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--- math_decimal_bernoulli [2010/03/16 11:39]
tyrtamos créée
+++ math_decimal_bernoulli [2010/03/16 14:05]
tyrtamos
@@ Ligne 15: / Ligne 15: @@
   * avec <m>B_0 = 1</m>
-  * et <m>B_i = 0</m> si i = nombre impair >1
+  * et <m>B_i = 0</m> si i est un nombre impair supérieur à 1
+Dans cette formule:
+  * <m>{C_{n+1}}^k</m> est le nombre de combinaison de n+1 objets pris k à k (voir analyse combinatoire sur ce site)
+  * <m>{B_k}</m> est le nombre de Bernoulli pour la valeur k
+Mais en quoi cette équation est-elle "récurrente"?
+Pour le savoir, développons pour n=3:
+  * <m>{{{C_{4}}^0}*{B_0}} ~+~ {{{C_{4}}^1}*{B_1}} ~+~ {{{C_{4}}^2}*{B_2}} ~+~ {{{C_{4}}^3}*{B_3}} ~=~ 0</m>
+Il faudrait donc que l'on puisse calculer <m>B_3</m>.
+Mais là, on voit bien le problème: on connait <m>{B_0}=1</m> par définition, mais pour calculer <m>B_3</m>, il faut calculer avant: <m>B_1</m> et <m>B_2</m>.
+Comment? En appliquant la formule pour n=1 et n=2 et, à chaque fois, en calculant le dernier B.
+Voilà le code qui fait cela (Python version 2.6):
+<code python>
+#!/usr/bin/python
+# -*- coding: utf-8 -*-
+from __future__ import division
+from decimal import *
+##############################################################################
+def combin(n, k):
+    """Nombre de combinaisons de n objets pris k a k"""
+    if k > n//2:
+        k = n-k
+    x = 1
+    y = 1
+    i = n-k+1
+    while i <= n:
+        x = (x*i)//y
+        y += 1
+        i += 1
+    return x
+##############################################################################
+def bernoulli(n):
+    """calcul du nombre de Bernoulli Bn """
+    prec = getcontext().prec
+    getcontext().prec = 100
+    B = [Decimal("1")]
+    for m in xrange(1,n+1):
+        if m>1 and m%2!=0:
+            B.append(Decimal("0"))
+        else:
+            c = Decimal("0")
+            for k in xrange(0,m):
+                c += Decimal(str(combin(m+1,k)))*B[k]
+            B.append(-c/Decimal(str(combin(m+1,m))))
+    getcontext().prec = prec
+    return float(B[n])
+</code>
+Pour que les calculs intermédiaires soient corrects, il est important qu'ils soient faits avec un nombre suffisant de chiffres significatifs. Ici, après essais, j'ai pris 100 chiffres! Au delà, on ne gagne plus rien (j'ai essayé avec 400 chiffres).
+Dans ce code, j'ai choisi que le résultat soit retourné en flottants. Mais on pourrait facilement continuer les calculs en mode Decimal() si c'était utile. Cependant, ici, en flottant, le nombre de Bernoulli maxi utilisable sera <m>B_258=1.3352784187354634e+306</m> puisqu'au delà, on dépasse les capacités de stockage des flottants (double précision du C).
