5
10*H1 - 0.51234*H1*(1+(1/2)*H1+(1/3)*H2+(1/4)*H3+(1/5)*H4) - 10;
10*H2 - 0.51234*H2*(1+(2/3)*H1+(2/4)*H2+(2/5)*H3+(2/6)*H4) - 10;
10*H3 - 0.51234*H3*(1+(3/4)*H1+(3/5)*H2+(3/6)*H3+(3/7)*H4) - 10;
10*H4 - 0.51234*H4*(1+(4/5)*H1+(4/6)*H2+(4/7)*H3+(4/8)*H4) - 10;
10*H5 - 0.51234*H5*(1+(5/6)*H1+(5/7)*H2+(5/8)*H3+(5/9)*H4) - 10;
TITLE : The Chandrasekhar H-Equation for n=5
ROOT COUNTS :
total degree : 32
2-homogeneous Bezout number : 16
with partition : {H1 H2 H3 H4 }{H5 }
generalized Bezout number : 16
based on the set structure :
{H1 }{H1 H2 H3 H4 }
{H1 H2 }{H2 H3 H4 }
{H1 H2 H3 }{H3 H4 }
{H1 H2 H3 H4 }{H4 }
{H1 H2 H3 H4 }{H5 }
mixed volume : 16
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 :
16 5
===========================================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.22116479888813E+02 -1.32021704023690E-110
H2 : -4.35531944616897E+01 1.36699638418230E-110
H3 : -3.78128171700921E+02 8.05644256837480E-110
H4 : 3.31749948667482E+02 -9.06479731564235E-110
H5 : -1.75146615917029E+02 5.19471100480994E-107
== err : 1.664E-10 = rco : 1.255E-04 = res : 3.813E-12 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 6.08785935000550E+00 9.84653213001885E-109
H2 : 1.73522013784859E+02 5.25858040241007E-107
H3 : -2.12414822095362E+02 -1.58183208056303E-106
H4 : 3.76564638397181E+01 1.09003771904209E-106
H5 : 1.45682454486059E+01 9.76669538301870E-108
== err : 7.316E-12 = rco : 8.580E-04 = res : 3.686E-13 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.80453689446262E+00 7.56365608369007E-124
H2 : 3.11885194140922E+00 3.78182804184504E-124
H3 : 6.15673509859360E+00 3.78182804184504E-124
H4 : 2.11049384592180E+01 -1.51273121673801E-123
H5 : -2.48270982679489E+01 3.02546243347603E-123
== err : 8.835E-14 = rco : 1.359E-02 = res : 1.243E-14 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.77834160295590E+01 -1.39688318725853E-112
H2 : -1.67884835368245E+01 -1.20196925415269E-112
H3 : -7.59901182309949E+01 1.81919670898786E-111
H4 : 1.98467006174816E+01 -1.74123113574552E-111
H5 : 9.04282448137846E+00 -8.51124174562176E-112
== err : 7.350E-13 = rco : 5.456E-03 = res : 1.110E-13 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 5.01862469733524E+00 -4.12415548563793E-116
H2 : 1.04633987544357E+02 5.52002349616154E-115
H3 : -3.59594177863615E+01 -1.19283266353836E-114
H4 : -6.88416795761099E+01 6.78899441481936E-115
H5 : 6.14647583265297E+01 -1.99989816780473E-113
== err : 4.577E-12 = rco : 1.502E-03 = res : 2.891E-13 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 4.08432090095311E+01 5.79092418907521E-124
H2 : -3.90230419773742E+00 -5.67274206276756E-124
H3 : -2.58525293007850E+00 -1.41818551569189E-124
H4 : -2.17058948803169E+00 -2.00909614723018E-124
H5 : -1.96981472978245E+00 -1.18182126307657E-125
== err : 1.567E-14 = rco : 1.375E-01 = res : 5.285E-14 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.39619978844554E+02 4.28975619052278E-112
H2 : -1.46116086702129E+03 -3.49941572296431E-111
H3 : 2.20851939952594E+03 8.49763837737375E-111
H4 : -1.08212699646998E+03 -5.93979907605355E-111
H5 : 4.33612672042733E+02 8.48425110370624E-106
