4
8*H1 - 0.51234*H1*(1 +(1/2)*H1+(1/3)*H2+(1/4)*H3) - 8;
8*H2 - 0.51234*H2*(1/2+(1/3)*H1+(1/4)*H2+(1/5)*H3) - 8;
8*H3 - 0.51234*H3*(1/3+(1/4)*H1+(1/5)*H2+(1/6)*H3) - 8;
8*H4 - 0.51234*H4*(1/4+(1/5)*H1+(1/6)*H2+(1/7)*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 : start residual : 9.326E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 2.51822165029661E+01 -2.80259692864963E-45
H2 : 2.76663658872307E+00 4.20389539297445E-45
H3 : 1.92497869539309E+00 7.88230386182710E-46
H4 : 1.62786627049800E+00 5.64017369306435E-47
== err : 5.821E-15 = rco : 3.032E-01 = res : 9.326E-15 = real regular ==
solution : 2 : start residual : 1.421E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -3.16331327268600E+00 -5.98699295060652E-49
H2 : 5.98637030800302E+01 3.42113882891801E-49
H3 : 4.71153264168264E+00 1.71056941445901E-48
H4 : 2.92029637979939E+00 0.00000000000000E+00
== err : 9.192E-15 = rco : 8.707E-02 = res : 1.421E-14 = real regular ==
solution : 3 : start residual : 1.954E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -2.04390936597088E+00 -1.71056941445901E-48
H2 : -5.35792480577045E+00 1.64214663788065E-47
H3 : 1.00248604763324E+02 -2.46321995682097E-47
H4 : 6.65820570096541E+00 2.73691106313441E-47
== err : 2.697E-14 = rco : 4.618E-02 = res : 1.954E-14 = real regular ==
solution : 4 : start residual : 1.878E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -9.16750879611222E+01 1.96715482662786E-48
H2 : 1.74858343204201E+02 -1.81320357932655E-47
H3 : 9.34554507528133E+00 2.01847190906163E-47
H4 : 4.84698612466881E+00 4.27642353614751E-49
== err : 2.920E-13 = rco : 8.642E-03 = res : 1.878E-13 = real regular ==
solution : 5 : start residual : 4.547E-12
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.60786418589275E+02 -2.11876327805912E-42
H2 : -8.54279410691036E+02 1.00893489431387E-41
H3 : 8.75536444248736E+02 -8.87862706996204E-42
H4 : 3.05801024813143E+01 -1.74882048347737E-42
== err : 1.468E-11 = rco : 6.658E-04 = res : 4.547E-12 = real regular ==
solution : 6 : start residual : 5.684E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : -5.92339596991102E+01 -2.72636629219036E-41
H2 : -1.56501821027881E+01 1.14794370197489E-41
H3 : 1.98847790262814E+02 3.44383110592467E-41
H4 : 1.10510121065134E+01 1.82953527502248E-41
== err : 1.867E-13 = rco : 7.146E-03 = res : 5.684E-14 = real regular ==
solution : 7 : start residual : 6.537E-13
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 7.44078945107443E+00 7.33159356534744E-42
H2 : -3.38634231314829E+02 -3.71288041107504E-41
H3 : 4.86695180986615E+02 3.39002124489460E-41
H4 : 1.98248659810970E+01 -5.42582765386569E-42
== err : 5.111E-12 = rco : 1.675E-03 = res : 6.537E-13 = real regular ==
solution : 8 : start residual : 2.012E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
H1 : 1.16536939222831E+00 -1.40129846432482E-45
H2 : 1.09668785507998E+00 7.00649232162409E-46
H3 : 1.07006151566156E+00 2.62743462060903E-46
H4 : 1.05533428697623E+00 8.75811540203011E-47
== err : 2.221E-16 = rco : 8.839E-01 = res : 2.012E-16 = real regular ==