6
12*H1 - 0.51234*H1*(1+(1/2)*H1+(1/3)*H2+(1/4)*H3+(1/5)*H4+(1/6)*H5) - 12;
12*H2 - 0.51234*H2*(1+(2/3)*H1+(2/4)*H2+(2/5)*H3+(2/6)*H4+(2/7)*H5) - 12;
12*H3 - 0.51234*H3*(1+(3/4)*H1+(3/5)*H2+(3/6)*H3+(3/7)*H4+(3/8)*H5) - 12;
12*H4 - 0.51234*H4*(1+(4/5)*H1+(4/6)*H2+(4/7)*H3+(4/8)*H4+(4/9)*H5) - 12;
12*H5 - 0.51234*H5*(1+(5/6)*H1+(5/7)*H2+(5/8)*H3+(5/9)*H4+(5/10)*H5) - 12;
12*H6 - 0.51234*H6*(1+(6/7)*H1+(6/8)*H2+(6/9)*H3+(6/10)*H4+(6/11)*H5) - 12;
TITLE : The Chandrasekhar H-Equation for n=6
ROOT COUNTS :
total degree : 64
2-homogeneous Bezout number : 32
with partition : {H1 H2 H3 H4 H5 }{H6 }
generalized Bezout number : 32
based on the set structure :
{H1 }{H1 H2 H3 H4 H5 }
{H1 H2 }{H2 H3 H4 H5 }
{H1 H2 H3 }{H3 H4 H5 }
{H1 H2 H3 H4 }{H4 H5 }
{H1 H2 H3 H4 H5 }{H5 }
{H1 H2 H3 H4 H5 }{H6 }
mixed volume : 32
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]
TIMING : black-box solver on juno, executed on 9/7/1998.
---------------------------------------------------------------------
| TIMING INFORMATION SUMMARY |
---------------------------------------------------------------------
| root counts | start system | continuation | total time |
---------------------------------------------------------------------
| 0h 0m 1s964ms | 0h 0m 0s113ms | 0h 2m10s594ms | 0h 2m14s755ms |
---------------------------------------------------------------------
THE SOLUTIONS :
32 6
===========================================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.52478437068955E+00 -6.43803527809232E-106
H2 : 1.46544365626337E+01 -5.01928305940161E-105
H3 : 2.76720166523013E+02 1.78766185922902E-104
H4 : -3.57186149421082E+02 -9.06911382243016E-104
H5 : 6.83219998395596E+01 8.91650854176427E-104
H6 : 2.40247645670325E+01 9.15086665135832E-104
== err : 1.469E-11 = rco : 3.164E-04 = res : 1.029E-12 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.18039077320382E+00 5.68786937493494E-121
H2 : 5.26610247476354E+00 -3.38851792549315E-121
H3 : 4.75105406270813E+01 1.12305165530630E-119
H4 : -1.05582510453516E+01 -1.31668125104877E-119
H5 : -5.59012797702533E+00 -5.80888787227398E-121
H6 : -4.11914439626348E+00 1.74266636168219E-120
== err : 5.308E-14 = rco : 2.807E-02 = res : 2.864E-14 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.06422178828185E+01 1.62145877294106E-121
H2 : -8.00514340830750E+00 -1.13454841255351E-121
H3 : -8.38935162117220E+00 -1.96655058175942E-121
H4 : -1.28876016124331E+01 -6.53499885630822E-121
H5 : -3.53248833660917E+01 8.83435030575001E-121
H6 : 4.99495886280820E+01 -2.17639665614532E-118
== err : 3.934E-12 = rco : 3.095E-03 = res : 2.136E-13 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 4.58804372895969E+00 6.81576177013480E-118
H2 : 1.04960253120326E+02 4.86273621563799E-116
H3 : -2.28718536317627E+01 1.16983256563768E-116
H4 : -2.41697569481786E+01 -2.50671325611430E-115
H5 : -5.64714483945308E+01 1.90742191210754E-115
H6 : 7.28492743509224E+01 9.13659061433634E-115
== err : 2.831E-12 = rco : 1.277E-03 = res : 1.967E-13 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.38740582816837E+02 4.20689238953442E-112
H2 : -1.58482351149297E+02 -1.31242048291267E-111
H3 : 1.42464797385588E+01 9.74569665529209E-112
H4 : 6.80738141178888E+00 -6.11136394425608E-113
H5 : 4.72314505692644E+00 -2.20293351478998E-113
H6 : 3.77476995479413E+00 -1.42949573986804E-113
== err : 8.823E-13 = rco : 4.484E-03 = res : 5.480E-13 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.13287093598037E+00 8.10841961720302E-111
H2 : 1.11527393390388E+01 -1.49693900625287E-110
