[racket] Math library kudos

Hi Luke,

Thanks for the knowledge.  Do you have some code that I could try out.  I
found gamma and bflog-gamma, but they work with floats and so I can't
imagine they are faster for exact answers... maybe for estimating nCr for
large numbers?

-Joe


On Tue, Feb 19, 2013 at 8:26 PM, Luke Vilnis <lvilnis at gmail.com> wrote:

> 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 <jgilray at 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
>>
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>>
>>
>
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