== err : 1.194E-08 = rco : 9.035E-06 = res : 3.940E-11 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.18899947055561E+00 -2.03125529591286E-126
H2 : 5.17221490140303E+00 5.07813823978215E-127
H3 : 3.63682693203126E+01 -2.36364252615315E-125
H4 : -1.15444212985878E+01 2.95455315769144E-125
H5 : -5.88446568719406E+00 -1.84659572355715E-126
== err : 2.984E-14 = rco : 2.707E-02 = res : 2.287E-14 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 6.41220091145328E+01 -3.41035934389290E-116
H2 : -1.01234762031905E+01 1.07069421261754E-116
H3 : -1.28642642831986E+01 1.75276608139612E-115
H4 : -3.62827537489224E+01 -1.55052384123503E-115
H5 : 3.81525025301074E+01 -1.44155096359529E-113
== err : 1.127E-12 = rco : 4.615E-03 = res : 1.297E-13 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.13589662856241E+02 -4.41804915039908E-111
H2 : -1.30917932781336E+02 1.32021704023690E-110
H3 : 1.50995923500665E+01 -9.19993764259574E-111
H4 : 7.08019245424940E+00 4.71042005005784E-112
H5 : 4.87669501314293E+00 6.23724585938694E-112
== err : 1.774E-13 = rco : 3.983E-03 = res : 1.461E-13 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.19666742019164E+00 0.00000000000000E+00
H2 : 4.03333440633433E+01 1.93629595742466E-121
H3 : -7.22654542456213E+00 -2.66240694145891E-121
H4 : -4.11840366528928E+00 1.08916647605137E-121
H5 : -3.17342712233808E+00 -6.05092486695206E-123
== err : 2.006E-14 = rco : 5.072E-02 = res : 1.132E-14 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 9.55766655450391E+00 2.45585498100182E-103
H2 : 4.50156084269161E+02 -2.20972446404212E-102
H3 : -1.05697990989999E+03 7.32766956248852E-102
H4 : 6.29451221470009E+02 -6.17920582170578E-102
H5 : -2.82166141279065E+02 1.11641293443496E-96
== err : 1.893E-09 = rco : 3.228E-05 = res : 2.048E-11 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.78330683940256E+02 2.65969203446269E-104
H2 : -3.39631281910615E+02 -7.98779640514041E-103
H3 : 7.51358384705354E+01 2.25681409350474E-102
H4 : 1.18349821893506E+02 -1.55831792314257E-102
H5 : -9.44546286584575E+01 -2.14812425246361E-100
== err : 2.110E-11 = rco : 3.097E-04 = res : 9.837E-13 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.14941934234002E+00 6.19614706375891E-120
H2 : 1.20222626880350E+00 0.00000000000000E+00
H3 : 1.23728159675204E+00 -1.54903676593973E-120
H4 : 1.26258767132566E+00 2.47845882550356E-119
H5 : 1.28182375002453E+00 0.00000000000000E+00
== err : 2.645E-16 = rco : 6.472E-01 = res : 5.829E-16 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.16324628178513E+02 3.31374904480557E-96
H2 : -5.63234808923598E+02 -2.14845053964547E-95
H3 : 4.43832707651702E+02 3.84482670629097E-95
H4 : -6.47374645129330E+01 -2.05463413456902E-95
H5 : -2.23874338973665E+01 -2.12092740326339E-96
== err : 1.063E-10 = rco : 1.202E-04 = res : 3.127E-12 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.43663114154226E+00 -9.74569665529209E-114
H2 : 1.34179171635312E+01 2.69630940796415E-112
H3 : 1.80969151070583E+02 -5.78244668213998E-112
H4 : -1.92972184496435E+02 3.42723665711105E-112
H5 : 1.13973709620809E+02 5.30116000081015E-107
== err : 9.814E-12 = rco : 4.122E-04 = res : 5.906E-13 ==