H3 : 1.58071147610527E+02 1.99591867500382E-110
H4 : -6.40738501585373E+01 3.69244954875707E-109
H5 : -1.02247669852195E+02 -4.25795984000815E-109
H6 : 1.13108398486882E+02 -2.47387466704474E-106
== err : 6.466E-12 = rco : 4.973E-04 = res : 5.711E-13 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 5.06454835814171E+00 7.04124123745707E-94
H2 : 3.10356875748233E+01 1.21079386555592E-91
H3 : 9.20668839208376E+02 -5.99554200687302E-91
H4 : -2.16762148564448E+03 1.03952690847802E-90
H5 : 1.24966106535584E+03 -5.90115941156242E-91
H6 : -6.59700749166740E+02 -4.16948944381597E-86
== err : 8.999E-09 = rco : 7.303E-06 = res : 5.323E-11 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.53149889674569E+02 3.25168417469372E-109
H2 : -4.68616318097814E+01 -7.91714407751516E-109
H3 : -5.73468603588809E+02 1.77054619128464E-108
H4 : 6.10507844195193E+02 -6.11083767663670E-108
H5 : -1.04518843618498E+02 5.28918448876012E-108
H6 : -3.41831622806402E+01 2.02120031155387E-107
== err : 2.461E-10 = rco : 9.160E-05 = res : 9.673E-12 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 6.15292600621423E+00 3.82551079375732E-110
H2 : 2.20797087828788E+02 -2.96393923238067E-108
H3 : -2.68430446519935E+02 8.56249111576639E-108
H4 : 3.38445546089952E+01 -6.50669488051246E-108
H5 : 1.36711159507517E+01 7.11877660751363E-109
H6 : 8.54778179300395E+00 -5.32244980001019E-110
== err : 3.234E-12 = rco : 9.179E-04 = res : 5.493E-13 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.14199064022975E+02 -7.60034396490713E-110
H2 : -2.22766014930415E+01 2.28699014844188E-110
H3 : -4.88628996216393E+01 8.43587502482084E-109
H4 : -4.35988193067735E+02 -1.33809714503381E-108
H5 : 4.31737285012113E+02 5.26839366922883E-109
H6 : -2.91329215867664E+02 -4.56759867957354E-104
== err : 2.200E-10 = rco : 6.809E-05 = res : 5.732E-12 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 9.94672041693105E+00 1.09521740091832E-120
H2 : 6.14431058322047E+02 -1.29013281819502E-119
H3 : -1.56344501410518E+03 3.12711797124082E-119
H4 : 1.14496293824953E+03 -2.23581673833879E-119
H5 : -1.67087048030661E+02 2.31674785843427E-120
H6 : -4.98546361554169E+01 -9.60886868871987E-119
== err : 5.018E-10 = rco : 1.819E-05 = res : 3.819E-11 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.20051765081231E+02 -3.72238612550373E-122
H2 : -9.92452325191326E+01 2.91658714961510E-121
H3 : -1.90797324323279E+03 -7.50252637885744E-121
H4 : 3.70493067096193E+03 8.03478913021555E-121
H5 : -1.91172872241648E+03 -3.11572403244350E-121
H6 : 9.38642195702171E+02 -1.53777720494730E-116
== err : 5.760E-08 = rco : 2.804E-06 = res : 9.682E-11 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.84126257293832E+02 -3.48117171047578E-103
H2 : -1.49531906105105E+03 2.76793277996357E-102
H3 : 1.85081599242891E+03 -6.66318256896047E-102
H4 : -3.51054285332415E+02 6.17876980661816E-102
H5 : -3.82533665464453E+02 -1.91061811393697E-102
H6 : 3.33959929087173E+02 7.40144331531329E-101
== err : 3.093E-09 = rco : 1.246E-05 = res : 5.007E-11 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 9.47367473427292E+01 7.26110984034247E-120
H2 : -1.68398255497713E+01 1.54903676593973E-120
H3 : -9.84597679724875E+01 -4.33730294463124E-119
H4 : 1.80463186901779E+01 1.60712564466247E-119
H5 : 8.55176536416566E+00 2.16865147231562E-119
H6 : 5.86084333779961E+00 -8.28976706772432E-121
== err : 3.160E-13 = rco : 5.725E-03 = res : 1.856E-13 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.67341159040361E+02 -3.46784372650810E-112
H2 : -7.06060024230957E+02 2.17329035413014E-111
