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+(1/3)*H1+(1/4)*H2+(1/5)*H3+(1/6)*H4) - 10;
10*H3 - 0.51234*H3*(1/3+(1/4)*H1+(1/5)*H2+(1/6)*H3+(1/7)*H4) - 10;
10*H4 - 0.51234*H4*(1/4+(1/5)*H1+(1/6)*H2+(1/7)*H3+(1/8)*H4) - 10;
10*H5 - 0.51234*H5*(1/5+(1/6)*H1+(1/7)*H2+(1/8)*H3+(1/9)*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]
---------------------------------------------------------------------
| TIMING INFORMATION SUMMARY |
---------------------------------------------------------------------
| root counts | start system | continuation | total time |
---------------------------------------------------------------------
| 0h 0m 1s300ms | 0h 0m 0s219ms | 0h 1m37s 97ms | 0h 1m42s298ms |
---------------------------------------------------------------------
THE SOLUTIONS :
16 5
===========================================================================
solution : 1 : start residual : 1.776E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -3.16674045924203E+00 1.87093529706454E-50
H2 : 7.34270964905843E+01 -6.41463530422127E-50
H3 : 4.79552337262059E+00 4.27642353614751E-50
H4 : 2.95302830145041E+00 -1.06910588403688E-50
H5 : 2.31562822401247E+00 0.00000000000000E+00
== err : 2.963E-15 = rco : 9.146E-02 = res : 1.776E-15 = real regular ==
solution : 2 : start residual : 5.684E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.70457128616768E+00 -4.59801058606581E-46
H2 : -3.12449633129058E+00 1.48887961834512E-45
H3 : -7.47259494596501E+00 2.45227231256843E-45
H4 : 1.68653887992348E+02 -5.60519385729927E-45
H5 : 8.69995153693113E+00 2.10194769648723E-45
== err : 6.720E-14 = rco : 3.350E-02 = res : 5.684E-14 = real regular ==
solution : 3 : start residual : 1.599E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -2.05916719163918E+00 2.08809742975953E-53
H2 : -5.34648695245143E+00 8.35238971903811E-53
H3 : 1.17839410737874E+02 -4.38500460249501E-52
H4 : 6.74455641760536E+00 4.17619485951906E-52
H5 : 3.94632099393055E+00 1.95759134039956E-54
== err : 1.965E-14 = rco : 4.388E-02 = res : 1.599E-14 = real regular ==
solution : 4 : start residual : 1.819E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.68862380532704E+02 1.25107926894920E-41
H2 : -6.20216226620403E+02 -4.34066212309255E-41
H3 : -6.69372678894961E+01 4.19716916034569E-41
H4 : 7.87222852963777E+02 -1.08516553077314E-41
H5 : 2.89233002193038E+01 -1.15481006445008E-41
== err : 3.074E-11 = rco : 3.592E-04 = res : 1.819E-12 = real regular ==
solution : 5 : start residual : 4.093E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.87802697322430E+00 -6.18813401845839E-42
H2 : 1.72732336898438E+01 -4.51778624898321E-42
H3 : -8.94431334194718E+02 5.99083119468146E-41
H4 : 1.14698159004317E+03 -5.04019031648350E-41
H5 : 3.44929045393284E+01 4.06040243022759E-41
== err : 1.032E-11 = rco : 5.876E-04 = res : 4.093E-12 = real regular ==
solution : 6 : start residual : 8.185E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.09802454091331E+02 -1.34524652575182E-41
H2 : 4.51601400819249E+01 9.10283482425401E-41
H3 : -1.64483573357954E+03 -1.75016573000312E-40
H4 : 1.79797428369885E+03 1.00355390821086E-40
H5 : 4.92914301551564E+01 4.61083246701438E-41
== err : 2.611E-11 = rco : 1.883E-04 = res : 8.185E-12 = real regular ==
solution : 7 : start residual : 6.217E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 3.22991878818205E+01 8.35238971903811E-53
H2 : 2.83832239787994E+00 -8.35238971903811E-53
H3 : 1.95548950755868E+00 -8.24480867695236E-55
H4 : 1.64709138471811E+00 1.95759134039956E-54
H5 : 1.48969896626639E+00 0.00000000000000E+00
== err : 4.263E-15 = rco : 3.087E-01 = res : 6.217E-15 = real regular ==
