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Tridimensional display of the rounding-off errors when computing the Verhulst dynamics [Visualisation tridimensionnelle des erreurs d'arrondi dans le calcul de la dynamique de Verhulst].




This picture displays the three following Verhulst dynamics (for 1000 iterations):
                    X  = [[R+1]X   ] - [R[X   X   ]]
                     n          n-1        n-1 n-1


Y = [[R+1]Y ] - [[RY ]Y ] n n-1 n-1 n-1

Z = Z + [R[Z - [Z Z ]]] n n-1 n-1 n-1 n-1
computed with the same initial conditions:
                    X  = 0.5
                     0


Y = 0.5 0

Z = 0.5 0
They are strictly equivalent mathematically speaking and thus the point set {Xn,Yn,Zn} should be on a straight line with equation:
                    Z = Y = X
in the tridimensional space. Unfortunately, when the R parameter increases (from 2.0 -picture 01- to 3.0 -picture 16-) the rounding-off errors become huge and give birth to three different numerical Verhulst dynamics. Then, the {Xn,Yn,Zn} points appear "anywhere" inside a cube.

This double precision (64 bits) computation was made on a SGI O200 under IRIX 6.5 and C compiler release 7.2.1.


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