FYI, log gamma is another fast way to calculate the number of combinations if you want to deal with really big numbers.
On Tue, Feb 19, 2013 at 7:28 PM, Joe Gilray <jgil...@gmail.com> wrote: > Racketeers, > > Thanks for putting together the fantastic math library. It will be a > wonderful resource. Here are some quick impressions (after playing mostly > with math/number-theory) > > 1) The functions passed all my tests and were very fast. If you need even > more speed you can keep a list of primes around and write functions to use > that, but that should be rarely necessary > > 2) I have a couple of functions to donate if you want them: > > 2a) Probablistic primality test: > > ; function that performs a Miller-Rabin probabalistic primality test k > times, returns #t if n is probably prime > ; algorithm from http://rosettacode.org/wiki/Miller-Rabin_primality_test, > code adapted from Lisp example > ; (module+ test (check-equal? (is-mr-prime? 1000000000000037 8) #t)) > (define (is-mr-prime? n k) > ; function that returns two values r and e such that number = divisor^e > * r, and r is not divisible by divisor > (define (factor-out number divisor) > (do ([e 0 (add1 e)] [r number (/ r divisor)]) > ((not (zero? (remainder r divisor))) (values r e)))) > > ; function that performs fast modular exponentiation by repeated squaring > (define (expt-mod base exponent modulus) > (let expt-mod-iter ([b base] [e exponent] [p 1]) > (cond > [(zero? e) p] > [(even? e) (expt-mod-iter (modulo (* b b) modulus) (/ e 2) p)] > [else (expt-mod-iter b (sub1 e) (modulo (* b p) modulus))]))) > > ; function to return a random, exact number in the passed range > (inclusive) > (define (shifted-rand lower upper) > (+ lower (random (add1 (- (modulo upper 4294967088) (modulo lower > 4294967088)))))) > > (cond > [(= n 1) #f] > [(< n 4) #t] > [(even? n) #f] > [else > (let-values ([(d s) (factor-out (- n 1) 2)]) ; represent n-1 as 2^s-d > (let lp ([a (shifted-rand 2 (- n 2))] [cnt k]) > (if (zero? cnt) #t > (let ([x (expt-mod a d n)]) > (if (or (= x 1) (= x (sub1 n))) (lp (shifted-rand 2 (- n > 2)) (sub1 cnt)) > (let ctestlp ([r 1] [ctest (modulo (* x x) n)]) > (cond > [(>= r s) #f] > [(= ctest 1) #f] > [(= ctest (sub1 n)) (lp (shifted-rand 2 (- n 2)) > (sub1 cnt))] > [else (ctestlp (add1 r) (modulo (* ctest ctest) > n))])))))))])) > > 2b) combinations calculator > > ; function that returns the number of combinations, not the combinations > themselves > ; faster than using n! / (r! (n-r)!) > (define (combinations n r) > (cond > [(or (< n 0) (< r 0)) (error "combinations: illegal arguments, n and r > must be >= 0")] > [(> r n) 0] > [else > (let lp ([mord n] [total 1] [mult #t]) > (cond > [(or (= 0 mord) (= 1 mord)) total] > [(and mult (= mord (- n r))) (lp r total #f)] > [(and mult (= mord r)) (lp (- n r) total #f)] > [mult (lp (sub1 mord) (* total mord) #t)] > [else (lp (sub1 mord) (/ total mord) #f)]))])) > > Thanks again! > -Joe > > ____________________ > Racket Users list: > http://lists.racket-lang.org/users > >
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