+Voilà les résultats:
+<code>
+B( 0 ) = 1.0
+B( 1 ) = -0.5
+B( 2 ) = 0.16666666666666666
+B( 3 ) = 0.0
+B( 4 ) = -0.033333333333333333
+B( 5 ) = 0.0
+B( 6 ) = 0.023809523809523808
+B( 7 ) = 0.0
+B( 8 ) = -0.033333333333333333
+B( 9 ) = 0.0
+B( 10 ) = 0.07575757575757576
+B( 11 ) = 0.0
+B( 12 ) = -0.2531135531135531
+B( 13 ) = 0.0
+B( 14 ) = 1.1666666666666667
+B( 15 ) = 0.0
+B( 16 ) = -7.0921568627450977
+B( 17 ) = 0.0
+B( 18 ) = 54.971177944862156
+B( 19 ) = 0.0
+B( 20 ) = -529.12424242424242
+B( 21 ) = 0.0
+B( 22 ) = 6192.123188405797
+B( 23 ) = 0.0
+B( 24 ) = -86580.253113553117
+B( 25 ) = 0.0
+B( 26 ) = 1425517.1666666667
+B( 27 ) = 0.0
+B( 28 ) = -27298231.067816094
+B( 29 ) = 0.0
+B( 30 ) = 601580873.9006424
+B( 31 ) = 0.0
+B( 32 ) = -15116315767.092157
+B( 33 ) = 0.0
+B( 34 ) = 429614643061.16669
+B( 35 ) = 0.0
+B( 36 ) = -13711655205088.332
+B( 37 ) = 0.0
+B( 38 ) = 488332318973593.19
+B( 39 ) = 0.0
+B( 40 ) = -19296579341940068.0
+B( 41 ) = 0.0
+B( 42 ) = 8.4169304757368256e+17
+B( 43 ) = 0.0
+B( 44 ) = -4.0338071854059454e+19
+B( 45 ) = 0.0
+B( 46 ) = 2.1150748638081993e+21
+B( 47 ) = 0.0
+B( 48 ) = -1.2086626522296526e+23
+B( 49 ) = 0.0
+B( 50 ) = 7.5008667460769642e+24
+B( 51 ) = 0.0
+B( 52 ) = -5.0387781014810688e+26
+B( 53 ) = 0.0
+B( 54 ) = 3.6528776484818122e+28
+B( 55 ) = 0.0
+B( 56 ) = -2.8498769302450882e+30
+B( 57 ) = 0.0
+B( 58 ) = 2.3865427499683627e+32
+B( 59 ) = 0.0
+B( 60 ) = -2.1399949257225335e+34
+B( 61 ) = 0.0
+B( 62 ) = 2.0500975723478097e+36
+B( 63 ) = 0.0
+B( 64 ) = -2.0938005911346379e+38
+B( 65 ) = 0.0
+B( 66 ) = 2.2752696488463515e+40
+B( 67 ) = 0.0
+B( 68 ) = -2.6257710286239577e+42
+B( 69 ) = 0.0
+B( 70 ) = 3.2125082102718032e+44
+B( 71 ) = 0.0
+B( 72 ) = -4.1598278166794712e+46
+B( 73 ) = 0.0
+B( 74 ) = 5.6920695482035283e+48
+B( 75 ) = 0.0
+B( 76 ) = -8.2183629419784578e+50
+B( 77 ) = 0.0
+B( 78 ) = 1.2502904327166994e+53
+B( 79 ) = 0.0
+B( 80 ) = -2.001558323324837e+55
+B( 81 ) = 0.0
+B( 82 ) = 3.3674982915364376e+57
+B( 83 ) = 0.0
+B( 84 ) = -5.947097050313545e+59
+B( 85 ) = 0.0
+B( 86 ) = 1.1011910323627977e+62
+B( 87 ) = 0.0
+B( 88 ) = -2.1355259545253502e+64
+B( 89 ) = 0.0
+B( 90 ) = 4.3328896986641194e+66
+B( 91 ) = 0.0
+B( 92 ) = -9.1885528241669332e+68
+B( 93 ) = 0.0
+B( 94 ) = 2.0346896776329074e+71
+B( 95 ) = 0.0
+B( 96 ) = -4.700383395803573e+73
+B( 97 ) = 0.0
+B( 98 ) = 1.1318043445484249e+76
+B( 99 ) = 0.0
+B( 100 ) = -2.8382249570693707e+78
+B( 101 ) = 0.0
+B( 102 ) = 7.4064248979678853e+80
+B( 103 ) = 0.0
