6
x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5 - u6;
x2 + x1*x3 + x2*x4 + x3*x5 - 2*u6;
x3 + x1*x4 + x2*x5 - 3*u6;
x4 + x1*x5 - 4*u6;
x5 - 5*u6;
x1 + x2 + x3 + x4 + x5 + 1;
TITLE : reduced 6-dimensional economics problem.
ROOT COUNTS :
total degree : 16
2-homogeneous Bezout number : 28
with partition : {x1 x2 x3 x4 }{x5 u6 }
generalized Bezout number : 16
based on the set structure :
{x1 x3 x5 u6 }{x2 x4 }
{x1 x2 x5 u6 }{x3 x4 }
{x1 x2 x3 u6 }{x4 x5 }
{x1 x4 u6 }{x5 }
{x5 u6 }
{x1 x2 x3 x4 x5 }
mixed volume : 16
REFERENCE :
This is the reduced economics problem (u6 = 1/x6).
Alexander Morgan:
`Solving polynomial systems using continuation for engineering
and scientific problems', Prentice-Hall, Englewood Cliffs, New Jersey, 1987,
(p 148).
THE SOLUTIONS :
16 6
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.00000000000000E+00 0.00000000000000E+00
x2 : 1.00000000000000E+00 -3.26265223399926E-55
x3 : 1.00000000000000E+00 -6.52530446799852E-55
x4 : 1.00000000000000E+00 0.00000000000000E+00
x5 : -5.00000000000000E+00 -1.95759134039956E-55
u6 : -1.00000000000000E+00 -2.03915764624954E-56
== err : 3.306E-39 = rco : 6.489E-02 = res : 2.133E-54 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -3.49983204654658E-01 -2.09424746927724E+00
x2 : -1.61089899852697E+00 2.31906369483777E-01
x3 : -2.07202455367483E-01 1.40762102476128E+00
x4 : 7.83588286261556E-01 6.17749478619368E-01
x5 : 3.84496372287553E-01 -1.63029403587178E-01
u6 : 7.68992744575105E-02 -3.26058807174356E-02
== err : 4.364E-15 = rco : 3.507E-02 = res : 4.578E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.99966409309316E-01 -3.34095588761525E-52
x2 : 1.38656942719869E+00 -5.84667280332668E-52
x3 : 1.19152225800455E-01 1.08581066347495E-51
x4 : -1.13367501000843E+00 -5.01143383142287E-52
x5 : -8.72080233681396E-01 1.67047794380762E-52
u6 : -1.74416046736279E-01 0.00000000000000E+00
== err : 3.078E-15 = rco : 5.229E-02 = res : 2.220E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 -8.78066024603626E-01
x2 : -1.65499971781608E-01 -2.14978127415289E+00
x3 : -1.62144990845010E+00 1.42121425610120E+00
x4 : -3.27974968334248E-01 9.07431740523421E-01
x5 : 7.14924848565961E-01 6.99201302131896E-01
u6 : 1.42984969713192E-01 1.39840260426379E-01
== err : 4.807E-15 = rco : 4.100E-02 = res : 4.003E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 3.31042891395086E+00
x2 : -5.25946979716094E+00 5.27706843576465E-01
x3 : 5.56851925799103E-01 -5.25646392480149E+00
x4 : 3.30816086148002E+00 4.18343529761887E-01
x5 : -5.54299011818328E-03 9.99984637512272E-01
u6 : -1.10859802363666E-03 1.99996927502454E-01
== err : 3.898E-15 = rco : 1.760E-02 = res : 5.403E-15 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.14998320465466E+00 -2.09424746927724E+00
x2 : -1.77249865678310E+00 -2.90939448721313E+00
x3 : -3.76795822665312E+00 9.94499923819594E-01
x4 : 1.18599548149207E+00 4.94385765452914E+00
x5 : 2.20447819728949E+00 -9.34715621858359E-01
u6 : 4.40895639457899E-01 -1.86943124371672E-01
== err : 6.999E-15 = rco : 2.110E-02 = res : 3.580E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -3.49983204654658E-01 2.09424746927724E+00
x2 : -1.61089899852697E+00 -2.31906369483777E-01
x3 : -2.07202455367483E-01 -1.40762102476128E+00
x4 : 7.83588286261556E-01 -6.17749478619368E-01
x5 : 3.84496372287553E-01 1.63029403587178E-01
u6 : 7.68992744575105E-02 3.26058807174356E-02
== err : 4.364E-15 = rco : 3.507E-02 = res : 4.578E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.49983204654658E-01 -1.21618144467362E+00
