diff options
author | zimmerma <zimmerma@280ebfd0-de03-0410-8827-d642c229c3f4> | 2016-02-17 12:34:05 +0000 |
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committer | zimmerma <zimmerma@280ebfd0-de03-0410-8827-d642c229c3f4> | 2016-02-17 12:34:05 +0000 |
commit | 3d3fffa4ec0a36f6e2b9de6cba7c17425de9742e (patch) | |
tree | b24d811954928642c0b54a561354b4d52e19b87d /src/gamma_inc.c | |
parent | 83ee5180a3bb1d4c4f240be5aaf99830f1e6afac (diff) | |
download | mpfr-3d3fffa4ec0a36f6e2b9de6cba7c17425de9742e.tar.gz |
now mpfr_gamma_inc(a,x) also works for 'a' a negative integer
(however a and x should not be too large, we should implement
Legendre's continued fraction for the general case)
git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@10049 280ebfd0-de03-0410-8827-d642c229c3f4
Diffstat (limited to 'src/gamma_inc.c')
-rw-r--r-- | src/gamma_inc.c | 192 |
1 files changed, 183 insertions, 9 deletions
diff --git a/src/gamma_inc.c b/src/gamma_inc.c index 544ba532e..78d7f0376 100644 --- a/src/gamma_inc.c +++ b/src/gamma_inc.c @@ -38,12 +38,11 @@ http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., gamma(a,x) = x^a * sum((-x)^k/k!/(a+k), k=0..infinity) gamma(a,x) = x^a * exp(-x) * sum(x^k/(a*(a+1)*...*(a+k)), k=0..infinity) - - For a negative integer, we have: - - gamma(-n,x) = (-1)^n/n [E_1(x) - exp(-x) sum((-1)^j*j!/x^(j+1), j=0..n-1)] */ +static int +mpfr_gamma_inc_negint (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t r); + int mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) { @@ -181,6 +180,9 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) MPFR_RET_NAN; } + if (mpfr_integer_p (a) && MPFR_SIGN(a) < 0) + return mpfr_gamma_inc_negint (y, a, x, rnd); + MPFR_SAVE_EXPO_MARK (expo); w = MPFR_PREC(y) + 13; /* working precision */ @@ -190,7 +192,7 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) MPFR_ZIV_INIT (loop, w); for (;;) { - mpfr_exp_t expu, precu; + mpfr_exp_t expu, precu, exps; mpfr_t s_abs; mpfr_exp_t decay = 0; MPFR_BLOCK_DECL (flags); @@ -207,7 +209,7 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) precu = MPFR_GET_EXP(a) <= 0 ? MPFR_ADD_PREC (MPFR_PREC(a), 1 - MPFR_EXP(a)) : (MPFR_EXP(a) <= MPFR_PREC(a)) ? MPFR_PREC(a) : MPFR_EXP(a); - MPFR_ASSERTN (precu + 1 <= MPFR_PREC_MAX); + MPFR_ASSERTD (precu + 1 <= MPFR_PREC_MAX); mpfr_set_prec (u, precu + 1); expu = (MPFR_EXP(a) > 0) ? MPFR_EXP(a) : 1; @@ -228,6 +230,8 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) mpfr_div (t, t, u, MPFR_RNDA); /* t = x^k/(a * ... * (a+k)) * (1 + theta)^(2k+1) */ mpfr_add (s, s, t, MPFR_RNDZ); + /* when s is zero, we consider ulp(s) = ulp(t) */ + exps = (MPFR_IS_ZERO(s)) ? MPFR_GET_EXP(t) : MPFR_GET_EXP(s); if (MPFR_IS_NEG(a)) { if (MPFR_IS_POS(t)) @@ -237,7 +241,8 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) } /* we stop when |t| < ulp(s), u > 0 and |x/u| < 1/2, which ensures that the tail is at most 2*ulp(s) */ - if (MPFR_GET_EXP(t) + w <= MPFR_GET_EXP(s) && MPFR_IS_POS(u) && + MPFR_ASSERTD (MPFR_IS_ZERO(t) == 0); + if (MPFR_GET_EXP(t) + w <= exps && MPFR_IS_POS(u) && MPFR_GET_EXP(x) + 1 < MPFR_GET_EXP(u)) break; @@ -245,7 +250,7 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) u