Cython wrapper for bernmm library¶
AUTHOR:
- David Harvey (2008-06): initial version
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sage.rings.bernmm.
bernmm_bern_modp
(p, k)¶ Computes \(B_k \mod p\), where \(B_k\) is the k-th Bernoulli number.
If \(B_k\) is not \(p\)-integral, returns -1.
INPUT:
p – a prime k – non-negative integerCOMPLEXITY:
Pretty much linear in $p$.EXAMPLES:
sage: from sage.rings.bernmm import bernmm_bern_modp sage: bernoulli(0) % 5, bernmm_bern_modp(5, 0) (1, 1) sage: bernoulli(1) % 5, bernmm_bern_modp(5, 1) (2, 2) sage: bernoulli(2) % 5, bernmm_bern_modp(5, 2) (1, 1) sage: bernoulli(3) % 5, bernmm_bern_modp(5, 3) (0, 0) sage: bernoulli(4), bernmm_bern_modp(5, 4) (-1/30, -1) sage: bernoulli(18) % 5, bernmm_bern_modp(5, 18) (4, 4) sage: bernoulli(19) % 5, bernmm_bern_modp(5, 19) (0, 0) sage: p = 10000019; k = 1000 sage: bernoulli(k) % p 1972762 sage: bernmm_bern_modp(p, k) 1972762
TESTS:
Check that bernmm works with the new NTL single precision modular arithmetic from trac ticket #19874:
sage: from sage.rings.bernmm import bernmm_bern_modp sage: bernmm_bern_modp(7, 128) == bernoulli(128) % 7 True
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sage.rings.bernmm.
bernmm_bern_rat
(k, num_threads=1)¶ Computes k-th Bernoulli number using a multimodular algorithm. (Wrapper for bernmm library.)
INPUT:
- k – non-negative integer
- num_threads – integer >= 1, number of threads to use
COMPLEXITY:
Pretty much quadratic in $k$. See the paper “A multimodular algorithm for computing Bernoulli numbers”, David Harvey, 2008, for more details.EXAMPLES:
sage: from sage.rings.bernmm import bernmm_bern_rat sage: bernmm_bern_rat(0) 1 sage: bernmm_bern_rat(1) -1/2 sage: bernmm_bern_rat(2) 1/6 sage: bernmm_bern_rat(3) 0 sage: bernmm_bern_rat(100) -94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330 sage: bernmm_bern_rat(100, 3) -94598037819122125295227433069493721872702841533066936133385696204311395415197247711/33330
TESTS:
sage: lst1 = [ bernoulli(2*k, algorithm='bernmm', num_threads=2) for k in [2932, 2957, 3443, 3962, 3973] ] sage: lst2 = [ bernoulli(2*k, algorithm='pari') for k in [2932, 2957, 3443, 3962, 3973] ] sage: lst1 == lst2 True sage: [ Zmod(101)(t) for t in lst1 ] [77, 72, 89, 98, 86] sage: [ Zmod(101)(t) for t in lst2 ] [77, 72, 89, 98, 86]