[racket] Math library kudos
Hi Joe,
There is a straight way to calculate product.
(define (combinations n r)
(/ (for/product ([i (in-range n (- n r) -1)]) i)
(for/product ([i (in-range r 1 -1 )]) i)))
Regards,
haiwei
On 20 February 2013 14:44, Joe Gilray <jgilray at gmail.com> wrote:
> 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
>>>
>>> ____________________
>>> Racket Users list:
>>> http://lists.racket-lang.org/users
>>>
>>>
>>
>
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