so that mpfr_add_ui (u, a, k) remains exact */ if (MPFR_EXP(u) > expu) /* exponent shift in u */ { - MPFR_ASSERTN(MPFR_EXP(u) == expu + 1); + MPFR_ASSERTD(MPFR_EXP(u) == expu + 1); expu = MPFR_EXP(u); mpfr_set_prec (u, mpfr_get_prec (u) + 1); } @@ -287,7 +292,9 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) e0 = MPFR_GET_EXP (t); e1 = (MPFR_IS_ZERO(s)) ? __gmpfr_emin : MPFR_GET_EXP (s); mpfr_sub (s, t, s, MPFR_RNDZ); - e2 = MPFR_GET_EXP (s); + /* if s is zero, we can assume ulp(s) = ulp(t), but anyway we won't + be able to round */ + e2 = (MPFR_IS_ZERO(s)) ? e0 : MPFR_GET_EXP (s); /* the final error is at most 1 ulp (for the final subtraction) + 2^(e0-e2) ulps # for the error in t + 2^(decay+1)*(2k+7) ulps * 2^(e1-e2) # for the error in gamma(a,x) */ @@ -330,3 +337,170 @@ mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd) MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inex, rnd); } + +/* For a negative integer, we have (formula 6.5.19): + + gamma(-n,x) = (-1)^n/n! [E_1(x) - exp(-x) sum((-1)^j*j!/x^(j+1), j=0..n-1)] + + See also http://arxiv.org/pdf/1407.0349v1.pdf. + + Assumes 'a' is a negative integer. +*/ +static int +mpfr_gamma_inc_negint (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, + mpfr_rnd_t rnd) +{ + mpfr_t s, t, abs_a, neg_x; + unsigned long j; + mpfr_prec_t w; + int inex; + mpfr_exp_t exp_s, new_exp_s, exp_t, err_s, logj; + MPFR_GROUP_DECL(group); + MPFR_ZIV_DECL(loop); + MPFR_SAVE_EXPO_DECL (expo); + + MPFR_ASSERTD(mpfr_integer_p (a)); + MPFR_ASSERTD(mpfr_cmp_ui (a, 0) < 0); + + MPFR_TMP_INIT_ABS(abs_a, a); + + /* below, theta represents any value such that |theta| <= 2^(-w) */ + + w = MPFR_PREC(y) + 10; /* initial working precision */ + + MPFR_SAVE_EXPO_MARK (expo); + MPFR_GROUP_INIT_2(group, w, s, t); + MPFR_ZIV_INIT (loop, w); + for (;;) + { + /* we require |a| <= 2^(w-3) for the error analysis below */ + if (MPFR_GET_EXP(a) + 3 > w) + w = MPFR_GET_EXP(a) + 3; + + mpfr_ui_div (t, 1, x, MPFR_RNDN); /* t = 1/x * (1 + theta) */ + mpfr_set (s, t, MPFR_RNDN); + MPFR_ASSERTD(MPFR_IS_ZERO(s) == 0); + exp_t = exp_s = MPFR_GET_EXP(s); /* max. exponent of s/t during loop */ + new_exp_s = exp_s; + + for (j = 1; mpfr_cmp_ui (abs_a, j) > 0; j++) + { + /* invariant: t = (-1)^(j-1)*(j-1)!/x^j * (1 + theta)^(2j-1) */ + mpfr_mul_ui (t, t, j, MPFR_RNDN); + mpfr_neg (t, t, MPFR_RNDN); /* exact */ + mpfr_div (t, t, x, MPFR_RNDN); + /* now t = (-1)^j*j!/x^(j+1) * (1 + theta)^(2j+1). + We have (1 + theta)^(2j+1) = exp((2j+1)*log(1+theta)). + For |u| <= 1/2, we have |log(1+u)| <= 1.4 |u| thus: + |(1+theta)^(2j+1)-1| <= max |exp(1.4*(2j+1)*u)-1| for |u|<=2^(-w). + Now for |v| <= 1/2 we have |exp(v)-1| <= 0.7*|v| thus: + |(1+theta)^(2j+1) - 1| <= 2*(2j+1)*2^(-w) + as long as 1.4*(2j+1)*2^(-w) <= 1/2, which is true when j<2^(w-3). + Since j < |a| it suffices that |a| <= 2^(w-3). + In that case the rel. error on t is bounded by 2*(2j+1)*2^(-w), + thus the error in ulps is bounded by 2*(2j+1) ulp(t). */ + if (MPFR_IS_ZERO(t)) /* underflow on t */ + break; + if (MPFR_GET_EXP(t) > exp_t) + exp_t = MPFR_GET_EXP(t); + mpfr_add (s, s, t, MPFR_RNDN); + /* if s is zero, we