! roots of recursively defined polynomial: ! which is defined by recurrence relation: ! p_{i+1}(x)=p_{i}(x)^2 +x where p_{1}(x)=x ! ! degree of p_{i} = 2^(i-1) ! degree = number of roots ! ! roots of p_{i}(x) are centers of hyperbolic components of period i and divisors(i) ! for mandelbrot set of function f(z)=z*z + x ! ! number_of_centers= number_of_roots - sum_of_numbers_of_roots_for_each_divisor_of_period_smaller_then_i ! here is axample of command for computing roots of polynomial for i=1 ! using MPSolve http://www.dm.unipi.it/cluster-pages/mpsolve/index.htm ! ./unisolve -Og Data/m_c_1.pol >Results/m_c_1g.res ! Adam Majewski fraktal.republika.pl 12.2007 ! Thx for Dario Bini for help (:-)) -------------------------------------------------------- ! ! below is a list of approximations of roots ---------------------------------------------------- ======================================================================================================== ======================================================================================================== period= 1 degree=1 0.00000000000000e+000 0.00000000000000e+000 ======================================================================================================== ======================================================================================================== period=2 degree=2 -1.00000000000000e+000 0.00000000000000e+000 0.00000000000000e+000 0.00000000000000e+000 ======================================================================================================== ======================================================================================================== period=3 degree=4 -1.75487766624669e+000 -0.45917748078996e-040 -0.12256116687665e+000 -0.74486176661974e+000 -0.12256116687665e+000 0.74486176661974e+000 0.00000000000000e+000 0.00000000000000e+000 ======================================================================================================== ======================================================================================================== period=4 degree=8 -1.94079980652948e+000 0.11754943508223e-037 -1.31070264133683e+000 0.39443045261051e-030 -1.00000000000000e+000 0.00000000000000e+000 -0.15652016683376e+000 -1.03224710892283e+000 -0.15652016683376e+000 1.03224710892283e+000 0.00000000000000e+000 0.00000000000000e+000 0.28227139076691e+000 -0.53006061757853e+000 0.28227139076691e+000 0.53006061757853e+000 ======================================================================================================== ======================================================================================================== period=5 degree=16 -1.98542425305421e+000 0.00000000000000e+000 -1.86078252220485e+000 0.00000000000000e+000 -1.62541372512330e+000 0.80779356694631e-027 -1.25636793006818e+000 0.38032096347272e+000 -1.25636793006818e+000 -0.38032096347272e+000 -0.50434017544624e+000 0.56276576145298e+000 -0.50434017544624e+000 -0.56276576145298e+000 -0.19804209936425e+000 -1.10026953729270e+000 -0.19804209936425e+000 1.10026953729270e+000 -0.44212357704071e-001 -0.98658097628089e+000 -0.44212357704071e-001 0.98658097628089e+000 0.00000000000000e+000 0.00000000000000e+000 0.35925922475801e+000 0.64251373713854e+000 0.35925922475801e+000 -0.64251373713854e+000 0.37951358801592e+000 0.33493230559750e+000 0.37951358801592e+000 -0.33493230559750e+000 ======================================================================================================== ======================================================================================================== period=6 degree=32 -1.99637613771119e+000 0.17795106716850e-046 -1.96677321639293e+000 0.00000000000000e+000 -1.90728009106530e+000 0.00000000000000e+000 -1.77289290338162e+000 0.00000000000000e+000 -1.75487766624669e+000 0.00000000000000e+000 -1.47601464272843e+000 -0.28381392745643e-034 -1.28408492552569e+000 0.42726889604069e+000 -1.28408492552569e+000 -0.42726889604069e+000 -1.13800066665096e+000 0.24033240126210e+000 -1.13800066665096e+000 -0.24033240126210e+000 -1.00000000000000e+000 0.00000000000000e+000 -0.59689164464513e+000 0.66298074457703e+000 -0.59689164464513e+000 -0.66298074457703e+000 -0.21752674703051e+000 1.11445426587329e+000 -0.21752674703051e+000 -1.11445426587329e+000 -0.16359826155202e+000 -1.09778064288827e+000 -0.16359826155202e+000 1.09778064288827e+000 -0.12256116687665e+000 -0.74486176661974e+000 -0.12256116687665e+000 0.74486176661974e+000 -0.11341865594944e+000 -0.86056947250157e+000 -0.11341865594944e+000 0.86056947250157e+000 -0.15570386020902e-001 1.02049736649829e+000 -0.15570386020902e-001 -1.02049736649829e+000 0.00000000000000e+000 0.00000000000000e+000 0.35989273901258e+000 0.68476202021181e+000 0.35989273901258e+000 -0.68476202021181e+000 0.38900684056977e+000 0.21585065087082e+000 0.38900684056977e+000 -0.21585065087082e+000 0.39653457003242e+000 0.60418181048899e+000 0.39653457003242e+000 -0.60418181048899e+000 0.44332563339962e+000 -0.37296241666285e+000 0.44332563339962e+000 0.37296241666285e+000 ======================================================================================================== ======================================================================================================== period=7 degree=64 -1.99909568232702e+000 0.21417030800082e-046 -1.99181417254912e+000 -0.76309787122495e-052 -1.97717958700626e+000 -0.23509887016446e-037 -1.95370589428440e+000 0.00000000000000e+000 -1.92714770936395e+000 -0.36354916859169e-047 -1.88480357158668e+000 -0.14012984643248e-044 -1.83231520275123e+000 -0.29582283945788e-030 -1.76926167027683e+000 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======================================================================================================== ======================================================================================================== period=15 degree=16 384 ======================================================================================================== ========================================================================================================= period=16 degree=32 768 ======================================================================================================== ========================================================================================================= period=17 degree=65 536 ======================================================================================================== ========================================================================================================= period=18 degree=131 072 ======================================================================================================== ========================================================================================================= period=19 degree=262 144 ======================================================================================================== ========================================================================================================= period=20 degree=524 288 ======================================================================================================== =========================================================================================================