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+(1/3)*H1+(1/4)*H2+(1/5)*H3+ (1/6)*H4+ (1/7)*H5) - 12;
12*H3 - 0.51234*H3*(1/3+(1/4)*H1+(1/5)*H2+(1/6)*H3+ (1/7)*H4+ (1/8)*H5) - 12;
12*H4 - 0.51234*H4*(1/4+(1/5)*H1+(1/6)*H2+(1/7)*H3+ (1/8)*H4+ (1/9)*H5) - 12;
12*H5 - 0.51234*H5*(1/5+(1/6)*H1+(1/7)*H2+(1/8)*H3+ (1/9)*H4+(1/10)*H5) - 12;
12*H6 - 0.51234*H6*(1/6+(1/7)*H1+(1/8)*H2+(1/9)*H3+(1/10)*H4+(1/11)*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 : start residual : 3.201E-10
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -4.57220913771979E+02 4.65219009326979E-58
H2 : 4.07309932574937E+03 -3.67212352777489E-57
H3 : -9.45189360192819E+03 8.95966403399155E-57
H4 : 6.18462414792198E+03 -8.52780135890005E-57
H5 : 1.16594553950104E+02 2.74504982675552E-57
H6 : 4.01323309123242E+01 -5.83359064992624E-59
== err : 2.969E-08 = rco : 1.135E-05 = res : 3.201E-10 = real regular ==
solution : 2 : start residual : 7.638E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -2.06523449568079E+00 -2.87375661629113E-47
H2 : -5.32545450172325E+00 -6.56858655152258E-47
H3 : 1.38187861009023E+02 -5.25486924121806E-46
H4 : 6.81069340432897E+00 7.00649232162409E-46
H5 : 3.97002136297403E+00 5.57315972037750E-49
H6 : 2.99520740050986E+00 2.05268329735081E-48
== err : 3.104E-14 = rco : 4.722E-02 = res : 7.638E-14 = real regular ==
solution : 3 : start residual : 2.410E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -7.97836660412259E+00 -4.20971132898361E-46
H2 : 6.27151468822791E+02 -1.78049749212209E-44
H3 : 1.43196327030901E+02 1.09695395410427E-43
H4 : -5.21505053922133E+03 -1.88003894748829E-43
H5 : 4.96504408821769E+03 9.78965718172547E-44
H6 : 1.02809321582547E+02 2.23750764981839E-45
== err : 6.890E-10 = rco : 3.276E-05 = res : 2.410E-11 = real regular ==
solution : 4 : start residual : 1.353E-10
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -5.21021604047962E+00 1.16469953940449E-56
H2 : -3.81090921164907E+01 -4.40377899153122E-56
H3 : 4.07767933150840E+03 3.07466738848563E-55
H4 : -1.19405704479503E+04 -6.93014894991506E-55
H5 : 8.46551838307313E+03 4.29218788156861E-55
H6 : 1.53873064561245E+02 -1.38005320611015E-55
== err : 3.108E-08 = rco : 1.155E-05 = res : 1.353E-10 = real regular ==
solution : 5 : start residual : 2.001E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.34211980147062E+02 1.06346222015768E-39
H2 : 4.55165850453470E+01 -3.48544406512126E-39
H3 : -1.93411782000459E+03 -9.21224820834849E-39
H4 : 2.07573035688293E+03 3.34510794755483E-38
H5 : 5.01157819618734E+01 -2.25054362772177E-38
H6 : 2.00955211050613E+01 -4.68414840266785E-40
== err : 1.826E-10 = rco : 1.592E-04 = res : 2.001E-11 = real regular ==
solution : 6 : start residual : 3.553E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.53680282276760E+00 -2.00457353256915E-51
H2 : -2.41516758763844E+00 -1.25285845785572E-52
H3 : -4.16306737552497E+00 2.83981250447296E-51
H4 : -9.54355566724332E+00 -2.53912647458759E-50
H5 : 2.53113472444298E+02 2.67276471009220E-50
H6 : 1.07276655049908E+01 3.34095588761524E-51
== err : 3.924E-14 = rco : 2.582E-02 = res : 3.553E-15 = real regular ==
solution : 7 : start residual : 3.453E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.41896532580621E+00 2.46628529721168E-42
H2 : 1.31328432405752E+01 3.81153182296350E-43
H3 : -5.79336016600813E+02 -2.44655501483398E-40
H4 : -6.51057190557389E+01 5.91191006517069E-40
H5 : 1.00403641419817E+03 -3.52992688357279E-40
H6 : 3.20649410293327E+01 5.17919912414452E-42
== err : 4.350E-12 = rco : 7.310E-04 = res : 3.453E-12 = real regular ==
solution : 8 : start residual : 1.070E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.05517846469115E+02 -4.89222852535276E-47