H3 : 5.56289175670613E+02 -3.37525960828283E-111
H4 : -5.78476128079951E+01 9.30714030580395E-112
H5 : -2.09140428193520E+01 6.46464544801042E-112
H6 : -1.21620426853564E+01 1.71361832855553E-112
== err : 8.068E-11 = rco : 1.352E-04 = res : 5.530E-12 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 5.16199523004502E+00 -1.01517673492626E-115
H2 : 1.37915298133592E+02 2.43642416382302E-114
H3 : -4.00395849675597E+01 -4.87284832764605E-114
H4 : -1.34736751348682E+02 2.37145285278774E-112
H5 : 3.77342808274185E+01 -2.69630940796415E-112
H6 : 1.54735341369303E+01 8.60869871217468E-113
== err : 1.421E-12 = rco : 1.556E-03 = res : 5.849E-13 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.22262596020634E+02 1.95130807439710E-94
H2 : -9.34013107408185E+02 -1.67083177003521E-93
H3 : 2.76071157906576E+02 4.46456361374335E-93
H4 : 1.39756349283305E+03 -4.72673748960611E-93
H5 : -1.05584890147724E+03 1.73162041145317E-93
H6 : 6.04547529200941E+02 -1.40136880498877E-89
== err : 4.122E-09 = rco : 8.591E-06 = res : 2.737E-11 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.13154550436088E+00 -4.84073989356165E-122
H2 : 1.18203024240421E+00 0.00000000000000E+00
H3 : 1.21673759622537E+00 -4.84073989356165E-122
H4 : 1.24247204712991E+00 -9.68147978712330E-122
H5 : 1.26245248469364E+00 -1.93629595742466E-121
H6 : 1.27847129867409E+00 -1.45222196806849E-121
== err : 4.215E-16 = rco : 6.287E-01 = res : 8.049E-16 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.24285776615435E+02 5.68781681796161E-103
H2 : -4.41022477694693E+02 -2.50970284432250E-102
H3 : 8.29771287294896E+01 1.36734331476639E-102
H4 : 2.30294046208341E+02 3.65729455493001E-102
H5 : -5.77258190059000E+01 -3.11837990663560E-102
H6 : -2.20162127700374E+01 1.74406035046734E-104
== err : 1.364E-11 = rco : 2.901E-04 = res : 2.507E-12 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.82923811525524E+00 2.71081434039452E-120
H2 : 3.28933728225724E+00 5.80888787227398E-121
H3 : 7.08676065981003E+00 8.42288741479727E-120
H4 : 4.20328901416938E+01 3.71768823825535E-119
H5 : -1.54295713463446E+01 -5.26672500419507E-119
H6 : -7.45663880688855E+00 6.97066544672877E-120
== err : 1.483E-14 = rco : 1.865E-02 = res : 1.066E-14 ==
solution 21 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.94793627290892E+01 -4.38556349488144E-113
H2 : -1.05185697150827E+01 -2.92370899658763E-113
H3 : -1.46864422301210E+01 1.07202663208213E-112
H4 : -7.18432368872510E+01 4.02822128418740E-112
H5 : 2.36041239781795E+01 -4.93781963868133E-112
H6 : 1.06095314146729E+01 -1.46185449829381E-113
== err : 6.239E-13 = rco : 5.664E-03 = res : 3.091E-13 ==
solution 22 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 6.20972279252226E+02 7.35387919477345E-107
H2 : -4.16115303994330E+03 -3.46256961536350E-106
H3 : 1.07798838506612E+04 7.70088628907849E-106
H4 : -1.18761836483198E+04 -1.07047438949592E-105
H5 : 4.67528921320353E+03 5.88249626388845E-106
H6 : -1.94780951950533E+03 -8.88420435832877E-98
== err : 9.365E-07 = rco : 3.252E-07 = res : 5.081E-09 ==
solution 23 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 4.91650129283578E+01 -6.92473396333930E-128
H2 : -3.77987005988417E+00 1.15412232722322E-127
H3 : -2.52153942738901E+00 8.65591745417413E-129
H4 : -2.12365158110310E+00 8.65591745417413E-129
H5 : -1.93129700730984E+00 6.60363175579126E-129
H6 : -1.81904766441278E+00 3.89641092067872E-129
== err : 1.198E-14 = rco : 1.490E-01 = res : 5.596E-14 ==
solution 24 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.62584893814356E+00 -3.46236698166965E-128