x2 : -1.18910100147303E+00 6.28630424774533E-01
x3 : 4.07202455367483E-01 2.59518299581422E-01
x4 : -5.83588286261556E-01 -1.13892369777438E-01
x5 : -1.84496372287553E-01 4.41925090095103E-01
u6 : -3.68992744575105E-02 8.83850180190205E-02
== err : 2.989E-15 = rco : 3.486E-02 = res : 1.570E-16 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 -3.31042891395086E+00
x2 : -5.25946979716095E+00 -5.27706843576464E-01
x3 : 5.56851925799105E-01 5.25646392480149E+00
x4 : 3.30816086148002E+00 -4.18343529761888E-01
x5 : -5.54299011818387E-03 -9.99984637512272E-01
u6 : -1.10859802363677E-03 -1.99996927502454E-01
== err : 5.822E-15 = rco : 1.760E-02 = res : 5.329E-15 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.14998320465466E+00 2.09424746927724E+00
x2 : -1.77249865678310E+00 2.90939448721313E+00
x3 : -3.76795822665312E+00 -9.94499923819594E-01
x4 : 1.18599548149207E+00 -4.94385765452914E+00
x5 : 2.20447819728949E+00 9.34715621858359E-01
u6 : 4.40895639457899E-01 1.86943124371672E-01
== err : 7.205E-15 = rco : 2.110E-02 = res : 1.897E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.29996640930932E+00 -1.45851920574871E-62
x2 : -1.36629889313586E-01 -3.28166821293459E-62
x3 : -1.58995626049845E+00 -4.86173068582902E-63
x4 : 5.73303223716882E-01 2.30932207576878E-62
x5 : -1.14668348321416E+00 2.63131441422336E-62
u6 : -2.29336696642832E-01 4.86173068582902E-63
== err : 3.968E-15 = rco : 3.472E-02 = res : 2.220E-16 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.50016795345342E-01 -1.21618144467362E+00
x2 : 1.72498656783105E-01 9.93444005801895E-01
x3 : 2.16795822665312E+00 -1.78564329356081E+00
x4 : -2.78599548149208E+00 8.14100532334898E-02
x5 : -8.04478197289493E-01 1.92697067919904E+00
u6 : -1.60895639457899E-01 3.85394135839809E-01
== err : 5.225E-15 = rco : 5.667E-02 = res : 2.318E-15 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.49983204654658E-01 1.21618144467362E+00
x2 : -1.18910100147303E+00 -6.28630424774533E-01
x3 : 4.07202455367483E-01 -2.59518299581422E-01
x4 : -5.83588286261556E-01 1.13892369777438E-01
x5 : -1.84496372287553E-01 -4.41925090095103E-01
u6 : -3.68992744575105E-02 -8.83850180190205E-02
== err : 2.989E-15 = rco : 3.486E-02 = res : 1.570E-16 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.00000000000000E-01 8.78066024603626E-01
x2 : -1.65499971781608E-01 2.14978127415289E+00
x3 : -1.62144990845010E+00 -1.42121425610120E+00
x4 : -3.27974968334248E-01 -9.07431740523421E-01
x5 : 7.14924848565961E-01 -6.99201302131896E-01
u6 : 1.42984969713192E-01 -1.39840260426379E-01
== err : 4.807E-15 = rco : 4.100E-02 = res : 4.003E-16 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.50016795345342E-01 1.21618144467362E+00
x2 : 1.72498656783105E-01 -9.93444005801895E-01
x3 : 2.16795822665312E+00 1.78564329356081E+00
x4 : -2.78599548149208E+00 -8.14100532334896E-02
x5 : -8.04478197289493E-01 -1.92697067919904E+00
u6 : -1.60895639457899E-01 -3.85394135839809E-01
== err : 5.141E-15 = rco : 5.667E-02 = res : 1.790E-15 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.00000000000000E-01 1.08209849026649E-50
x2 : -2.00000000000000E-01 -1.29926062296149E-52
x3 : -2.00000000000000E-01 0.00000000000000E+00
x4 : -2.00000000000000E-01 -3.64659148177856E-51
x5 : -2.00000000000000E-01 -3.47211934098088E-51
u6 : -4.00000000000000E-02 -6.94423868196176E-52
== err : 1.850E-17 = rco : 5.947E-02 = res : 2.776E-17 ==