H2 : -7.49054663378506E+02 1.54635475067094E-46
H3 : -6.79206345946955E+01 -1.23503111723940E-46
H4 : 9.06576345253170E+02 1.41635147517206E-46
H5 : 2.93930100552015E+01 -1.50187994589501E-46
H6 : 1.34266962028177E+01 1.75333364982048E-48
== err : 7.410E-12 = rco : 3.964E-04 = res : 1.070E-12 = real regular ==
solution : 9 : start residual : 7.304E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.80775232975565E+00 -3.61357788804465E-47
H2 : 1.69969977459810E+01 8.92917234347601E-47
H3 : -1.03636644124761E+03 8.67600807013608E-46
H4 : 1.30998760500262E+03 -3.03797128007919E-45
H5 : 3.47722339222578E+01 2.24016170517551E-45
H6 : 1.48367453873608E+01 5.73896038550996E-47
== err : 1.785E-11 = rco : 5.013E-04 = res : 7.304E-12 = real regular ==
solution : 10 : start residual : 7.742E-09
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 6.25623604870343E+02 2.39176933263890E-69
H2 : -9.13232558627337E+03 -3.12761577397370E-68
H3 : 3.71893469822835E+04 1.13950369001096E-67
H4 : -5.63730069281478E+04 -1.55366588015603E-67
H5 : 2.83856752499103E+04 7.08876435865771E-68
H6 : 4.16215590704210E+02 3.39447828822257E-66
== err : 7.554E-06 = rco : 7.483E-07 = res : 7.742E-09 = real regular ==
solution : 11 : start residual : 1.450E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.03564914423259E+00 2.11221641163380E-38
H2 : -4.76553878517979E+02 -1.39130776679357E-37
H3 : 6.75314060924763E+02 2.76241172443238E-37
H4 : 2.02924135848416E+01 -1.91017832008622E-37
H5 : 9.23626953160546E+00 3.21424236552969E-38
H6 : 5.98166396977795E+00 -1.89410710825857E-39
== err : 8.298E-12 = rco : 1.577E-03 = res : 1.450E-12 = real regular ==
solution : 12 : start residual : 5.139E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -9.62700534381646E+00 -1.49808462000668E-46
H2 : 1.06848491282599E+03 -1.32445968622732E-44
H3 : -2.65708544672153E+03 7.48162008218422E-44
H4 : -1.84965867648922E+02 -1.22547929762906E-43
H5 : 2.24862773227698E+03 6.17666088728173E-44
H6 : 6.20363120194779E+01 7.62742901907271E-46
== err : 5.426E-10 = rco : 5.301E-05 = res : 5.139E-11 = real regular ==
solution : 13 : start residual : 1.151E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 5.23543718036476E+00 -1.09772116501349E-40
H2 : -2.16123803289949E+02 1.66451836786359E-40
H3 : -2.03446085259661E+01 -5.73971850987445E-40
H4 : -2.84349576720872E+01 1.17090257601439E-39
H5 : 5.88869439187304E+02 -6.54327910125687E-40
H6 : 2.14239889431730E+01 -1.87258316384654E-40
== err : 1.573E-12 = rco : 4.011E-03 = res : 1.151E-12 = real regular ==
solution : 14 : start residual : 3.197E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -3.16247883043659E+00 4.27642353614751E-49
H2 : 8.76396268558204E+01 -2.02715979615730E-48
H3 : 4.85275876988387E+00 1.36845553156720E-48
H4 : 2.97457399255287E+00 1.28292706084425E-49
H5 : 2.32842576276792E+00 1.20274411954149E-50
H6 : 2.00122901122237E+00 2.13821176807376E-50
== err : 3.584E-14 = rco : 9.684E-02 = res : 3.197E-14 = real regular ==
solution : 15 : start residual : 3.695E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 5.83078394225069E+00 5.30923962163387E-40
H2 : -2.79715194195156E+02 -1.54283633545425E-38
H3 : -3.63941977237016E+01 5.71216786102705E-38
H4 : 5.72137788192116E+02 -4.76626225059974E-38
H5 : 2.03939873091585E+01 1.65303893084384E-39
H6 : 9.91307823833911E+00 -4.57742551162488E-40
== err : 1.245E-12 = rco : 2.056E-03 = res : 3.695E-13 = real regular ==
solution : 16 : start residual : 1.364E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 9.98711528257885E+01 2.15211014456624E-47
H2 : 2.06424035480832E+01 -1.47215880231878E-47
H3 : 5.82675116185308E+01 2.87119076216944E-46