+B( 104 ) = -2.0096454802756605e+83
+B( 105 ) = 0.0
+B( 106 ) = 5.6657170050805942e+85
+B( 107 ) = 0.0
+B( 108 ) = -1.6584511154136216e+88
+B( 109 ) = 0.0
+B( 110 ) = 5.0368859950492378e+90
+B( 111 ) = 0.0
+B( 112 ) = -1.5861468237658186e+93
+B( 113 ) = 0.0
+B( 114 ) = 5.1756743617545625e+95
+B( 115 ) = 0.0
+B( 116 ) = -1.7488921840217116e+98
+B( 117 ) = 0.0
+B( 118 ) = 6.1160519994952182e+100
+B( 119 ) = 0.0
+B( 120 ) = -2.2122776912707833e+103
+B( 121 ) = 0.0
+B( 122 ) = 8.2722776798770969e+105
+B( 123 ) = 0.0
+B( 124 ) = -3.1958925111415708e+108
+B( 125 ) = 0.0
+B( 126 ) = 1.2750082223387793e+111
+B( 127 ) = 0.0
+B( 128 ) = -5.2500923086774131e+113
+B( 129 ) = 0.0
+B( 130 ) = 2.2301817894241627e+116
+B( 131 ) = 0.0
+B( 132 ) = -9.7684521930955207e+118
+B( 133 ) = 0.0
+B( 134 ) = 4.409836197845295e+121
+B( 135 ) = 0.0
+B( 136 ) = -2.0508570886464089e+124
+B( 137 ) = 0.0
+B( 138 ) = 9.8214433279791277e+126
+B( 139 ) = 0.0
+B( 140 ) = -4.8412600798208881e+129
+B( 141 ) = 0.0
+B( 142 ) = 2.4553088801480982e+132
+B( 143 ) = 0.0
+B( 144 ) = -1.2806926804084748e+135
+B( 145 ) = 0.0
+B( 146 ) = 6.8676167104668579e+137
+B( 147 ) = 0.0
+B( 148 ) = -3.7846468581969106e+140
+B( 149 ) = 0.0
+B( 150 ) = 2.1426101250665291e+143
+B( 151 ) = 0.0
+B( 152 ) = -1.2456727137183695e+146
+B( 153 ) = 0.0
+B( 154 ) = 7.4345787551000156e+148
+B( 155 ) = 0.0
+B( 156 ) = -4.5535795304641704e+151
+B( 157 ) = 0.0
+B( 158 ) = 2.861211281685887e+154
+B( 159 ) = 0.0
+B( 160 ) = -1.843772355203387e+157
+B( 161 ) = 0.0
+B( 162 ) = 1.2181154536221047e+160
+B( 163 ) = 0.0
+B( 164 ) = -8.2482187185314122e+162
+B( 165 ) = 0.0
+B( 166 ) = 5.7225877937832942e+165
+B( 167 ) = 0.0
+B( 168 ) = -4.0668530525059105e+168
+B( 169 ) = 0.0
+B( 170 ) = 2.9596092064642052e+171
+B( 171 ) = 0.0
+B( 172 ) = -2.2049522565189457e+174
+B( 173 ) = 0.0
+B( 174 ) = 1.6812597072889599e+177
+B( 175 ) = 0.0
+B( 176 ) = -1.3116736213556958e+180
+B( 177 ) = 0.0
+B( 178 ) = 1.0467894009478039e+183
+B( 179 ) = 0.0
+B( 180 ) = -8.543289357883371e+185
+B( 181 ) = 0.0
+B( 182 ) = 7.1287821322486547e+188
+B( 183 ) = 0.0
+B( 184 ) = -6.0802931455535905e+191
+B( 185 ) = 0.0
+B( 186 ) = 5.2996776424849921e+194
+B( 187 ) = 0.0
+B( 188 ) = -4.719425916874586e+197
+B( 189 ) = 0.0
+B( 190 ) = 4.2928413791402983e+200
+B( 191 ) = 0.0
+B( 192 ) = -3.9876744968232205e+203
+B( 193 ) = 0.0
+B( 194 ) = 3.781978041935888e+206
+B( 195 ) = 0.0
+B( 196 ) = -3.6614233683681192e+209
+B( 197 ) = 0.0
+B( 198 ) = 3.617609027237286e+212
+B( 199 ) = 0.0
