqrt(p)-1)/2 # check up to sqrt(p) | |
| 79 if (sieve(i)) # if i is prime, eliminate multiples of i | |
| 80 sieve(3*i+1:2*i+1:len) = 0; # do it | |
| 81 endif | |
| 82 endfor | |
| 83 x = [2, 1+2*find(sieve)]; # primes remaining after sieve | |
| 84 else | |
| 85 a = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ... | |
| 10549 | 86 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, ... |
| 87 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, ... | |
| 88 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, ... | |
| 89 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, ... | |
| 90 293, 307, 311, 313, 317, 331, 337, 347, 349]; | |
| 5827 | 91 x = a(a <= p); |
| 92 endif | |
| 93 | |
| 94 endfunction |