can assume its ulp is that of t */ + new_exp_s = (MPFR_IS_ZERO(s)) ? MPFR_GET_EXP(t) : MPFR_GET_EXP(s); + if (new_exp_s > exp_s) + exp_s = new_exp_s; + } + + /* the error on s is bounded by (j-1) * 2^(exp_s - EXP(s)) * 1/2 + for the mpfr_add roundings, plus + sum(2*(2i+1), i=1..j-1) * 2^(exp_t - EXP(s)) for the error on t. + The latter sum is (2*j^2-2) * 2^(exp_t - EXP(s)). */ + + logj = MPFR_INT_CEIL_LOG2(j); + exp_s += logj - 1; + exp_t += 1 + 2 * logj; + + /* now the error on s is bounded by 2^(exp_s-EXP(s))+2^(exp_t-EXP(s)) */ + + exp_s = (exp_s >= exp_t) ? exp_s + 1 : exp_t + 1; + err_s = exp_s - new_exp_s; + + /* now the error on the sum S := sum((-1)^j*j!/x^(j+1), j=0..n-1) + is bounded by 2^err_s ulp(s) */ + + MPFR_TMP_INIT_NEG(neg_x, x); + + mpfr_exp (t, neg_x, MPFR_RNDN); /* t = exp(-x) * (1 + theta) */ + mpfr_mul (s, s, t, MPFR_RNDN); + if (MPFR_IS_ZERO(s)) + { + MPFR_ASSERTD(MPFR_IS_ZERO(t) == 0); + new_exp_s += MPFR_GET_EXP(t); + } + /* s = exp(-x) * (S +/- 2^err_s ulp(S)) * (1 + theta)^2. + = exp(-x) * (S +/- 2^err_s ulp(S)) * (1 +/- 3 ulp(1)) + The error on s is bounded by: + exp(-x) * [2^err_s*ulp(S) + S*3*ulp(1) + 2^err_s*ulp(S)*3*ulp(1)] + <= ulp(s) * [2^(err_s+1) + 6 + 1] + <= ulp(s) * 2^(err_s+2) as long as err_s >= 2. */ + + err_s = (err_s >= 2) ? err_s + 2 : 4; + /* now the error on s is bounded by 2^err_s ulp(s) */ + + mpfr_eint (t, neg_x, MPFR_RNDN); /* t = -E1(-x) * (1 + theta) */ + mpfr_neg (t, t, MPFR_RNDN); /* exact */ + + exp_s = (MPFR_IS_ZERO(s)) ? new_exp_s : MPFR_GET_EXP(s); + MPFR_ASSERTD(MPFR_IS_ZERO(t) == 0); + exp_t = MPFR_GET_EXP(t); + mpfr_sub (s, t, s, MPFR_RNDN); /* E_1(x) - exp(-x) * S */ + if (MPFR_IS_ZERO(s)) /* cancellation: increase working precision */ + goto next_w; + + /* err(s) <= 1/2 * ulp(s) [mpfr_sub] + + 2^err_s * 2^(exp_s-EXP(s)) * ulp(s) [previous error on s] + + 1/2 * 2^(exp_t-EXP(s)) * ulp(s) [error on t] */ + + exp_s += err_s; + exp_t -= 1; + exp_s = (exp_s >= exp_t) ? exp_s + 1 : exp_t + 1; + MPFR_ASSERTD(MPFR_IS_ZERO(s) == 0); + err_s = exp_s - MPFR_GET_EXP(s); + /* err(s) <= 1/2 * ulp(s) + 2^err_s * ulp(s) */ + + /* divide by n! */ + mpfr_gamma (t, abs_a, MPFR_RNDN); /* t = (n-1)! * (1 + theta) */ + mpfr_mul (t, t, abs_a, MPFR_RNDN); /* t = n! * (1 + theta)^2 */ + mpfr_div (s, s, t, MPFR_RNDN); + /* since (1 + theta)^2 converts to an error of at most 3 ulps + for w >= 2, the final error is at most: + 2 * (1/2 + 2^err_s) * ulp(s) [error on previous s] + + 2 * 3 * ulp(s) [error on t] + + 1 * ulp(s) [product of errors] + = ulp(s) * (2^(err_s+1) + 8) */ + err_s = (err_s >= 2) ? err_s + 1 : 4; + + /* the final error is bounded by 2^err_s * ulp(s) */ + + /* Is there a better way to compute (-1)^n? */ + mpfr_set_si (t, -1, MPFR_RNDN); + mpfr_pow (t, t, abs_a, MPFR_RNDN); + if (MPFR_IS_NEG(t)) + mpfr_neg (s, s, MPFR_RNDN); + + if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err_s, MPFR_PREC(y), rnd))) + break; + + next_w: + MPFR_ZIV_NEXT (loop, w); + MPFR_GROUP_REPREC_2(group, w, s, t); + } + MPFR_ZIV_FREE (loop); + + inex = mpfr_set (y, s, rnd); + MPFR_GROUP_CLEAR(group); + + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (y, inex, rnd); +} + + |