+B( 200 ) = -3.6470772645191356e+215
+B( 201 ) = 0.0
+B( 202 ) = 3.7508755436454406e+218
+B( 203 ) = 0.0
+B( 204 ) = -3.934586729643903e+221
+B( 205 ) = 0.0
+B( 206 ) = 4.2088211148190084e+224
+B( 207 ) = 0.0
+B( 208 ) = -4.5902296220617918e+227
+B( 209 ) = 0.0
+B( 210 ) = 5.1031725772629572e+230
+B( 211 ) = 0.0
+B( 212 ) = -5.7822762303656954e+233
+B( 213 ) = 0.0
+B( 214 ) = 6.6762482167835884e+236
+B( 215 ) = 0.0
+B( 216 ) = -7.8535307644450417e+239
+B( 217 ) = 0.0
+B( 218 ) = 9.4106894067058724e+242
+B( 219 ) = 0.0
+B( 220 ) = -1.1484933873465185e+246
+B( 221 ) = 0.0
+B( 222 ) = 1.4272958742848785e+249
+B( 223 ) = 0.0
+B( 224 ) = -1.8059559586909309e+252
+B( 225 ) = 0.0
+B( 226 ) = 2.3261535307660807e+255
+B( 227 ) = 0.0
+B( 228 ) = -3.0495751715499594e+258
+B( 229 ) = 0.0
+B( 230 ) = 4.0685806076433976e+261
+B( 231 ) = 0.0
+B( 232 ) = -5.5231031321974362e+264
+B( 233 ) = 0.0
+B( 234 ) = 7.6277279396434395e+267
+B( 235 ) = 0.0
+B( 236 ) = -1.0715571119697886e+271
+B( 237 ) = 0.0
+B( 238 ) = 1.5310200895969188e+274
+B( 239 ) = 0.0
+B( 240 ) = -2.2244891682179836e+277
+B( 241 ) = 0.0
+B( 242 ) = 3.286267919069014e+280
+B( 243 ) = 0.0
+B( 244 ) = -4.9355928955960348e+283
+B( 245 ) = 0.0
+B( 246 ) = 7.5349571200832511e+286
+B( 247 ) = 0.0
+B( 248 ) = -1.1691485154584178e+290
+B( 249 ) = 0.0
+B( 250 ) = 1.8435261467838938e+293
+B( 251 ) = 0.0
+B( 252 ) = -2.9536826172968082e+296
+B( 253 ) = 0.0
+B( 254 ) = 4.8079321277501568e+299
+B( 255 ) = 0.0
+B( 256 ) = -7.9502125045885252e+302
+B( 257 ) = 0.0
+B( 258 ) = 1.3352784187354634e+306
+</code>
+Et voilà une liste apte à être intégrée dans un code par copier-coller:
+<code>
+bernoulli = [1.0, -0.5, 0.166666666667, 0.0, -0.0333333333333, 0.0, 0.0238095238095, 0.0, -0.0333333333333, 0.0,
+.0757575757576, 0.0, -0.253113553114, 0.0, 1.16666666667, 0.0, -7.09215686275, 0.0, 54.9711779449, 0.0, -529.124242424,
+.0, 6192.12318841, 0.0, -86580.2531136, 0.0, 1425517.16667, 0.0, -27298231.0678, 0.0, 601580873.901, 0.0, -15116315767.1,
+.0, 429614643061.0, 0.0, -1.37116552051e+13, 0.0, 4.88332318974e+14, 0.0, -1.92965793419e+16, 0.0, 8.41693047574e+17,
+.0, -4.03380718541e+19, 0.0, 2.11507486381e+21, 0.0, -1.20866265223e+23, 0.0, 7.50086674608e+24, 0.0, -5.03877810148e+26,
+.0, 3.65287764848e+28, 0.0, -2.84987693025e+30, 0.0, 2.38654274997e+32, 0.0, -2.13999492572e+34, 0.0, 2.05009757235e+36,
+.0, -2.09380059113e+38, 0.0, 2.27526964885e+40, 0.0, -2.62577102862e+42, 0.0, 3.21250821027e+44, 0.0, -4.15982781668e+46,