H4 : -2.90863896450069E+03 -1.04276311505421E-45
H5 : 3.19519177275311E+03 7.85151361236684E-46
H6 : 7.19743242243930E+01 -2.76043139258322E-46
== err : 3.282E-10 = rco : 1.171E-04 = res : 1.364E-11 = real regular ==
solution : 17 : start residual : 7.276E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -4.10536798363169E+02 -1.35672596277157E-38
H2 : 3.14710725668885E+03 1.10546978500182E-37
H3 : -5.28367396968053E+03 -2.77274321775015E-37
H4 : -3.07372681010780E+02 2.72820300211352E-37
H5 : 3.36662804364178E+03 -9.18010578469320E-38
H6 : 8.67333630437480E+01 9.85079189257203E-40
== err : 1.913E-09 = rco : 1.322E-05 = res : 7.276E-11 = real regular ==
solution : 18 : start residual : 2.274E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.34861661292549E+02 2.53912647458759E-50
H2 : 2.58133083902869E+02 -1.92439059126638E-49
H3 : 9.64981958905874E+00 2.13821176807376E-49
H4 : 4.94308920113334E+00 5.26200552299401E-51
H5 : 3.48610103751311E+00 1.33638235504610E-51
H6 : 2.79793102964639E+00 1.23719772713252E-51
== err : 2.682E-13 = rco : 9.562E-03 = res : 2.274E-13 = real regular ==
solution : 19 : start residual : 4.441E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.71156006894815E+00 9.86514118884671E-43
H2 : -3.12579669849605E+00 8.96831017167883E-43
H3 : -7.44725547353586E+00 3.58732406867153E-42
H4 : 1.92025213960664E+02 -3.01335221768409E-41
H5 : 8.76593791644231E+00 3.01335221768409E-41
H6 : 4.96379025156254E+00 6.50202487446715E-43
== err : 6.639E-14 = rco : 3.121E-02 = res : 4.441E-14 = real regular ==
solution : 20 : start residual : 4.547E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.20508390959360E+02 1.89231741474527E-48
H2 : 3.51686918584344E+01 -9.23707483807863E-48
H3 : -1.08118525347965E+03 1.55447995538962E-47
H4 : -1.03162745154704E+02 -1.23160997841048E-47
H5 : 1.44707613920418E+03 4.27642353614751E-48
H6 : 4.34301244891843E+01 -2.54981753342795E-48
== err : 1.164E-11 = rco : 2.620E-04 = res : 4.547E-12 = real regular ==
solution : 21 : start residual : 4.857E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.12313279973486E+00 3.42113882891801E-49
H2 : 1.07718646032708E+00 -5.83070078730661E-51
H3 : 1.05806832021072E+00 0.00000000000000E+00
H4 : 1.04701359840710E+00 -8.55284707229503E-50
H5 : 1.03966709118573E+00 0.00000000000000E+00
H6 : 1.03438273717615E+00 -7.09846340082770E-51
== err : 1.513E-16 = rco : 9.070E-01 = res : 4.857E-16 = real regular ==
solution : 22 : start residual : 9.095E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.84533638236230E+02 6.99528193390949E-42
H2 : -5.78762062558846E+02 -3.03128883802744E-41
H3 : -3.79681050302224E+01 -8.44142194909270E-42
H4 : -4.50563842064075E+01 1.23762680369168E-40
H5 : 8.48713156718530E+02 -9.57815526335299E-41
H6 : 2.90175648851437E+01 -2.22771423366038E-42
== err : 3.299E-12 = rco : 1.187E-03 = res : 9.095E-13 = real regular ==
solution : 23 : start residual : 1.137E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -7.29882622703505E+01 -4.20389539297445E-45
H2 : -9.20669759084681E+00 1.36626600271670E-44
H3 : -1.48090344402121E+01 8.75811540203011E-45
H4 : 3.19103765396552E+02 -2.10194769648723E-44
H5 : 1.31242944284201E+01 -1.57646077236542E-45
H6 : 6.93990677309839E+00 -5.91172789637032E-46
== err : 1.882E-13 = rco : 9.766E-03 = res : 1.137E-13 = real regular ==
solution : 24 : start residual : 1.137E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -6.55358637543527E+01 -1.76324152623343E-38
H2 : -7.11361606508209E+00 3.23260946476129E-38
H3 : -8.27835279186435E+00 8.81620763116716E-39
H4 : -1.58592951721058E+01 2.35098870164458E-38
H5 : 3.78959532661966E+02 -4.11423022787801E-38