solution : 8 : start residual : 3.864E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -7.34830645308781E+01 -5.34552942018439E-49
H2 : -1.56911709496064E+01 2.03130117967007E-48
H3 : 2.34062076413485E+02 -4.19089506542456E-48
H4 : 1.12015413215740E+01 3.25008188747211E-48
H5 : 5.90624488166850E+00 -2.67276471009220E-51
== err : 2.158E-13 = rco : 7.789E-03 = res : 3.864E-14 = real regular ==
solution : 9 : start residual : 7.276E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -3.84174526335672E+02 -3.30725335603851E-35
H2 : 3.42875736910369E+03 2.11518453486962E-34
H3 : -8.01204805809732E+03 -3.85891285487941E-34
H4 : 5.35374624539852E+03 2.11859346848701E-34
H5 : 1.14672895497658E+02 2.90391185691261E-35
== err : 7.027E-09 = rco : 2.361E-05 = res : 7.276E-11 = real regular ==
solution : 10 : start residual : 3.268E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 5.96391831027843E+00 -7.00649232162409E-45
H2 : -2.37225566643728E+02 -2.79699173479234E-42
H3 : -3.63992516720591E+01 1.07619722060146E-41
H4 : 5.02193011211055E+02 -8.65441931567007E-42
H5 : 2.02397988917478E+01 2.44526582024681E-43
== err : 2.925E-12 = rco : 1.871E-03 = res : 3.268E-13 = real regular ==
solution : 11 : start residual : 2.709E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -6.08290197743639E+01 -2.67276471009220E-50
H2 : -9.16994776228140E+00 4.27642353614751E-50
H3 : -1.48426666282313E+01 1.23615367841764E-50
H4 : 2.80104928836987E+02 -2.94004118110142E-50
H5 : 1.30207462380258E+01 1.57024926717917E-50
== err : 2.809E-13 = rco : 1.063E-02 = res : 2.709E-14 = real regular ==
solution : 12 : start residual : 4.485E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.13007721987734E+02 -4.65274880732849E-47
H2 : 2.15497977197668E+02 6.56858655152258E-47
H3 : 9.52525263879487E+00 -8.21073318940323E-48
H4 : 4.90446909986979E+00 1.19739859012130E-48
H5 : 3.46567533836083E+00 6.41463530422127E-49
== err : 9.449E-14 = rco : 9.020E-03 = res : 4.485E-14 = real regular ==
solution : 13 : start residual : 1.751E-11
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -1.07654680101267E+01 2.45923979274081E-37
H2 : 1.16828798793342E+03 -1.16263738136017E-37
H3 : -4.03369497714726E+03 -1.42528690037202E-36
H4 : 3.22354241804999E+03 1.39332814770904E-36
H5 : 7.66199275517539E+01 1.08127113176419E-36
== err : 1.596E-09 = rco : 5.782E-05 = res : 1.751E-11 = real regular ==
solution : 14 : start residual : 2.188E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.20457103659868E+00 -3.37954625861408E-37
H2 : -4.05928912172771E+02 1.06382238749417E-36
H3 : 5.73999581049296E+02 -7.99336158559156E-37
H4 : 2.00829588748873E+01 3.37954625861408E-38
H5 : 9.18082623108647E+00 1.01937400735370E-38
== err : 2.355E-12 = rco : 1.441E-03 = res : 2.188E-12 = real regular ==
solution : 15 : start residual : 6.366E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.03990221304732E+02 -1.43633092593294E-44
H2 : -1.06128399963747E+03 -1.96181785005474E-44
H3 : 1.05557015488455E+03 1.81818475746145E-43
H4 : 3.14814500172305E+01 -1.59222538008907E-43
H5 : 1.31196853665994E+01 -4.02599617387072E-45
== err : 2.292E-11 = rco : 3.454E-04 = res : 6.366E-12 = real regular ==
solution : 16 : start residual : 2.567E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.14074974785872E+00 -5.39493951306578E-50
H2 : 1.08562564192107E+00 1.74601041054954E-49
H3 : 1.06335912671548E+00 -1.09476442525376E-47
H4 : 1.05072887444418E+00 2.18952885050753E-47
H5 : 1.04245391286127E+00 6.73893727863618E-50
== err : 2.915E-16 = rco : 8.959E-01 = res : 2.567E-16 = real regular ==