Calcul de la Dynamique de Verhulst





(R+1)X-R(XX)



Avec R=3 :



                    IBM ES9000 :
                              (R+1)X-R(XX)   (R+1)X-(RX)X   ((R+1)-(RX))X  RX+(1-(RX))X   X+R(X-(XX))
                    X(00)   = 0.500000       0.500000       0.500000       0.500000       0.500000
                    X(10)   = 0.384631       0.384631       0.384631       0.384631       0.384631
                    X(20)   = 0.418895       0.418895       0.418895       0.418895       0.418895
                    X(30)   = 0.046399       0.046399       0.046399       0.046399       0.046399
                    X(40)   = 0.320185       0.320183       0.320188       0.320182       0.320189
                    X(50)   = 0.063406       0.064521       0.061895       0.064941       0.061244
                    X(60)   = 1.040381       0.846041       0.529794       1.319900       1.214070
                    X(70)   = 0.004104       1.199452       0.873553       0.573637       0.000009
                    X(80)   = 0.108044       0.121414       1.260726       0.395871       0.280590
                    X(90)   = 0.096374       0.089244       0.582157       0.344503       1.023735


                    IBM RS6000 :
                              (R+1)X-R(XX)   (R+1)X-(RX)X   ((R+1)-(RX))X  RX+(1-(RX))X   X+R(X-(XX))
                    X(00)   = 0.500000       0.500000       0.500000       0.500000       0.500000
                    X(10)   = 0.384631       0.384631       0.384631       0.384631       0.384631
                    X(20)   = 0.418895       0.418895       0.418895       0.418895       0.418895
                    X(30)   = 0.046399       0.046399       0.046399       0.046399       0.046399
                    X(40)   = 0.320177       0.320184       0.320188       0.320190       0.320189
                    X(50)   = 0.067567       0.063747       0.061859       0.060822       0.061486
                    X(60)   = 0.001145       0.271115       0.616781       0.298613       1.307350
                    X(70)   = 1.296775       1.328462       0.486629       0.938605       1.054669
                    X(80)   = 0.553038       0.817163       1.277151       1.325437       0.617058
                    X(90)   = 0.094852       0.154184       1.174162       0.148151       0.237355




                    Power-Challenge M Silicon Graphics (R8000, IRIX 6.2, cc 7.0) :
                              option '-O2'   option '-O3'
                    X(00)   = 0.500000       0.500000
                    X(10)   = 0.384631       0.384631
                    X(20)   = 0.418895       0.418895
                    X(30)   = 0.046399       0.046399
                    X(40)   = 0.320184       0.320188
                    X(50)   = 0.063747       0.061859
                    X(60)   = 0.271115       0.616781
                    X(70)   = 1.328462       0.486629
                    X(80)   = 0.817163       1.277151




                    O200 Silicon Graphics (R10000, IRIX 6.5.5m, cc 7.2.1) :
                              option '-O2'   option '-O3'
                    X(00)   = 0.500000       0.500000
                    X(10)   = 0.384631       0.384631
                    X(20)   = 0.418895       0.418895
                    X(30)   = 0.046399       0.046399
                    X(40)   = 0.320184       0.320192
                    X(50)   = 0.063747       0.059988
                    X(60)   = 0.271115       1.000531
                    X(70)   = 1.328462       1.329692
                    X(80)   = 0.817163       0.021952




(calculs effectués en double précision, c'est-à-dire sur 64 bits).




                                        O200 Silicon Graphics (processeur R10000, IRIX 6.5.5m, Java) :
                              (R+1)X-R(XX)         (R+1)X-(RX)X         ((R+1)-(RX))X        RX+(1-(RX))X         X+R(X-(XX))
                    X(0000) = 0.5                  0.5                  0.5                  0.5                  0.5
                    X(0500) = 1.288736212247168    0.007057813075738616 1.2767485100695732   1.246534177059494    0.03910723014701789
                    X(1000) = 1.3327294162589722   0.916560711983132    1.207710752523091    0.27770146115891703  0.26663342726567785
                    X(1500) = 1.1448646685382955   0.4481000759915065   0.3102077001456977   0.015374092695375374 0.9841637252962943
                    X(2000) = 1.0548628914440754   0.896126931497168    0.6851138190159249   0.009229885271816535 0.3860923315999224
                    X(2500) = 1.292802584458599    0.06063433547953646  1.174118726001978    0.6922411856638806   0.020878761210912034
                    X(3000) = 1.0497821908090537   0.0219606878364607   1.3287403237319588   0.11354602472378028  0.13270749449424302
                    X(3500) = 0.8115039383609847   1.3213031319440816   0.6545151597367076   0.5760786099237328   1.324039473116061
                    X(4000) = 0.04922223042798102  1.3203298564077224   0.09243804931690679  0.9496284087750142   1.316597313359563
                    X(4500) = 0.4745896653599724   0.32865616721789603  0.018965010461877246 0.25384661313701296  0.18512853535354462


                                        PC (processeur Pentium II, LINUX Mandrake 7.0, Java) :
                              (R+1)X-R(XX)         (R+1)X-(RX)X         ((R+1)-(RX))X        RX+(1-(RX))X         X+R(X-(XX))
                    X(0000) = 0.5                  0.5                  0.5                  0.5                  0.5
                    X(0500) = 1.2887362122471679   0.00705781307573862  1.2767485100695732   1.2465341770675666   0.03910723014701789
                    X(1000) = 1.3327294162589722   0.91656071198313205  1.207710752523091    0.6676224369769922   0.26663342726567785
                    X(1500) = 1.1448646685382955   0.44810007599150647  0.31020770014569771  0.41049165176544455  0.98416372529629426
                    X(2000) = 1.0548628914440754   0.89612693149716804  0.68511381901592494  1.0026346845706315   0.3860923315999224
                    X(2500) = 1.3328681064703032   0.06063433547953646  1.1741187260019781   0.0154001182074282   0.02087876121091203
                    X(3000) = 1.2956769824648844   0.0219606878364607   1.3287403237319588   0.50504896336548377  0.13270749449424302
                    X(3500) = 0.19193027175727995  0.37986077053509781  0.6545151597367076   0.38299434265835819  1.324039473116061
                    X(4000) = 1.2491385720940165   0.96017143401896088  0.09243804931690679  0.6565274346305322   1.316597313359563
                    X(4500) = 0.00644889182443986  1.3185465795235045   0.01896501046187725  0.94966313327336349  0.18512853535354462




Les valeurs correctes sont (avec 16 décimales) :



                    
                    X(00) = 0.5000000000000000
                    X(10) = 0.3846309658187955
                    X(20) = 0.4188950250152601
                    X(30) = 0.0463994768425783
                    X(40) = 0.3201870617215119
                    X(50) = 0.0622361944358458
                    X(60) = 0.0049027225204761
                    X(70) = 0.5530823826833586
                    X(80) = 0.1196398066817574
                    X(90) = 0.3109290853617026












Jean-François COLONNA
www.lactamme.polytechnique.fr
jean-francois.colonna@polytechnique.edu
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France
france telecom, France Telecom R&D

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