+.0, 5.6920695482e+48, 0.0, -8.21836294198e+50, 0.0, 1.25029043272e+53, 0.0, -2.00155832332e+55, 0.0, 3.36749829154e+57,
+.0, -5.94709705031e+59, 0.0, 1.10119103236e+62, 0.0, -2.13552595453e+64, 0.0, 4.33288969866e+66, 0.0, -9.18855282417e+68,
+.0, 2.03468967763e+71, 0.0, -4.7003833958e+73, 0.0, 1.13180434455e+76, 0.0, -2.83822495707e+78, 0.0, 7.40642489797e+80,
+.0, -2.00964548028e+83, 0.0, 5.66571700508e+85, 0.0, -1.65845111541e+88, 0.0, 5.03688599505e+90, 0.0, -1.58614682377e+93,
+.0, 5.17567436175e+95, 0.0, -1.74889218402e+98, 0.0, 6.1160519995e+100, 0.0, -2.21227769127e+103, 0.0, 8.27227767988e+105,
+.0, -3.19589251114e+108, 0.0, 1.27500822234e+111, 0.0, -5.25009230868e+113, 0.0, 2.23018178942e+116, 0.0, -9.7684521931e+118,
+.0, 4.40983619785e+121, 0.0, -2.05085708865e+124, 0.0, 9.82144332798e+126, 0.0, -4.84126007982e+129, 0.0, 2.45530888015e+132,
+.0, -1.28069268041e+135, 0.0, 6.86761671047e+137, 0.0, -3.7846468582e+140, 0.0, 2.14261012507e+143, 0.0, -1.24567271372e+146,
+.0, 7.4345787551e+148, 0.0, -4.55357953046e+151, 0.0, 2.86121128169e+154, 0.0, -1.8437723552e+157, 0.0, 1.21811545362e+160,
+.0, -8.24821871853e+162, 0.0, 5.72258779378e+165, 0.0, -4.06685305251e+168, 0.0, 2.95960920646e+171, 0.0, -2.20495225652e+174,
+.0, 1.68125970729e+177, 0.0, -1.31167362136e+180, 0.0, 1.04678940095e+183, 0.0, -8.54328935788e+185, 0.0, 7.12878213225e+188,
+.0, -6.08029314555e+191, 0.0, 5.29967764248e+194, 0.0, -4.71942591687e+197, 0.0, 4.29284137914e+200, 0.0, -3.98767449682e+203,
+.0, 3.78197804194e+206, 0.0, -3.66142336837e+209, 0.0, 3.61760902724e+212, 0.0, -3.64707726452e+215, 0.0, 3.75087554365e+218,
+.0, -3.93458672964e+221, 0.0, 4.20882111482e+224, 0.0, -4.59022962206e+227, 0.0, 5.10317257726e+230, 0.0, -5.78227623037e+233,
+.0, 6.67624821678e+236, 0.0, -7.85353076445e+239, 0.0, 9.41068940671e+242, 0.0, -1.14849338735e+246, 0.0, 1.42729587428e+249,
+.0, -1.80595595869e+252, 0.0, 2.32615353077e+255, 0.0, -3.04957517155e+258, 0.0, 4.06858060764e+261, 0.0, -5.5231031322e+264,
+.0, 7.62772793964e+267, 0.0, -1.07155711197e+271, 0.0, 1.5310200896e+274, 0.0, -2.22448916822e+277, 0.0, 3.28626791907e+280,
+.0, -4.9355928956e+283, 0.0, 7.53495712008e+286, 0.0, -1.16914851546e+290, 0.0, 1.84352614678e+293, 0.0, -2.9536826173e+296,
+.0, 4.80793212775e+299, 0.0, -7.95021250459e+302, 0.0, 1.33527841874e+306, 0.0]
+</code>
+\\
+Amusez-vous bien!
+<html>
+<head>
+<style type="text/css">
+<!--
+body {background-image:url(fondcorps.jpg);}
+-->
+</style>
+</head>
+<body>
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