H2 : 2.50334573768182E+00 4.61648930889287E-128
H3 : 4.04817763884991E+00 -4.61648930889287E-128
H4 : 7.54007149782880E+00 1.78888960719599E-127
H5 : 2.30912110401676E+01 -6.92473396333930E-128
H6 : -3.51057955714387E+01 1.84659572355715E-126
== err : 2.290E-14 = rco : 8.167E-03 = res : 1.305E-14 ==
solution 25 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.99347908131701E+02 -8.17528289281565E-105
H2 : -3.35639566582015E+02 6.01700820911232E-104
H3 : 4.73991112700689E+01 5.31938406892538E-104
H4 : 4.13113056961279E+01 -4.58687872172910E-103
H5 : 8.63898963367872E+01 3.74972975350478E-103
H6 : -1.03652152770022E+02 5.42053956925249E-102
== err : 1.898E-11 = rco : 3.596E-04 = res : 2.442E-12 ==
solution 26 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 8.84076528528998E+00 -5.15745385620987E-107
H2 : 4.67611934855825E+02 -8.86134667203696E-105
H3 : -8.93088316297117E+02 3.83842305697215E-104
H4 : 2.05389217530589E+02 -5.14642574022425E-104
H5 : 2.50055053478088E+02 2.19267899921060E-104
H6 : -2.34715226321462E+02 1.23910037712109E-103
== err : 6.830E-10 = rco : 4.683E-05 = res : 1.192E-11 ==
solution 27 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.62832103157822E+00 3.51281686800672E-108
H2 : 6.96627562471526E+00 1.71915128540329E-107
H3 : 2.35781865571705E+01 1.11771445800214E-106
H4 : 2.55080987665569E+02 -3.50855890816672E-106
H5 : -2.82218533004220E+02 2.14601175936411E-106
H6 : 2.04753315835934E+02 7.84478345640209E-102
== err : 2.009E-10 = rco : 1.664E-04 = res : 1.785E-12 ==
solution 28 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.19315044191211E+00 4.84073989356165E-122
H2 : 4.95601512692032E+01 2.42036994678082E-122
H3 : -6.87446221283792E+00 0.00000000000000E+00
H4 : -3.98275366522494E+00 -1.93629595742466E-121
H5 : -3.08743098038080E+00 -1.49382207652879E-122
H6 : -2.65300087552096E+00 2.42036994678082E-122
== err : 1.542E-14 = rco : 5.413E-02 = res : 1.377E-14 ==
solution 29 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 4.32179382984064E+02 -2.21505468443320E-143
H2 : -1.96481399366718E+03 1.06490799832789E-142
H3 : 3.24004802502049E+03 2.26235389755690E-143
H4 : -1.95698757145588E+03 -4.22759866378864E-142
H5 : 2.55609394993315E+02 3.26880106950545E-142
H6 : 7.09346854735957E+01 -1.32818192534447E-142
== err : 1.896E-08 = rco : 4.075E-06 = res : 1.257E-10 ==
solution 30 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.42918378144617E+01 4.18208726722381E-119
H2 : 1.30126397429125E+03 -2.25297347397912E-118
H3 : -5.20169933548546E+03 6.36643521682711E-118
H4 : 6.94832727772565E+03 1.00421527274741E-118
H5 : -3.05614851647160E+03 -8.00674808686401E-118
H6 : 1.36896828770925E+03 -2.34266783371164E-111
== err : 5.440E-07 = rco : 9.123E-07 = res : 9.696E-10 ==
solution 31 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.41695710081988E+00 6.30451306976616E-99
H2 : 2.92081929343525E+02 3.61032212632294E-97
H3 : -1.33214715348052E+02 -1.57416509730112E-96
H4 : -8.17664060049490E+02 2.09853913773665E-96
H5 : 6.90188543805867E+02 -8.88915581542616E-97
H6 : -4.24890826053001E+02 1.33263917436313E-92
== err : 1.233E-09 = rco : 2.472E-05 = res : 1.854E-11 ==
solution 32 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.36121472337384E+02 1.46042690601032E-113
H2 : -3.56639821901531E+01 -1.18252227482426E-113
H3 : -3.27583064967396E+02 -1.79813179173814E-112
H4 : 1.09515971414293E+02 3.58052834408492E-112
H5 : 1.56418258258544E+02 -1.80016214520799E-112
H6 : -1.60934053276268E+02 1.30429906903326E-111
== err : 6.656E-11 = rco : 1.766E-04 = res : 1.643E-12 ==