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include error analysis

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This commit is contained in:
Fredrik Johansson 2012-04-19 13:02:05 +02:00
parent 775334ae53
commit 470d5dfe89
1 changed files with 18 additions and 1 deletions
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19
arb/zeta_ui_euler_product.c
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@ -24,7 +24,23 @@
******************************************************************************/ ******************************************************************************/
#include "arb.h" #include "arb.h"
#include "arb.h"
/*
Let P(a,b) = prod_{a <= p <= b} (1 - p^(-s)).
Then 1/zeta(s) = P(a,M) * P(M+1,inf).
According to the analysis in S. Fillebrown,
"Faster Computation of Bernoulli Numbers", Journal of Algorithms 13,
431-445 (1992), it holds for all s >= 6 and M >= 1 that
(1/P(M+1,inf) - 1) <= 2 * M^(1-s) / (s/2 - 1).
Writing 1/zeta(s) = P(a,M) * (1 - eps) and solving for eps gives
1/(1-eps) <= 1 + 2 * M^(1-s) / (s/2 - 1), so we have
eps <= 2 * M^(1-s) / (s/2 - 1).
Since 0 < P(a,M) <= 1, this bounds the absolute error of 1/zeta(s).
*/
static __inline__ void static __inline__ void
arb_mul_ui(arb_t y, const arb_t x, ulong c) arb_mul_ui(arb_t y, const arb_t x, ulong c)
@ -68,6 +84,7 @@ arb_zeta_inv_ui_euler_product(arb_t z, ulong s)
} }
prec = arb_prec(z); prec = arb_prec(z);
/* heuristic */
wp = prec + FLINT_BIT_COUNT(prec) + (prec/s) + 4; wp = prec + FLINT_BIT_COUNT(prec) + (prec/s) + 4;
arb_init(t, wp); arb_init(t, wp);