H6 : 1.49984174843374E+01 -1.41426664083306E-38
== err : 1.458E-13 = rco : 9.390E-03 = res : 1.137E-13 = real regular ==
solution : 25 : start residual : 3.056E-10
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -2.22185958732431E+02 8.16763287600941E-55
H2 : -1.12246227256862E+02 -8.28248484597762E-55
H3 : 8.10855675234779E+03 -1.48262691801453E-53
H4 : -1.98426077093258E+04 3.48552579236669E-53
H5 : 1.26745086273289E+04 -2.03751676158107E-53
H6 : 2.15130911826897E+02 2.08212134372789E-54
== err : 1.748E-07 = rco : 4.080E-06 = res : 3.056E-10 = real regular ==
solution : 26 : start residual : 5.562E-09
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.77496802919043E+01 -1.17817644860004E-65
H2 : -3.41023205607909E+03 6.25486983016619E-65
H3 : 1.99273233438124E+04 -8.48840236315595E-65
H4 : -3.55768465242109E+04 8.70388405797245E-66
H5 : 1.96950601425490E+04 2.67178755541703E-65
H6 : 3.07296571872084E+02 -1.26814860474698E-64
== err : 8.740E-07 = rco : 1.487E-06 = res : 5.562E-09 = real regular ==
solution : 27 : start residual : 1.160E-10
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.07217384598114E+01 -2.12305300018044E-36
H2 : 1.38287157793907E+03 -8.89014622583388E-34
H3 : -4.75322457020208E+03 5.49501291212790E-33
H4 : 3.72168524490599E+03 -9.76751169940062E-33
H5 : 7.78754718505708E+01 5.26282926795348E-33
H6 : 2.87047765147015E+01 8.10650291685820E-35
== err : 3.544E-09 = rco : 3.925E-05 = res : 1.160E-10 = real regular ==
solution : 28 : start residual : 1.310E-10
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -3.40231771443673E+02 -1.44484956161694E-44
H2 : 1.84720711999059E+03 3.44128249434268E-43
H3 : 2.84749106063236E+02 -1.54865375596397E-42
H4 : -8.66626954593298E+03 2.36267679200566E-42
H5 : 7.43362559545616E+03 -1.15693609696392E-42
H6 : 1.43738367462758E+02 -7.59875987568637E-44
== err : 4.778E-09 = rco : 9.539E-06 = res : 1.310E-10 = real regular ==
solution : 29 : start residual : 1.776E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.95871012519871E+01 3.66412035743838E-53
H2 : 2.88461820516351E+00 0.00000000000000E+00
H3 : 1.97461892961171E+00 -5.22024357439882E-54
H4 : 1.65903700308561E+00 7.27868617827564E-55
H5 : 1.49842915963576E+00 1.95759134039956E-54
H6 : 1.40101212111789E+00 4.07831529249908E-55
== err : 1.283E-14 = rco : 3.191E-01 = res : 1.776E-15 = real regular ==
solution : 30 : start residual : 1.705E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.83346251527689E+00 -1.48452079539393E-37
H2 : 7.70837456783719E+00 -1.27320731873439E-36
H3 : 3.12217244636650E+01 3.65402418953108E-35
H4 : -1.83563389063952E+03 -9.86709958080228E-35
H5 : 2.21694546905795E+03 6.60392726291961E-35
H6 : 5.31394392492683E+01 8.38848380781131E-37
== err : 1.252E-10 = rco : 2.835E-04 = res : 1.705E-11 = real regular ==
solution : 31 : start residual : 9.095E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.47985772574652E+02 -8.01230713732598E-46
H2 : -1.27617273735208E+03 6.91891116760379E-45
H3 : 1.26030418136836E+03 -1.55347071943509E-44
H4 : 3.21541812545569E+01 1.02743641310066E-44
H5 : 1.33118531015810E+01 -4.90591308066843E-46
H6 : 8.10182094588233E+00 2.89727694573994E-47
== err : 1.566E-10 = rco : 3.473E-04 = res : 9.095E-12 = real regular ==
solution : 32 : start residual : 7.550E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -8.80704567460267E+01 7.00649232162409E-45
H2 : -1.56855527919553E+01 -1.82168800362226E-44
H3 : 2.74789659113203E+02 7.00649232162409E-45
H4 : 1.13178778216492E+01 4.37905770101505E-45
H5 : 5.94388526948817E+00 5.52856034753151E-46
H6 : 4.18761854796129E+00 -3.77693726712548E-46
== err : 2.498E-13 = rco : 8.460E-03 = res : 7.550E-14 = real regular ==