4 8*H1 - 0.51234*H1*(1+(1/2)*H1+(1/3)*H2+(1/4)*H3) - 8; 8*H2 - 0.51234*H2*(1+(2/3)*H1+(2/4)*H2+(2/5)*H3) - 8; 8*H3 - 0.51234*H3*(1+(3/4)*H1+(3/5)*H2+(3/6)*H3) - 8; 8*H4 - 0.51234*H4*(1+(4/5)*H1+(4/6)*H2+(4/7)*H3) - 8; TITLE : The Chandrasekhar H-Equation for n=4 ROOT COUNTS : total degree : 16 2-homogeneous Bezout number : 8 with partition : {H1 H2 H3 }{H4 } generalized Bezout number : 8 based on the set structure : {H1 }{H1 H2 H3 } {H1 H2 }{H2 H3 } {H1 H2 H3 }{H3 } {H1 H2 H3 }{H4 } mixed volume : 8 REFERENCES : Laureano Gonzalez-Vega: "Some examples on problem solving by using the symbolic viewpoint when dealing with polynomial systems of equations". in: "Computer Algebra in Science and Engineering", Editors: J. Fleischer, J. Grabmeier, F.W. Hehl and W. Kuechlin. Pages 102-116 World Scientific Publishing, 1995. S. Chandrasekhar: "Radiative Transfer", Dover, NY, 1960. C.T. Kelley: "Solution of the Chandrasekhar H-equation by Newton's method". J. Math. Phys., 21 (1980), pp. 1625-1628. Jorge J. More': "A collection of nonlinear model problems" in: "Computational Solution of Nonlinear Systems of Equations", Editors: Eugene L. Allgower and Kurt Georg. Pages 723-762. Lectures in Applied Mathematics, Volume 26, AMS, 1990. NOTE : the parameter c equals 0.51234. In general c can be any number in the interval (0,1] THE SOLUTIONS : 8 4 =========================================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 3.23234611912785E+01 3.30909953661441E-124 H2 : -4.08520581599796E+00 -4.90455824176778E-124 H3 : -2.67375455108960E+00 -1.06363913676892E-124 H4 : -2.23295583879165E+00 -2.83637103138378E-124 == err : 8.626E-15 = rco : 1.257E-01 = res : 3.553E-15 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 2.13878357402332E+00 6.89486635735363E-124 H2 : 4.65724101587131E+00 -7.56365608369007E-124 H3 : 1.87684762342963E+01 1.01063204129602E-123 H4 : -1.68612350186289E+01 2.42036994678082E-122 == err : 4.651E-14 = rco : 2.608E-02 = res : 1.132E-14 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 8.76084766994284E+01 -2.47845882550356E-118 H2 : -1.00582925744765E+02 2.13147458993306E-117 H3 : 1.66392100394692E+01 -2.35453588422839E-117 H4 : 7.52507489032550E+00 -1.23922941275178E-118 == err : 5.633E-13 = rco : 3.587E-03 = res : 6.839E-14 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 5.88966639781217E+01 -5.40561307813535E-110 H2 : -1.54901332806343E+01 2.44500037687968E-109 H3 : -3.97417697033545E+01 -2.39510241000458E-109 H4 : 2.92607213491990E+01 -1.83092273120350E-106 == err : 1.295E-12 = rco : 5.299E-03 = res : 1.896E-13 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 1.17379972280208E+00 -3.17242729664456E-117 H2 : 1.22825205825231E+00 -5.07588367463130E-116 H3 : 1.26270921307842E+00 -2.53794183731565E-116 H4 : 1.28672129216381E+00 0.00000000000000E+00 == err : 1.663E-16 = rco : 6.745E-01 = res : 3.053E-16 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 5.79689067953214E+00 -6.19614706375891E-119 H2 : 1.14667154697669E+02 2.16865147231562E-118 H3 : -1.16799284383068E+02 -1.67295970721491E-118 H4 : 5.68224655652938E+01 -4.10036228091310E-115 == err : 1.842E-12 = rco : 1.250E-03 = res : 4.268E-13 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 1.59631636700878E+02 4.61988642640884E-107 H2 : -3.81386641388061E+02 -3.03592536592581E-106 H3 : 2.47319505511373E+02 3.15747634092770E-106 H4 : -9.86088106502519E+01 4.44700508162162E-101 == err : 1.787E-10 = rco : 2.390E-04 = res : 3.943E-12 == solution 8 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : H1 : 3.18142927381331E+00 0.00000000000000E+00 H2 : 3.02411166378081E+01 -3.17383639986385E-128 H3 : -7.85804508743045E+00 2.88530581805804E-128 H4 : -4.33625865693282E+00 -7.21326454514511E-129 == err : 6.367E-15 = rco : 4.877E-02 = res : 6.217E-15 ==